title

College Algebra - Full Course

description

Learn Algebra in this full college course. These concepts are often used in programming.
This course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https://www.youtube.com/channel/UCkyLJh6hQS1TlhUZxOMjTFw
⭐️ Lecture Notes ⭐️
🔗 Algebra Notes: http://lindagreen.web.unc.edu/files/2020/08/classNotes_m110_2018F.pdf
⭐️ Course Contents ⭐️
⌨️ (0:00:00) Exponent Rules
⌨️ (0:10:14) Simplifying using Exponent Rules
⌨️ (0:21:18) Simplifying Radicals
⌨️ (0:31:46) Factoring
⌨️ (0:45:08) Factoring - Additional Examples
⌨️ (0:55:37) Rational Expressions
⌨️ (1:05:00) Solving Quadratic Equations
⌨️ (1:15:22) Rational Equations
⌨️ (1:25:31) Solving Radical Equations
⌨️ (1:37:01) Absolute Value Equations
⌨️ (1:42:23) Interval Notation
⌨️ (1:49:35) Absolute Value Inequalities
⌨️ (1:56:55) Compound Linear Inequalities
⌨️ (2:05:59) Polynomial and Rational Inequalities
⌨️ (2:16:20) Distance Formula
⌨️ (2:20:59) Midpoint Formula
⌨️ (2:23:30) Circles: Graphs and Equations
⌨️ (2:33:06) Lines: Graphs and Equations
⌨️ (2:41:35) Parallel and Perpendicular Lines
⌨️ (2:49:05) Functions
⌨️ (3:00:53) Toolkit Functions
⌨️ (3:08:00) Transformations of Functions
⌨️ (3:20:29) Introduction to Quadratic Functions
⌨️ (3:23:54) Graphing Quadratic Functions
⌨️ (3:33:02) Standard Form and Vertex Form for Quadratic Functions
⌨️ (3:37:18) Justification of the Vertex Formula
⌨️ (3:41:11) Polynomials
⌨️ (3:49:06) Exponential Functions
⌨️ (3:56:53) Exponential Function Applications
⌨️ (4:08:38) Exponential Functions Interpretations
⌨️ (4:18:17) Compound Interest
⌨️ (4:29:33) Logarithms: Introduction
⌨️ (4:38:15) Log Functions and Their Graphs
⌨️ (4:48:59) Combining Logs and Exponents
⌨️ (4:53:38) Log Rules
⌨️ (5:02:10) Solving Exponential Equations Using Logs
⌨️ (5:10:20) Solving Log Equations
⌨️ (5:19:27) Doubling Time and Half Life
⌨️ (5:35:34) Systems of Linear Equations
⌨️ (5:47:36) Distance, Rate, and Time Problems
⌨️ (5:53:20) Mixture Problems
⌨️ (5:59:48) Rational Functions and Graphs
⌨️ (6:13:13) Combining Functions
⌨️ (6:17:10) Composition of Functions
⌨️ (6:29:32) Inverse Functions
--
Learn to code for free and get a developer job: https://www.freecodecamp.org
Read hundreds of articles on programming: https://www.freecodecamp.org/news
Learn Algebra in this full college course. These concepts are often used in programming.
This course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https://www.youtube.com/channel/UCkyLJh6hQS1TlhUZxOMjTFw
⌨️ (0:00:00) Exponent Rules
⌨️ (0:10:14) Simplifying using Exponent Rules
⌨️ (0:21:18) Simplifying Radicals
⌨️ (0:31:46) Factoring
⌨️ (0:45:08) Factoring - Additional Examples
⌨️ (0:55:37) Rational Expressions
⌨️ (1:05:00) Solving Quadratic Equations
⌨️ (1:15:22) Rational Equations
⌨️ (1:25:31) Solving Radical Equations
⌨️ (1:37:01) Absolute Value Equations
⌨️ (1:42:23) Interval Notation
⌨️ (1:49:35) Absolute Value Inequalities
⌨️ (1:56:55) Compound Linear Inequalities
⌨️ (2:05:59) Polynomial and Rational Inequalities
⌨️ (2:16:20) Distance Formula
⌨️ (2:20:59) Midpoint Formula
⌨️ (2:23:30) Circles: Graphs and Equations
⌨️ (2:33:06) Lines: Graphs and Equations
⌨️ (2:41:35) Parallel and Perpendicular Lines
⌨️ (2:49:05) Functions
⌨️ (3:00:53) Toolkit Functions
⌨️ (3:08:00) Transformations of Functions
⌨️ (3:20:29) Introduction to Quadratic Functions
⌨️ (3:23:54) Graphing Quadratic Functions
⌨️ (3:33:02) Standard Form and Vertex Form for Quadratic Functions
⌨️ (3:37:18) Justification of the Vertex Formula
⌨️ (3:41:11) Polynomials
⌨️ (3:49:06) Exponential Functions
⌨️ (3:56:53) Exponential Function Applications
⌨️ (4:08:38) Exponential Functions Interpretations
⌨️ (4:18:17) Compound Interest
⌨️ (4:29:33) Logarithms: Introduction
⌨️ (4:38:15) Log Functions and Their Graphs
⌨️ (4:48:59) Combining Logs and Exponents
⌨️ (4:53:38) Log Rules
⌨️ (5:02:10) Solving Exponential Equations Using Logs
⌨️ (5:10:20) Solving Log Equations
⌨️ (5:19:27) Doubling Time and Half Life
⌨️ (5:35:34) Systems of Linear Equations
⌨️ (5:47:36) Distance, Rate, and Time Problems
⌨️ (5:53:20) Mixture Problems
⌨️ (5:59:48) Rational Functions and Graphs
⌨️ (6:13:13) Combining Functions
⌨️ (6:17:10) Composition of Functions
⌨️ (6:29:32) Inverse Functions
--
Learn to code for free and get a developer job: https://www.freecodecamp.org
Read hundreds of articles on programming: https://www.freecodecamp.org/news

detail

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It means the nth root of x.', 'start': 361.418, 'duration': 9.229}, {'end': 378.029, 'text': 'For example, 64 to the 1 third power means the cube root of 64, which happens to be 4.', 'start': 370.647, 'duration': 7.382}, {'end': 385.611, 'text': 'And 9 to the 1 half means the square root of 9, which is usually written without that little superscript up there.', 'start': 378.029, 'duration': 7.582}, {'end': 390.41, 'text': 'Now the square root of 9 is just 3.', 'start': 386.831, 'duration': 3.579}], 'summary': 'Fractional exponents represent roots, for example, 64^(1/3) equals 4.', 'duration': 31.513, 'max_score': 358.897, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU358897.jpg'}, {'end': 452.012, 'src': 'embed', 'start': 421.364, 'weight': 5, 'content': [{'end': 427.447, 'text': 'The cube root of 5 is also a number that when you cube it, you get 5.', 'start': 421.364, 'duration': 6.083}, {'end': 431.128, 'text': 'The next rule tells us we can distribute an exponent over a product.', 'start': 427.447, 'duration': 3.681}, {'end': 442.782, 'text': "In other words, if we have a product x times y all raised to the nth power, that's equal to x to the n times y to the n.", 'start': 432.611, 'duration': 10.171}, {'end': 452.012, 'text': 'For example, 5 times 7 all raised to the third power is equal to 5 cubed times 7 cubed.', 'start': 442.782, 'duration': 9.23}], 'summary': 'Cube root of 5 is a number cubed to get 5. rule: distribute exponent over a product.', 'duration': 30.648, 'max_score': 421.364, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU421364.jpg'}, {'end': 580.249, 'src': 'embed', 'start': 549.551, 'weight': 6, 'content': [{'end': 555.236, 'text': 'a minus b to the n is not generally equal to a to the n minus b to the n.', 'start': 549.551, 'duration': 5.685}, {'end': 558.579, 'text': "And if you're not sure, just try an example with numbers.", 'start': 555.236, 'duration': 3.343}, {'end': 568.207, 'text': 'For example, two plus three squared is not the same thing as two squared plus three squared.', 'start': 560.28, 'duration': 7.927}, {'end': 574.792, 'text': 'And two minus three squared is definitely not equal to two squared minus three squared.', 'start': 568.727, 'duration': 6.065}, {'end': 580.249, 'text': "In this video, I gave eight exponent rules, which I'll list again here.", 'start': 576.368, 'duration': 3.881}], 'summary': 'A^n - b^n ≠ (a - b)^n, demonstrated with examples.', 'duration': 30.698, 'max_score': 549.551, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU549551.jpg'}, {'end': 646.177, 'src': 'embed', 'start': 614.665, 'weight': 7, 'content': [{'end': 619.488, 'text': "In this video I'll work out some examples of simplifying expressions using exponent rules.", 'start': 614.665, 'duration': 4.823}, {'end': 623.15, 'text': "I'll start by reviewing the exponent rules.", 'start': 621.389, 'duration': 1.761}, {'end': 631.435, 'text': 'The product rule says that when you multiply two expressions with the same base you add the exponents.', 'start': 624.631, 'duration': 6.804}, {'end': 638.751, 'text': 'The quotient rule says that when you divide two expressions with the same base, you subtract the exponents.', 'start': 632.435, 'duration': 6.316}, {'end': 646.177, 'text': 'The power rule says that when you take a power to a power, you multiply the exponents.', 'start': 639.492, 'duration': 6.685}], 'summary': 'Video demonstrates simplifying expressions with exponent rules.', 'duration': 31.512, 'max_score': 614.665, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU614665.jpg'}, {'end': 921.945, 'src': 'embed', 'start': 895.045, 'weight': 8, 'content': [{'end': 900.386, 'text': "I'd like to show you one more method to solve these two problems, kind of a shortcut method before we go on.", 'start': 895.045, 'duration': 5.341}, {'end': 911.088, 'text': 'That shortcut relies on the principle that a negative exponent in the numerator corresponds to a positive exponent in the denominator.', 'start': 901.366, 'duration': 9.722}, {'end': 921.945, 'text': 'For example, the x to the negative 2 in the numerator here, after some manipulations, became an x to the positive 2 in the denominator.', 'start': 912.289, 'duration': 9.656}], 'summary': 'Negative exponent in numerator becomes positive in denominator.', 'duration': 26.9, 'max_score': 895.045, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU895045.jpg'}, {'end': 1346.768, 'src': 'embed', 'start': 1318.147, 'weight': 9, 'content': [{'end': 1325.752, 'text': 'First, if we have the radical of a product, we can rewrite that as the product of two radicals.', 'start': 1318.147, 'duration': 7.605}, {'end': 1337.798, 'text': 'For example, the square root of nine times 16 is the same thing as the square root of nine times the square root of 16.', 'start': 1327.613, 'duration': 10.185}, {'end': 1341.866, 'text': 'You can check that both of these evaluate to 12.', 'start': 1337.798, 'duration': 4.068}, {'end': 1346.768, 'text': "Similarly, it's possible to distribute a radical sign across division.", 'start': 1341.866, 'duration': 4.902}], 'summary': 'Radicals can be rewritten as the product of two radicals. for example, √(9*16) = √9 * √16 = 12.', 'duration': 28.621, 'max_score': 1318.147, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU1318147.jpg'}], 'start': 17.677, 'title': 'Exponent rules and simplifying expressions', 'summary': 'Explains 8 exponent rules, such as raising a number to the 0th power being equal to 1, and negative exponents being the reciprocal of the positive exponent. it covers various rules including product, quotient, power, zero, negative, and fractional exponents, with examples demonstrating simplifications and tips for simplifying expressions with radicals.', 'chapters': [{'end': 199.981, 'start': 17.677, 'title': 'Exponent rules explained', 'summary': 'Explains the exponent rules, including the product rule, quotient rule, and power rule, demonstrating examples such as 2 cubed times 2 to the fourth equals 2 to the seventh, and 3 to the 6 divided by 3 squared equals 3 to the 4th.', 'duration': 182.304, 'highlights': ['The product rule states that x to the power of n times x to the power of m equals x to the n plus m power, demonstrated by the example 2 cubed times 2 to the fourth equals 2 to the seventh.', 'The quotient rule states that x to the n power divided by x to the m power equals x to the n minus m power, illustrated with the example 3 to the 6 divided by 3 squared equals 3 to the 4th.', 'The power rule explains that x to the n power raised to the m power equals x to the n times m power.']}, {'end': 580.249, 'start': 201.442, 'title': 'Exponent rules explained', 'summary': 'Delves into 8 exponent rules, such as the rule for raising a number to the 0th power being equal to 1, and the rule for negative exponents being the reciprocal of the positive exponent. it also explains fractional exponents and the distribution of exponents over multiplication and division.', 'duration': 378.807, 'highlights': ['The rule for raising a number to the 0th power being equal to 1 is explained with the example of 2^3 divided by 2^3, which equals 1, supporting the consistency with the quotient rule.', 'Explanation of the rule for negative exponents, where x to the n is equal to 1 over x to the n, is clarified using the example of 5^7 times 5^-7 equals 1, derived from the product rule and consistent with the quotient rule.', 'Fractional exponents are exemplified with 64^(1/3) being the cube root of 64, equating to 4, and 9^(1/2) being the square root of 9, which is 3, demonstrating the meaning of 1/n power as the nth root of x.', 'The rule for distributing an exponent over a product is explained through the example of (x times y)^n equaling x^n times y^n, demonstrated with 5 times 7 all raised to the third power being equal to 5 cubed times 7 cubed.', 'The chapter concludes with a cautionary note that exponents cannot be distributed over addition or subtraction, illustrated with the examples of a plus b to the n not being equal to a to the n plus b to the n, and a minus b to the n not being equal to a to the n minus b to the n.']}, {'end': 893.265, 'start': 581.249, 'title': 'Exponent rules examples', 'summary': 'Covers various exponent rules including product rule, quotient rule, power rule, zero exponent, negative exponent, and fractional exponent, and demonstrates their application in simplifying expressions. the examples illustrate the use of these rules in different scenarios, resulting in simplifications like 3 over x to the 6th and 4y to the 8th.', 'duration': 312.016, 'highlights': ['The product rule states that when multiplying two expressions with the same base, the exponents are added, demonstrated by examples like 3 times x to the -2 divided by x to the fourth resulting in 3 over x to the 6th.', 'The quotient rule dictates that when dividing two expressions with the same base, the exponents are subtracted, showcased through examples like 4y cubed divided by y to the -5 resulting in 4y to the 8th.', 'The video reviews various exponent rules including the power rule, power of zero rule, negative exponents, and distributing exponents across multiplication and division.', 'The examples cover different approaches to simplifying expressions using exponent rules, showcasing the application of rules like the power rule, negative exponent rule, and product rule in obtaining simplified forms like 3 over x to the 6th and 4y to the 8th.']}, {'end': 1432.981, 'start': 895.045, 'title': 'Exponent rules and simplifying expressions', 'summary': 'Demonstrates how to simplify expressions using exponent rules, including converting negative exponents and applying the product and power rule, and provides tips for simplifying expressions with radicals, with examples and explanations.', 'duration': 537.936, 'highlights': ['The chapter demonstrates how to simplify expressions using exponent rules, including converting negative exponents and applying the product and power rule, and provides tips for simplifying expressions with radicals, with examples and explanations.', 'The shortcut method relies on the principle that a negative exponent in the numerator corresponds to a positive exponent in the denominator, and a negative exponent in the denominator is equivalent to a positive exponent in the numerator.', 'In the first problem, 3x to the minus 2 over x to the 4, we can move the negative exponent in the numerator and make it a positive exponent in the denominator, resulting in 3 over x to the 6. In the second example for y cubed over y to the minus 5, the y to the minus 5 in the denominator can be changed into a y to the 5 in the numerator, resulting in 4y to the 8.', "I have y's in the numerator and in the denominator and also z's in the numerator and in the denominator. I'm going to try to get all my y's either in the numerator or the denominator and similarly for the z's. By using the principle that a positive exponent in the numerator corresponds to a negative exponent in the denominator, I move the y to the 3 downstairs and make it a y to the negative 3. Since I have a positive exponent on z in the numerator and a negative exponent in the denominator and I want to get rid of negative exponents, I'm going to pass the z's to the numerator. A z to the minus 2 in the denominator becomes a z to the plus 2 in the numerator.", 'Using the product rule, I can rewrite the expression on the inside of the parentheses as 25x to the 10th over y to the 8th. When I distribute my 3 halves power, I get 25 to the 3 halves times x to the 15th over y to the 12th. Finally, I rewrite 25 to the 3 halves as 125. Therefore, the original expression is 125x to the 15 over y to the 12th.', 'The chapter provides a review of rules for radical expressions, including distributing a radical sign across division and product and provides caution about distributing a radical sign across addition or subtraction.']}], 'duration': 1415.304, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17677.jpg', 'highlights': ['The product rule states x^n * x^m = x^(n+m), demonstrated by 2^3 * 2^4 = 2^7', 'The quotient rule states x^n / x^m = x^(n-m), illustrated with 3^6 / 3^2 = 3^4', 'Raising a number to the 0th power equals 1, demonstrated by 2^3 / 2^3 = 1', 'Negative exponents: x^n = 1 / x^-n, shown with 5^7 * 5^-7 = 1', 'Fractional exponents: 64^(1/3) = 4, 9^(1/2) = 3, demonstrating 1/n power as the nth root of x', 'Distributing an exponent over a product: (x * y)^n = x^n * y^n, shown with 5 * 7^3 = 5^3 * 7^3', 'Exponents cannot be distributed over addition or subtraction, illustrated with a^n + b^n and a^n - b^n', 'Simplifying expressions using exponent rules, converting negative exponents, applying product and power rule', 'Shortcut method: negative exponent in numerator = positive exponent in denominator, and vice versa', 'Review of rules for radical expressions, caution about distributing radical sign across addition or subtraction']}, {'end': 3072.212, 'segs': [{'end': 1457.164, 'src': 'embed', 'start': 1432.981, 'weight': 0, 'content': [{'end': 1440.824, 'text': "You might notice that these rules for radicals, the ones that hold and the ones that don't hold, remind you of rules for exponents.", 'start': 1432.981, 'duration': 7.843}, {'end': 1445.926, 'text': "And that's no coincidence, because radicals can be written in terms of exponents.", 'start': 1441.164, 'duration': 4.762}, {'end': 1450.128, 'text': 'For example, if we look at the first rule, we could rewrite this.', 'start': 1446.386, 'duration': 3.742}, {'end': 1457.164, 'text': 'The nth root of a times b is the same thing as the 1 over nth power.', 'start': 1451.021, 'duration': 6.143}], 'summary': 'Radicals can be written in terms of exponents, such as the nth root of a times b being the same as the 1 over nth power.', 'duration': 24.183, 'max_score': 1432.981, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU1432981.jpg'}, {'end': 1917.686, 'src': 'embed', 'start': 1887.779, 'weight': 2, 'content': [{'end': 1897.748, 'text': 'In this video, we went over the rules for radicals and we simplified some radical expressions by working with fractional exponents,', 'start': 1887.779, 'duration': 9.969}, {'end': 1903.253, 'text': 'pulling things out of the radical sign and rationalizing the denominator.', 'start': 1897.748, 'duration': 5.505}, {'end': 1908.898, 'text': 'This video goes over some common methods of factoring.', 'start': 1906.175, 'duration': 2.723}, {'end': 1912.641, 'text': 'Recall that factoring an expression means to write it as a product.', 'start': 1909.458, 'duration': 3.183}, {'end': 1917.686, 'text': 'So we could factor the number 30 by writing it as 6 times 5.', 'start': 1913.162, 'duration': 4.524}], 'summary': 'Reviewed rules for radicals, simplified expressions, and explained factoring methods.', 'duration': 29.907, 'max_score': 1887.779, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU1887779.jpg'}, {'end': 2665.444, 'src': 'heatmap', 'start': 2178.125, 'weight': 0.822, 'content': [{'end': 2181.788, 'text': 'So I factor out the 4 from those two terms.', 'start': 2178.125, 'duration': 3.663}, {'end': 2189.384, 'text': 'Notice that the factor of x plus 3 now appears in both pieces.', 'start': 2183.703, 'duration': 5.681}, {'end': 2193.585, 'text': 'So I can factor out the greatest common factor of x plus 3.', 'start': 2190.064, 'duration': 3.521}, {'end': 2196.665, 'text': "I think I'll factor it out on the left side instead of the right.", 'start': 2193.585, 'duration': 3.08}, {'end': 2210.128, 'text': 'And now I have an x squared from this first piece, and I have a 4 from this second piece, and that completes my factoring by grouping.', 'start': 2197.105, 'duration': 13.023}, {'end': 2216.429, 'text': 'You might wonder if we could factor further by factoring the expression x squared plus 4.', 'start': 2211.288, 'duration': 5.141}, {'end': 2221.11, 'text': "But in fact, as we'll see later, this expression, which is a sum of two squares,", 'start': 2216.429, 'duration': 4.681}, {'end': 2225.791, 'text': 'x squared plus 2 squared does not factor any further over the integers.', 'start': 2221.11, 'duration': 4.681}, {'end': 2229.991, 'text': "Next, we'll do some factoring of quadratics.", 'start': 2227.351, 'duration': 2.64}, {'end': 2238.873, 'text': "A quadratic is an expression with a squared term, just a term with x in it, and a constant term with no x's in it.", 'start': 2230.551, 'duration': 8.322}, {'end': 2250.979, 'text': "I'd like to factor this expression as a product of x plus or minus some number times x plus or minus some other number.", 'start': 2240.39, 'duration': 10.589}, {'end': 2259.853, 'text': 'The key idea is that if I can find those two numbers, then if I were to distribute out this expression, those two numbers would have to multiply,', 'start': 2252.25, 'duration': 7.603}, {'end': 2262.594, 'text': 'to give me my constant term of 8..', 'start': 2259.853, 'duration': 2.741}, {'end': 2271.017, 'text': 'And these two numbers would end up having to add to give me my negative 6, because when I multiply out,', 'start': 2262.594, 'duration': 8.423}, {'end': 2275.178, 'text': 'this number will be a coefficient of x and this number will be another coefficient of x.', 'start': 2271.017, 'duration': 4.161}, {'end': 2279.417, 'text': "They'll add together to the negative 6.", 'start': 2275.178, 'duration': 4.239}, {'end': 2290.663, 'text': 'So if I look at all the pairs of numbers that multiply together, to give me eight, so that could be one and eight, two and four, four and two,', 'start': 2279.417, 'duration': 11.246}, {'end': 2292.443, 'text': "but that's really the same thing as I had before.", 'start': 2290.663, 'duration': 1.78}, {'end': 2294.925, 'text': "And that's sort of the same thing I had before.", 'start': 2292.684, 'duration': 2.241}, {'end': 2296.525, 'text': "I shouldn't forget the negatives.", 'start': 2294.925, 'duration': 1.6}, {'end': 2302.889, 'text': 'I could have negative one, negative eight, or I could have negative two, negative four, those all to multiply together to give me eight.', 'start': 2296.525, 'duration': 6.364}, {'end': 2310.793, 'text': "Now I just have to find, see if there's a pair of these numbers that add to negative 6.", 'start': 2303.749, 'duration': 7.044}, {'end': 2313.334, 'text': "And it's not hard to see that these ones will work.", 'start': 2310.793, 'duration': 2.541}, {'end': 2322.739, 'text': 'So now I can write out my factoring as that would be x minus 2 times x minus 4.', 'start': 2313.774, 'duration': 8.965}, {'end': 2325.6, 'text': "And it's always a good idea to check by multiplying out.", 'start': 2322.739, 'duration': 2.861}, {'end': 2331.363, 'text': "I'm going to get x squared minus 4x minus 2x plus 8.", 'start': 2326.02, 'duration': 5.343}, {'end': 2333.987, 'text': 'And that works out to just what I want.', 'start': 2331.363, 'duration': 2.624}, {'end': 2345.529, 'text': "Now the second example is a bit more complicated, because now my leading coefficient, my coefficient of x squared is not just one, it's the number 10.", 'start': 2334.928, 'duration': 10.601}, {'end': 2351.351, 'text': "Now there are lots of different methods for approaching a problem like this, and I'm just going to show you one method, my favorite method,", 'start': 2345.529, 'duration': 5.822}, {'end': 2353.152, 'text': 'that uses factoring by grouping.', 'start': 2351.351, 'duration': 1.801}, {'end': 2358.234, 'text': "But to start out, I'm going to multiply my coefficient of x squared by my constant term.", 'start': 2353.792, 'duration': 4.442}, {'end': 2361.295, 'text': "So I'm multiplying 10 by negative 6.", 'start': 2358.254, 'duration': 3.041}, {'end': 2364.383, 'text': 'That gives me negative 60.', 'start': 2361.295, 'duration': 3.088}, {'end': 2369.885, 'text': "And I'll also take my coefficient of x, the number 11, and write that down here.", 'start': 2364.383, 'duration': 5.502}, {'end': 2377.609, 'text': "Now I'm going to look for two numbers that multiply to give me negative 60 and add to give me 11.", 'start': 2369.905, 'duration': 7.704}, {'end': 2381.37, 'text': 'You might notice that this is exactly what we were doing in the previous problem.', 'start': 2377.609, 'duration': 3.761}, {'end': 2388.757, 'text': "It's just here we didn't have to multiply the coefficient of x squared by 8 because the coefficient of x squared was just 1.", 'start': 2381.61, 'duration': 7.147}, {'end': 2397.343, 'text': 'So to find the two numbers that multiply to negative 60 and add to 11, you might just be able to come up with them in your head thinking about it,', 'start': 2388.757, 'duration': 8.586}, {'end': 2406.369, 'text': 'but if not, you can figure it out pretty systematically by writing out all the factors, pairs of factors that multiply to negative 60..', 'start': 2397.343, 'duration': 9.026}, {'end': 2412.754, 'text': 'So I can start with negative 1 and 60, negative 2 and 30, negative 3 and 20,', 'start': 2406.369, 'duration': 6.385}, {'end': 2421.719, 'text': 'and keep going like this until I have found factors that actually add together to give me the number 11..', 'start': 2412.754, 'duration': 8.965}, {'end': 2425.962, 'text': "And now that I look at it, I've already found them.", 'start': 2421.719, 'duration': 4.243}, {'end': 2431.925, 'text': "15 minus 4 gives me 11, so I don't have to continue with my chart of factors.", 'start': 2425.982, 'duration': 5.943}, {'end': 2443.011, 'text': "Now, once I've found those factors, I write out my expression, 10x squared, but instead of writing 11x, I write negative 4x plus 15x.", 'start': 2432.785, 'duration': 10.226}, {'end': 2445.672, 'text': 'Now I copy down the negative 6.', 'start': 2443.371, 'duration': 2.301}, {'end': 2449.754, 'text': 'Notice that negative 4x plus 15x equals 11x.', 'start': 2445.672, 'duration': 4.082}, {'end': 2451.435, 'text': "That's how I chose those numbers.", 'start': 2449.774, 'duration': 1.661}, {'end': 2457.158, 'text': 'And so this expression evaluates as the same as this expression.', 'start': 2451.835, 'duration': 5.323}, {'end': 2458.579, 'text': "I haven't changed my expression.", 'start': 2457.178, 'duration': 1.401}, {'end': 2462.981, 'text': 'But I have turned it into something that I can apply factoring by grouping on.', 'start': 2459.139, 'duration': 3.842}, {'end': 2464.562, 'text': "Look, I've got four terms here.", 'start': 2463.001, 'duration': 1.561}, {'end': 2471.046, 'text': "And so if I factor out my greatest common factor of my first two terms, that's, let's see, I think it's 2x.", 'start': 2464.962, 'duration': 6.084}, {'end': 2473.443, 'text': 'So I factor out the 2x.', 'start': 2472.082, 'duration': 1.361}, {'end': 2476.565, 'text': 'I get 5x minus 2.', 'start': 2473.523, 'duration': 3.042}, {'end': 2480.868, 'text': 'And then I factor out the greatest common factor of 15x and negative 6.', 'start': 2476.565, 'duration': 4.303}, {'end': 2481.469, 'text': 'That would be 3.', 'start': 2480.868, 'duration': 0.601}, {'end': 2485.211, 'text': 'And I get a 5x minus 2.', 'start': 2481.469, 'duration': 3.742}, {'end': 2486.532, 'text': 'Again, this is working beautifully.', 'start': 2485.211, 'duration': 1.321}, {'end': 2489.834, 'text': 'So I have a 5x minus 2 in each part.', 'start': 2486.892, 'duration': 2.942}, {'end': 2493.917, 'text': 'And so I put the 5x minus 2 on the right.', 'start': 2490.215, 'duration': 3.702}, {'end': 2498.02, 'text': "And I put what's left from these terms in here.", 'start': 2494.298, 'duration': 3.722}, {'end': 2501.102, 'text': "So that's 2x plus 3.", 'start': 2498.04, 'duration': 3.062}, {'end': 2503.503, 'text': 'and I have factored my expression.', 'start': 2501.102, 'duration': 2.401}, {'end': 2512.165, 'text': "There are a couple of special kinds of expressions that appear frequently that it's handy to just memorize the formula for.", 'start': 2506.183, 'duration': 5.982}, {'end': 2514.466, 'text': 'So the first one is the difference of squares.', 'start': 2512.385, 'duration': 2.081}, {'end': 2523.888, 'text': 'If you see something of the form a squared minus b squared, then you can factor that as a plus b times a minus b.', 'start': 2514.826, 'duration': 9.062}, {'end': 2525.469, 'text': "And let's just check that that works.", 'start': 2523.888, 'duration': 1.581}, {'end': 2538.553, 'text': 'If I do a plus b times a minus b and multiply that out, I get a squared minus ab plus ba minus b squared and those middle two terms cancel out.', 'start': 2525.569, 'duration': 12.984}, {'end': 2541.835, 'text': 'So it gives me back the difference of squares just like I want it.', 'start': 2538.613, 'duration': 3.222}, {'end': 2550.978, 'text': 'So for this first example, if I think of x squared minus 16 as x squared minus four squared,', 'start': 2542.635, 'duration': 8.343}, {'end': 2557.581, 'text': "then I can see that's a difference of squares and I can immediately write it as x plus four times x minus four.", 'start': 2550.978, 'duration': 6.603}, {'end': 2566.926, 'text': "And the second example, 9p squared minus 1, that's the same thing as 3p squared minus 1 squared.", 'start': 2558.161, 'duration': 8.765}, {'end': 2573.91, 'text': "So that's 3p plus 1 times 3p minus 1.", 'start': 2567.306, 'duration': 6.604}, {'end': 2587.178, 'text': 'Notice that if I have a sum of squares, for example, x squared plus 4, which is x squared plus 2 squared, then that does not factor.', 'start': 2573.91, 'duration': 13.268}, {'end': 2594.573, 'text': "The difference of squares formula doesn't apply, and there is no formula that applies for a sum of squares.", 'start': 2589.332, 'duration': 5.241}, {'end': 2600.895, 'text': 'There is, however, a formula for both a difference of cubes and a sum of cubes.', 'start': 2595.834, 'duration': 5.061}, {'end': 2611.058, 'text': 'The difference of cubes formula, a cubed minus b cubed, is a minus b times a squared plus ab plus b squared.', 'start': 2601.635, 'duration': 9.423}, {'end': 2614.979, 'text': 'The formula for the sum of cubes is pretty much the same.', 'start': 2612.138, 'duration': 2.841}, {'end': 2618.24, 'text': 'You just switch the negative and positive sign here and here.', 'start': 2614.999, 'duration': 3.241}, {'end': 2625.644, 'text': 'So that gives us a plus b times a squared minus ab plus b squared.', 'start': 2618.979, 'duration': 6.665}, {'end': 2629.848, 'text': 'As usual, you can check these formulas by multiplying out.', 'start': 2626.285, 'duration': 3.563}, {'end': 2633.222, 'text': "Let's look at one example of using these formulas.", 'start': 2630.799, 'duration': 2.423}, {'end': 2641.01, 'text': "y cubed plus 27 is actually a sum of two cubes because it's y cubed plus 3 cubed.", 'start': 2633.822, 'duration': 7.188}, {'end': 2648.319, 'text': 'So I can factor it using the sum of cubes formula by plugging in y for a and 3 for b.', 'start': 2641.491, 'duration': 6.828}, {'end': 2657.071, 'text': 'That gives me y plus 3 times y squared minus y times 3 plus 3 squared.', 'start': 2649.74, 'duration': 7.331}, {'end': 2665.444, 'text': 'And I can clean that up a little bit to read y plus 3 times y squared minus 3y plus 9.', 'start': 2657.471, 'duration': 7.973}], 'summary': 'Factored expressions using grouping and special formulas for sums and differences of squares and cubes.', 'duration': 487.319, 'max_score': 2178.125, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU2178125.jpg'}, {'end': 2614.979, 'src': 'embed', 'start': 2589.332, 'weight': 4, 'content': [{'end': 2594.573, 'text': "The difference of squares formula doesn't apply, and there is no formula that applies for a sum of squares.", 'start': 2589.332, 'duration': 5.241}, {'end': 2600.895, 'text': 'There is, however, a formula for both a difference of cubes and a sum of cubes.', 'start': 2595.834, 'duration': 5.061}, {'end': 2611.058, 'text': 'The difference of cubes formula, a cubed minus b cubed, is a minus b times a squared plus ab plus b squared.', 'start': 2601.635, 'duration': 9.423}, {'end': 2614.979, 'text': 'The formula for the sum of cubes is pretty much the same.', 'start': 2612.138, 'duration': 2.841}], 'summary': 'No difference of squares formula, but formulas for difference and sum of cubes exist.', 'duration': 25.647, 'max_score': 2589.332, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU2589332.jpg'}, {'end': 2745.14, 'src': 'embed', 'start': 2707.712, 'weight': 3, 'content': [{'end': 2710.695, 'text': 'This video gives some additional examples of factoring.', 'start': 2707.712, 'duration': 2.983}, {'end': 2718.921, 'text': 'Please pause the video and decide which of these first five expressions factor and which one does not.', 'start': 2711.695, 'duration': 7.226}, {'end': 2726.763, 'text': 'The first expression can be factored by pulling out a common factor of x from each term.', 'start': 2721.018, 'duration': 5.745}, {'end': 2732.208, 'text': 'So that becomes x times x plus one.', 'start': 2728.285, 'duration': 3.923}, {'end': 2745.14, 'text': 'The second example can be factored as a difference of two squares, since x squared minus 25 is something squared minus something else squared.', 'start': 2734.15, 'duration': 10.99}], 'summary': 'Video provides examples of factoring, including x times x plus one and a difference of squares.', 'duration': 37.428, 'max_score': 2707.712, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU2707712.jpg'}], 'start': 1432.981, 'title': 'Factoring techniques in algebra', 'summary': 'Covers the relationship between radicals and exponents, rationalizing the denominator, simplifying radical expressions, and factoring techniques including greatest common factor, factoring by grouping, and factoring of quadratics. it also discusses factoring expressions, special cases like the difference of squares and the sum of squares, factoring formulas for difference and sum of cubes, and the factoring by grouping method, providing examples for each concept.', 'chapters': [{'end': 1811.327, 'start': 1432.981, 'title': 'Radicals and exponents relationship', 'summary': 'Discusses the relationship between radicals and exponents, explaining how radicals can be rewritten in terms of exponents and provides examples for simplifying expressions involving exponents and radicals.', 'duration': 378.346, 'highlights': ["The relationship between radicals and exponents is explained, showing how radicals can be rewritten in terms of exponents and vice versa, providing a mnemonic 'flower over root' to remember the relationship.", 'Examples are provided for simplifying expressions involving exponents and radicals, demonstrating the application of the rules and the mnemonic in computation.', 'The process of rewriting radical expressions in terms of squares and simplifying them using exponent rules is demonstrated through an example, emphasizing the cancellation of square roots and squares to simplify the expression.']}, {'end': 2250.979, 'start': 1812.108, 'title': 'Radical expressions and factoring', 'summary': 'Covers rationalizing the denominator, simplifying radical expressions, and factoring techniques including greatest common factor, factoring by grouping, and factoring of quadratics.', 'duration': 438.871, 'highlights': ['The chapter covers rationalizing the denominator, simplifying radical expressions, and factoring techniques including greatest common factor, factoring by grouping, and factoring of quadratics.', "The process of rationalizing the denominator is explained, resulting in the expression 'three times the square root of x' as the final answer.", 'The concept of factoring is introduced, with examples demonstrating factoring as the opposite of distributing out and the terminology involving terms and factors.', 'The technique of factoring by grouping is explained, with an example demonstrating the process and the completion of factoring by grouping.', 'The method of factoring quadratics as a product of x plus or minus some number is introduced.']}, {'end': 2587.178, 'start': 2252.25, 'title': 'Factoring expressions and special cases', 'summary': 'Discusses factoring expressions using the method of finding two numbers that multiply to give a constant term and add to give a coefficient of x, and also highlights the technique for factoring special cases like the difference of squares and the sum of squares, with examples demonstrating the process.', 'duration': 334.928, 'highlights': ['The technique of factoring expressions by finding two numbers that multiply to give a constant term and add to give a coefficient of x is demonstrated with examples.', 'The process of factoring special cases like the difference of squares and the sum of squares is explained, with examples illustrating the method.', 'The chapter also emphasizes the usefulness of memorizing the formulas for special cases like the difference of squares.']}, {'end': 2821.092, 'start': 2589.332, 'title': 'Factoring formulas and examples', 'summary': 'Covers factoring formulas for difference and sum of cubes and provides examples of factoring using these formulas. it also discusses different factoring techniques and their applications in solving problems.', 'duration': 231.76, 'highlights': ['The formula for the difference of cubes, a^3 - b^3, is a - b times a^2 + ab + b^2, and the formula for the sum of cubes, a^3 + b^3, is a + b times a^2 - ab + b^2.', 'The example of using the sum of cubes formula for y^3 + 27 demonstrates the application of the formula, resulting in the factored form y + 3 times y^2 - 3y + 9.', 'Discussion on various factoring techniques such as factoring out the greatest common factor, factoring by grouping, factoring quadratics, difference of squares, and difference and sum of cubes.']}, {'end': 3072.212, 'start': 2821.412, 'title': 'Factoring by grouping method', 'summary': 'Explains the method of factoring quadratic expressions using the factoring by grouping technique, demonstrating it with an example and detailing the steps involved, such as finding suitable factors and checking the solution.', 'duration': 250.8, 'highlights': ['The method of factoring a quadratic involves using the factoring by grouping technique, which is demonstrated through an example involving the expression 5x^2 - 14x + 8, where the steps of finding suitable factors and checking the solution are explained in detail.', 'The process of finding suitable factors for the given expression, 5x^2 - 14x + 8, is described, emphasizing the importance of identifying the factors that multiply to the constant term and add up to the coefficient of the x term, leading to the selection of -4 and -10 as the suitable factors.', 'The technique of factoring by grouping is illustrated with the expression 5x^2 - 14x + 8, where the steps of grouping the terms, factoring out common factors, and checking the solution are detailed, providing a comprehensive understanding of the method.', 'The explanation highlights the significance of identifying expressions with four terms as potential candidates for factoring by grouping, providing a clear criterion for determining the suitability of the method.', 'The importance of verifying the factored expression through distribution and multiplication is emphasized as a crucial step in ensuring the accuracy of the factoring process, reinforcing the understanding of the factoring by grouping method.']}], 'duration': 1639.231, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU1432981.jpg', 'highlights': ["The relationship between radicals and exponents is explained, showing how radicals can be rewritten in terms of exponents and vice versa, providing a mnemonic 'flower over root' to remember the relationship.", 'The process of rewriting radical expressions in terms of squares and simplifying them using exponent rules is demonstrated through an example, emphasizing the cancellation of square roots and squares to simplify the expression.', 'The chapter covers rationalizing the denominator, simplifying radical expressions, and factoring techniques including greatest common factor, factoring by grouping, and factoring of quadratics.', 'The technique of factoring expressions by finding two numbers that multiply to give a constant term and add to give a coefficient of x is demonstrated with examples.', 'The formula for the difference of cubes, a^3 - b^3, is a - b times a^2 + ab + b^2, and the formula for the sum of cubes, a^3 + b^3, is a + b times a^2 - ab + b^2.']}, {'end': 3892.906, 'segs': [{'end': 3185.28, 'src': 'embed', 'start': 3136.951, 'weight': 0, 'content': [{'end': 3139.553, 'text': "That'll simplify things in making the rest of factoring easier.", 'start': 3136.951, 'duration': 2.602}, {'end': 3145.518, 'text': 'One more tip is that you might need to do several of these factoring techniques in one problem.', 'start': 3140.314, 'duration': 5.204}, {'end': 3151.862, 'text': 'For example, you might have to first pull out a common factor, then factor a difference of squares,', 'start': 3145.798, 'duration': 6.064}, {'end': 3157.607, 'text': 'and then you might notice that one of your factors is itself a difference of squares and you have to apply a difference of squares again.', 'start': 3151.862, 'duration': 5.745}, {'end': 3163.315, 'text': "So don't stop when you factor a little bit, keep factoring as far as you can go.", 'start': 3158.427, 'duration': 4.888}, {'end': 3168.604, 'text': 'Here are some extra examples of factoring quadratics for you to practice.', 'start': 3165.258, 'duration': 3.346}, {'end': 3170.967, 'text': 'Please pause the video and give these a try.', 'start': 3169.004, 'duration': 1.963}, {'end': 3178.175, 'text': "For the first one, let's multiply 2 times negative 14.", 'start': 3172.571, 'duration': 5.604}, {'end': 3181.777, 'text': 'That gives us negative 28.', 'start': 3178.175, 'duration': 3.602}, {'end': 3185.28, 'text': "And then we'll bring the 3 down in the bottom of the X.", 'start': 3181.777, 'duration': 3.503}], 'summary': 'Tips for factoring equations, including multiple techniques in one problem and practicing with extra examples.', 'duration': 48.329, 'max_score': 3136.951, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3136951.jpg'}, {'end': 3452.595, 'src': 'embed', 'start': 3393.47, 'weight': 4, 'content': [{'end': 3401.356, 'text': 'If we want to reduce a rational expression with variables in it to lowest terms, we proceed the same way.', 'start': 3393.47, 'duration': 7.886}, {'end': 3408.863, 'text': "First, we'll factor the numerator, that's three times x plus two, and then factor the denominator.", 'start': 3402.137, 'duration': 6.726}, {'end': 3416.549, 'text': 'In this case, it factors to x plus two times x plus two, we could also write that as x plus two squared.', 'start': 3410.064, 'duration': 6.485}, {'end': 3423.374, 'text': "Now we cancel the common factors, and we're left with three over x plus two.", 'start': 3417.37, 'duration': 6.004}, {'end': 3427.397, 'text': 'Definitely a simpler way of writing that rational expression.', 'start': 3424.335, 'duration': 3.062}, {'end': 3431.408, 'text': "Next, let's practice multiplying and dividing.", 'start': 3429.306, 'duration': 2.102}, {'end': 3440.315, 'text': 'Recall that if we multiply two fractions with just numbers in them, we simply multiply the numerators and multiply the denominators.', 'start': 3432.348, 'duration': 7.967}, {'end': 3443.257, 'text': 'So in this case, we would get 4 times 2 over 3 times 5, or 8 15ths.', 'start': 3440.695, 'duration': 2.562}, {'end': 3452.595, 'text': 'If we want to divide two fractions, like in the second example,', 'start': 3448.553, 'duration': 4.042}], 'summary': 'Reducing rational expression and multiplying/dividing fractions simplifies the expression.', 'duration': 59.125, 'max_score': 3393.47, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3393470.jpg'}, {'end': 3699.281, 'src': 'embed', 'start': 3666.809, 'weight': 3, 'content': [{'end': 3671.812, 'text': 'And now that I have a common denominator, I can just subtract my two numerators, and I get 27 30th.', 'start': 3666.809, 'duration': 5.003}, {'end': 3685.335, 'text': 'If I factor, I can reduce this to 3 squared over 2 times 5, which is 9 tenths.', 'start': 3672.112, 'duration': 13.223}, {'end': 3693.004, 'text': 'The process for finding the sum of two rational expressions with variables in them follows the exact same process.', 'start': 3686.596, 'duration': 6.408}, {'end': 3695.827, 'text': 'First, we have to find the least common denominator.', 'start': 3693.444, 'duration': 2.383}, {'end': 3699.281, 'text': "I'll do that by factoring the two denominators.", 'start': 3696.277, 'duration': 3.004}], 'summary': 'Subtracting numerators yields 27/30. factoring reduces it to 9/10. same process for sum of rational expressions with variables.', 'duration': 32.472, 'max_score': 3666.809, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3666809.jpg'}], 'start': 3072.212, 'title': 'Factoring and rational expressions', 'summary': 'Covers factoring techniques such as common factor extraction, difference of squares, and factoring quadratics. it also explains working with rational expressions, including simplifying, multiplying, dividing, and finding the sum of rational expressions.', 'chapters': [{'end': 3319.045, 'start': 3072.212, 'title': 'Factoring techniques and tips', 'summary': 'Covers various factoring techniques including common factor extraction, difference of squares, factoring by grouping, factoring quadratics, and factoring sums and differences of cubes, while emphasizing the importance of pulling out common factors first and the possibility of using multiple factoring techniques within one problem.', 'duration': 246.833, 'highlights': ['The importance of pulling out common factors first is emphasized to simplify the factoring process.', 'The possibility of using multiple factoring techniques within one problem is highlighted, demonstrating the need to continue factoring as far as possible.', 'Explains the process of factoring quadratics using examples, demonstrating the steps involved in identifying suitable numbers for factoring.']}, {'end': 3892.906, 'start': 3319.666, 'title': 'Working with rational expressions', 'summary': 'Covers simplifying rational expressions by factoring and canceling common factors, multiplying and dividing rational expressions using the rules for fractions, and finding the sum of rational expressions by finding the least common denominator and adding the numerators.', 'duration': 573.24, 'highlights': ['Simplifying to lowest terms', 'Multiplying and dividing rational expressions', 'Finding the sum of rational expressions']}], 'duration': 820.694, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3072212.jpg', 'highlights': ['The importance of pulling out common factors first is emphasized to simplify the factoring process.', 'The possibility of using multiple factoring techniques within one problem is highlighted, demonstrating the need to continue factoring as far as possible.', 'Explains the process of factoring quadratics using examples, demonstrating the steps involved in identifying suitable numbers for factoring.', 'Finding the sum of rational expressions', 'Multiplying and dividing rational expressions', 'Simplifying to lowest terms']}, {'end': 5800.014, 'segs': [{'end': 3976.646, 'src': 'embed', 'start': 3947.604, 'weight': 4, 'content': [{'end': 3956.026, 'text': "The equation 3x squared equals minus 7x plus 2 is also a quadratic equation, it's just not in standard form.", 'start': 3947.604, 'duration': 8.422}, {'end': 3969.53, 'text': 'The key steps to solving quadratic equations are usually to write the equation in standard form and then either factor it or use the quadratic formula,', 'start': 3956.787, 'duration': 12.743}, {'end': 3971.191, 'text': "which I'll show you later in this video.", 'start': 3969.53, 'duration': 1.661}, {'end': 3976.646, 'text': "Let's start with the example y squared equals 18 minus 7y.", 'start': 3974.403, 'duration': 2.243}], 'summary': 'Steps to solve quadratic equations: write in standard form, then factor or use quadratic formula.', 'duration': 29.042, 'max_score': 3947.604, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3947604.jpg'}, {'end': 5180.972, 'src': 'embed', 'start': 5153.355, 'weight': 3, 'content': [{'end': 5160.598, 'text': 'I want to get the term with the square root in it on one side of the equation by itself and everything else on the other side of the equation.', 'start': 5153.355, 'duration': 7.243}, {'end': 5169.402, 'text': 'If I start with my original equation, x plus the square root of x equals 12, and I subtract x from both sides,', 'start': 5161.618, 'duration': 7.784}, {'end': 5174.086, 'text': 'then that does isolate the square root term on the left side with everything else on the right.', 'start': 5169.402, 'duration': 4.684}, {'end': 5180.972, 'text': "Once I've isolated the term with the square root, I'm going to get rid of the square root.", 'start': 5175.207, 'duration': 5.765}], 'summary': 'Isolate square root term in x + sqrt(x) = 12 and solve for x.', 'duration': 27.617, 'max_score': 5153.355, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5153355.jpg'}, {'end': 5535.624, 'src': 'embed', 'start': 5509.757, 'weight': 0, 'content': [{'end': 5519.505, 'text': "There's something that you need to be careful about, though when taking an even root or the one over an even number power,", 'start': 5509.757, 'duration': 9.748}, {'end': 5523.689, 'text': 'you always have to consider plus or minus your answer.', 'start': 5519.505, 'duration': 4.184}, {'end': 5531.737, 'text': "It's kind of like when you write x squared equals four and you take the square root of both sides.", 'start': 5525.571, 'duration': 6.166}, {'end': 5535.624, 'text': 'x could equal plus or minus a square root of four.', 'start': 5532.603, 'duration': 3.021}], 'summary': 'When taking an even root or an even number power, consider plus or minus your answer.', 'duration': 25.867, 'max_score': 5509.757, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5509757.jpg'}, {'end': 5742.913, 'src': 'embed', 'start': 5711.969, 'weight': 1, 'content': [{'end': 5718.951, 'text': 'Similarly, we can check that the P equals negative one 32nd actually does satisfy the equation.', 'start': 5711.969, 'duration': 6.982}, {'end': 5720.551, 'text': "I'll leave that step to the viewer.", 'start': 5719.191, 'duration': 1.36}, {'end': 5730.345, 'text': 'So our two solutions are P equals one over 32 and P equals minus one over 32.', 'start': 5723.972, 'duration': 6.373}, {'end': 5734.267, 'text': 'I do want to point out an alternate approach to getting rid of the fractional exponent.', 'start': 5730.345, 'duration': 3.922}, {'end': 5742.913, 'text': 'We could have gotten rid of it all in one fell swoop by raising both sides of our equation to the 5 fourths power.', 'start': 5734.948, 'duration': 7.965}], 'summary': 'Solutions for p are 1/32 and -1/32. an alternate approach is raising both sides to the 5/4 power.', 'duration': 30.944, 'max_score': 5711.969, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5711969.jpg'}, {'end': 5786.481, 'src': 'embed', 'start': 5762.486, 'weight': 2, 'content': [{'end': 5769.714, 'text': 'In other words, p to the 1 power, which is just p, is plus or minus 1 sixteenth to the 5 fourths.', 'start': 5762.486, 'duration': 7.228}, {'end': 5774.336, 'text': "So that's an alternate and possibly faster way to get to the solution.", 'start': 5770.254, 'duration': 4.082}, {'end': 5783.62, 'text': "Once again, the plus or minus comes from the fact that when we take the 5, fourth power, we're really taking an even root, a fourth root,", 'start': 5774.776, 'duration': 8.844}, {'end': 5786.481, 'text': 'and so we need to consider both positive and negative answers.', 'start': 5783.62, 'duration': 2.861}], 'summary': 'Using alternate method, p to the 1 power is ± 1 sixteenth to the 5 fourths.', 'duration': 23.995, 'max_score': 5762.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5762486.jpg'}], 'start': 3892.906, 'title': 'Solving equations', 'summary': 'Covers solving quadratic, rational, and radical equations, demonstrating methods such as factoring, quadratic formula, and finding least common denominators, with examples yielding specific solutions and emphasizing the importance of checking for extraneous solutions, providing a comprehensive understanding of solving equations.', 'chapters': [{'end': 4066.527, 'start': 3892.906, 'title': 'Solving quadratic equations', 'summary': 'Demonstrates solving quadratic equations in standard form, factoring, and finding the values of the variable using specific examples, including the equation y squared equals 18 minus 7y.', 'duration': 173.621, 'highlights': ['The standard form for a quadratic equation is the form ax squared plus bx plus c equals 0, where a, b, and c represent real numbers.', 'The key steps to solving quadratic equations are usually to write the equation in standard form and then either factor it or use the quadratic formula.', 'The equation y squared minus 18 plus 7y equals 0 is derived from y squared equals 18 minus 7y by rearranging it in standard form.', 'The equation y squared plus 7y minus 18 equals 0 is factored as y plus nine times y minus two equals zero, leading to the values y equals negative 9 or y equals 2.']}, {'end': 4419.291, 'start': 4066.527, 'title': 'Quadratic equations and solutions', 'summary': 'Covers solving quadratic equations using factoring and the quadratic formula, including examples with real solutions and the importance of considering all possible solutions, emphasizing the method of factoring and the use of the quadratic formula over integers.', 'duration': 352.764, 'highlights': ['The chapter emphasizes the method of factoring to solve quadratic equations, as demonstrated in the example of finding solutions for w squared equals 121, where the factors w plus 11 and w minus 11 equal to zero lead to the solutions w equals -11 or w equals 11.', 'The use of the quadratic formula to solve quadratic equations is highlighted, as shown in the example of finding solutions for x times x plus 2 equals 7, where the application of the quadratic formula results in the solutions x equals -1 plus 2 square root of 2 and x equals -1 minus 2 square root of 2.', 'The significance of considering all possible solutions for quadratic equations is emphasized, particularly in the example of finding solutions for x times x plus 2 equals 7, where the faulty reasoning of assuming whole numbers for the solutions is corrected by demonstrating the use of the quadratic formula to find the correct solutions.', 'The importance of using the correct method to solve quadratic equations is highlighted, with an emphasis on factoring and the quadratic formula over integers, as demonstrated through the examples provided in the chapter.']}, {'end': 4739.242, 'start': 4419.291, 'title': 'Solving quadratic and rational equations', 'summary': 'Covers solving quadratic equations using the quadratic formula and solving rational equations by finding the least common denominator, with an emphasis on the importance of checking for extraneous solutions.', 'duration': 319.951, 'highlights': ['Solving quadratic equations using the quadratic formula', 'Importance of checking for extraneous solutions in rational equations', 'Solving rational equations by finding the least common denominator']}, {'end': 5130.658, 'start': 4740.899, 'title': 'Solving rational equations', 'summary': 'Demonstrates how to solve rational equations by finding the least common denominator and then clearing the denominators, leading to the solution c equals negative two, using the method of finding the least common denominator and clearing the denominator.', 'duration': 389.759, 'highlights': ['The chapter demonstrates how to solve rational equations by finding the least common denominator and then clearing the denominators.', 'The solution C equals negative two is obtained using the method of finding the least common denominator and clearing the denominator.', 'It is essential to check the solutions and eliminate any extraneous solutions that make the denominators of the original equations go to 0.']}, {'end': 5508.856, 'start': 5130.658, 'title': 'Solving radical equations', 'summary': 'Discusses solving radical equations by isolating the radical term, getting rid of the radical, and checking for extraneous solutions, resulting in the solutions x=9 and identifying a fractional exponent as a radical in disguise.', 'duration': 378.198, 'highlights': ['The chapter emphasizes isolating the radical term by getting it on one side of the equation and the remaining terms on the other side, exemplified by isolating the square root term x and subtracting x from both sides to yield 12 minus x on the right side.', 'It demonstrates getting rid of the radical by squaring both sides of the equation, illustrated through the process of squaring the equation x + square root of x = 12 to obtain a quadratic equation without any radical signs.', 'An important step of checking for extraneous solutions is highlighted, wherein x=16 is identified as an extraneous solution by plugging it into the original equation and finding it to be untrue, leading to the elimination of x=16 as a solution.', 'The concept of a fractional exponent as a radical in disguise is introduced, and the process of isolating and subsequently getting rid of the fractional exponent is demonstrated through the equation 2 times p to the 4 fifths equals 1 eighth.']}, {'end': 5800.014, 'start': 5509.757, 'title': 'Radical equations solving', 'summary': 'Explains the concept of solving radical equations, emphasizing the importance of considering plus or minus solutions for even roots or even number powers, and demonstrates the process of solving an equation involving fractional exponents, resulting in two solutions: p equals 1/32 and p equals -1/32.', 'duration': 290.257, 'highlights': ['The importance of considering plus or minus solutions for even roots or even number powers is emphasized, exemplified by the equation x squared equals four and the cube root of negative 8, highlighting the need to include both positive and negative answers. (Relevance: 5)', 'The process of solving an equation involving fractional exponents is demonstrated, resulting in two solutions: P equals 1/32 and P equals -1/32, which are then verified by plugging them back into the original equation. (Relevance: 4)', 'An alternate approach to getting rid of the fractional exponent is presented, involving raising both sides of the equation to the 5 fourths power, resulting in the expression p to the 1 power, which is just p, equaling plus or minus 1 sixteenth to the 5 fourths. (Relevance: 3)']}], 'duration': 1907.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU3892906.jpg', 'highlights': ['The importance of considering plus or minus solutions for even roots or even number powers is emphasized, exemplified by the equation x squared equals four and the cube root of negative 8, highlighting the need to include both positive and negative answers.', 'The process of solving an equation involving fractional exponents is demonstrated, resulting in two solutions: P equals 1/32 and P equals -1/32, which are then verified by plugging them back into the original equation.', 'An alternate approach to getting rid of the fractional exponent is presented, involving raising both sides of the equation to the 5 fourths power, resulting in the expression p to the 1 power, which is just p, equaling plus or minus 1 sixteenth to the 5 fourths.', 'The chapter emphasizes isolating the radical term by getting it on one side of the equation and the remaining terms on the other side, exemplified by isolating the square root term x and subtracting x from both sides to yield 12 minus x on the right side.', 'The use of the quadratic formula to solve quadratic equations is highlighted, as shown in the example of finding solutions for x times x plus 2 equals 7, where the application of the quadratic formula results in the solutions x equals -1 plus 2 square root of 2 and x equals -1 minus 2 square root of 2.']}, {'end': 8166.467, 'segs': [{'end': 5992.071, 'src': 'embed', 'start': 5960.077, 'weight': 1, 'content': [{'end': 5967.163, 'text': 'The second example is a little different because the absolute value sign is around a more complicated expression, not just around the x.', 'start': 5960.077, 'duration': 7.086}, {'end': 5972.807, 'text': 'I would start by isolating the absolute value part.', 'start': 5969.286, 'duration': 3.521}, {'end': 5976.928, 'text': "But it's already isolated.", 'start': 5975.827, 'duration': 1.101}, {'end': 5982.569, 'text': "So I'll just go ahead and jump to thinking about distance on the number line.", 'start': 5977.788, 'duration': 4.781}, {'end': 5992.071, 'text': 'So on my number line, the whole expression, three x plus two is supposed to be at a distance of four from zero.', 'start': 5983.609, 'duration': 8.462}], 'summary': 'The expression |3x + 2| should be at a distance of 4 from zero on the number line.', 'duration': 31.994, 'max_score': 5960.077, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5960077.jpg'}, {'end': 6162.612, 'src': 'embed', 'start': 6133.043, 'weight': 5, 'content': [{'end': 6136.365, 'text': 'In many cases, an absolute value equation will have two solutions.', 'start': 6133.043, 'duration': 3.322}, {'end': 6139.427, 'text': "But in some cases, it'll have no solutions.", 'start': 6137.386, 'duration': 2.041}, {'end': 6142.748, 'text': "And occasionally, it'll have just one solution.", 'start': 6140.167, 'duration': 2.581}, {'end': 6150.973, 'text': 'This video is about interval notation, an easy and well-known way to record inequalities.', 'start': 6145.09, 'duration': 5.883}, {'end': 6157.757, 'text': 'Before dealing with interval notation, it is important to know how to deal with inequalities.', 'start': 6152.854, 'duration': 4.903}, {'end': 6162.612, 'text': 'Our first example of an inequality is written here.', 'start': 6159.122, 'duration': 3.49}], 'summary': 'An absolute value equation can have two, one, or no solutions, and interval notation is used to record inequalities.', 'duration': 29.569, 'max_score': 6133.043, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU6133043.jpg'}, {'end': 6567.78, 'src': 'embed', 'start': 6542.471, 'weight': 0, 'content': [{'end': 6547.372, 'text': 'It is still identical, just written in an easier form to transform it into interval notation.', 'start': 6542.471, 'duration': 4.901}, {'end': 6550.953, 'text': 'Then we can see that 0 has a soft bracket.', 'start': 6548.092, 'duration': 2.861}, {'end': 6560.237, 'text': 'because it is not including 0, put our comma, 4, our other key value, and a hard bracket, because it is or equal to 4.', 'start': 6551.693, 'duration': 8.544}, {'end': 6567.78, 'text': 'For interval notation, you must always have the smaller number on the left side.', 'start': 6560.237, 'duration': 7.543}], 'summary': 'Transcript explains interval notation with example of [0, 4]', 'duration': 25.309, 'max_score': 6542.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU6542471.jpg'}, {'end': 7054.012, 'src': 'embed', 'start': 7028.837, 'weight': 2, 'content': [{'end': 7035.584, 'text': "The good news is we can solve linear inequalities just like we'd solve linear equations by distributing,", 'start': 7028.837, 'duration': 6.747}, {'end': 7040.409, 'text': 'adding and subtracting terms to both sides and multiplying and dividing by numbers on both sides.', 'start': 7035.584, 'duration': 4.825}, {'end': 7050.98, 'text': "The only thing that's different is that if you multiply or divide by a negative number, then you need to reverse the direction of the inequality.", 'start': 7041.029, 'duration': 9.951}, {'end': 7054.012, 'text': 'For example, if we had the inequality,', 'start': 7051.991, 'duration': 2.021}], 'summary': 'Linear inequalities can be solved like equations by adding, subtracting, multiplying, and dividing; reversing inequality if multiplied or divided by a negative number.', 'duration': 25.175, 'max_score': 7028.837, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU7028837.jpg'}, {'end': 7434.914, 'src': 'embed', 'start': 7409.335, 'weight': 8, 'content': [{'end': 7414.379, 'text': 'As my final example, I have an inequality that has two inequality signs in it.', 'start': 7409.335, 'duration': 5.044}, {'end': 7418.142, 'text': 'Negative 3 is less than or equal to 6x minus 2 is less than 10.', 'start': 7414.479, 'duration': 3.663}, {'end': 7423.005, 'text': 'I can think of this as being a compound inequality with two parts.', 'start': 7418.142, 'duration': 4.863}, {'end': 7432.432, 'text': 'Negative 3x is less than or equal to 6x minus 2, and at the same time, 6x minus 2 is less than 10.', 'start': 7423.546, 'duration': 8.886}, {'end': 7434.914, 'text': 'I could solve this in two pieces as before.', 'start': 7432.432, 'duration': 2.482}], 'summary': 'The compound inequality -3 ≤ 6x - 2 < 10 can be solved in two parts.', 'duration': 25.579, 'max_score': 7409.335, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU7409335.jpg'}, {'end': 7536.117, 'src': 'embed', 'start': 7484.847, 'weight': 4, 'content': [{'end': 7488.628, 'text': "Either way we do it, let's see if what it looks like on the number line.", 'start': 7484.847, 'duration': 3.781}, {'end': 7501.193, 'text': "So on the number line, we're looking for things that are between two and negative one six, including the negative one six, but not including the two.", 'start': 7489.489, 'duration': 11.704}, {'end': 7507.935, 'text': 'In interval notation, we can write this as hard bracket negative one six to soft bracket.', 'start': 7502.093, 'duration': 5.842}, {'end': 7513.537, 'text': 'In this video, we solve linear inequalities, including some compound inequalities.', 'start': 7509.175, 'duration': 4.362}, {'end': 7517.022, 'text': 'joined by the conjunctions and and or.', 'start': 7513.939, 'duration': 3.083}, {'end': 7523.967, 'text': "Remember, when we're working with and we're looking for places on the number line where both statements are true.", 'start': 7518.303, 'duration': 5.664}, {'end': 7528.711, 'text': "That is, we're looking for the overlap on the number line.", 'start': 7525.268, 'duration': 3.443}, {'end': 7536.117, 'text': 'In this case, the points on the number line that are colored both red and blue at the same time.', 'start': 7530.713, 'duration': 5.404}], 'summary': 'Solving linear inequalities, including compound inequalities, on the number line.', 'duration': 51.27, 'max_score': 7484.847, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU7484847.jpg'}, {'end': 7670.454, 'src': 'embed', 'start': 7640.981, 'weight': 6, 'content': [{'end': 7645.423, 'text': 'So somehow our solution to this inequality should take this into account.', 'start': 7640.981, 'duration': 4.442}, {'end': 7655.082, 'text': 'In fact, a good way to solve an inequality involving x squared or higher power terms is to solve the associated equation first.', 'start': 7646.715, 'duration': 8.367}, {'end': 7662.848, 'text': 'But before we even do that, I like to pull everything over to one side so that my inequality has zero on the other side.', 'start': 7655.942, 'duration': 6.906}, {'end': 7670.454, 'text': "So for our equation, I'll subtract four from both sides to get x squared minus four is less than zero.", 'start': 7664.389, 'duration': 6.065}], 'summary': 'To solve x^2 - 4 < 0, first solve x^2 - 4 = 0.', 'duration': 29.473, 'max_score': 7640.981, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU7640981.jpg'}, {'end': 7774.476, 'src': 'embed', 'start': 7745.46, 'weight': 7, 'content': [{'end': 7749.223, 'text': 'And in fact everywhere on this region of the number line.', 'start': 7745.46, 'duration': 3.763}, {'end': 7755.027, 'text': "my expression is going to be positive because it can't jump from positive to negative without going through a place where it's.", 'start': 7749.223, 'duration': 5.804}, {'end': 7764.628, 'text': 'I can figure out whether x squared minus four is positive or negative on this region and on this region of the number line by plugging in test values similarly.', 'start': 7756.502, 'duration': 8.126}, {'end': 7770.273, 'text': 'A value to plug in between negative two and two, a nice value is x equals zero.', 'start': 7764.648, 'duration': 5.625}, {'end': 7774.476, 'text': "Zero squared minus four, that's negative four, a negative number.", 'start': 7771.093, 'duration': 3.383}], 'summary': 'The expression x squared minus four is negative for x between -2 and 2.', 'duration': 29.016, 'max_score': 7745.46, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU7745460.jpg'}], 'start': 5800.014, 'title': 'Solving equations and inequalities', 'summary': "Covers techniques for solving absolute value equations, converting equations from interval notation to inequality notation, and explains how to solve absolute value inequalities, linear inequalities, polynomial, and rational inequalities. it emphasizes the use of interval notation, reversing inequalities when multiplying or dividing by a negative number, and the use of 'and' and 'or' conjunctions in compound inequalities.", 'chapters': [{'end': 6390.04, 'start': 5800.014, 'title': 'Solving absolute value equations', 'summary': 'Covers solving absolute value equations, providing techniques to isolate the absolute value part, understand the concept of distance on a number line, and check solutions, with emphasis on common mistakes and scenarios where equations may have no or one solution. additionally, it introduces interval notation and demonstrates how to represent inequalities in interval notation, emphasizing the use of soft and hard brackets.', 'duration': 590.026, 'highlights': ['Solving absolute value equations involves isolating the absolute value part, understanding the concept of distance on a number line, and checking solutions.', 'Understanding the concept of distance on a number line is crucial in solving absolute value equations.', 'Interval notation is introduced as a method to represent inequalities, emphasizing the use of soft and hard brackets.']}, {'end': 6703.375, 'start': 6390.96, 'title': 'Interval notation & inequalities', 'summary': 'Discusses converting equations from interval notation to inequality notation, using key values such as 5 and negative infinity, and the importance of the direction of inequalities, highlighting the process of converting absolute value inequalities to interval notation and discussing the implications on the number line.', 'duration': 312.415, 'highlights': ['The importance of the direction of inequalities, emphasizing the need for the lower value to be on the left side, demonstrated by the example of flipping the inequality to maintain equivalence, which is crucial for transforming it into interval notation and has practical implications on the number line.', "Converting absolute value inequalities to interval notation, highlighting how the original equation 'absolute value of x is less than 5' can be expressed as 'negative five is less than x, which is less than five', showing the equivalence of different formulations and their representation on the number line.", 'Explaining the significance of interval notation as an alternative way to write inequalities, demonstrated through examples of inequalities with absolute value signs, showcasing the process of expressing the inequalities in interval notation and its equivalence to the original expressions.']}, {'end': 7275.114, 'start': 6704.096, 'title': 'Solving absolute value inequalities', 'summary': 'Explains how to solve absolute value inequalities using distance on the number line and provides examples of compound inequalities, with a focus on reversing the inequality when multiplying or dividing by a negative number.', 'duration': 571.018, 'highlights': ['Solving absolute value inequalities involves thinking in terms of distance on the number line and rewriting the inequality without absolute value signs to find the range of values for the variable.', 'The process of solving absolute value inequalities involves considering the distance between the expression and zero, with less than indicating proximity to zero and greater than indicating distance from zero.', 'Examples illustrate the process of solving compound inequalities by addressing each part separately and then combining the solutions at the end.', 'Emphasis is placed on reversing the inequality when multiplying or dividing by a negative number in the context of solving linear inequalities.']}, {'end': 7557.145, 'start': 7275.114, 'title': 'Solving linear inequalities', 'summary': "Demonstrates solving linear inequalities, including compound inequalities, using examples and graphical representations, emphasizing the use of 'and' and 'or' conjunctions to find overlapping or inclusive solutions on the number line.", 'duration': 282.031, 'highlights': ['Solving linear inequalities using examples and graphical representations', "Emphasizing the use of 'and' and 'or' conjunctions in finding overlapping or inclusive solutions on the number line", 'Illustrating the process of finding overlapping solutions on the number line']}, {'end': 8166.467, 'start': 7559.086, 'title': 'Solving polynomial and rational inequalities', 'summary': 'Discusses solving polynomial and rational inequalities, emphasizing the importance of solving associated equations first, using test values, and considering where the expression is positive or negative to find the solutions, with examples demonstrating the process.', 'duration': 607.381, 'highlights': ['Solving associated equation first is crucial in solving inequalities involving x squared or higher power terms.', 'Using test values to determine where the expression is positive or negative is an effective approach in finding solutions.', 'Consideration of where the expression is positive or negative on the number line is critical in finding solutions.']}], 'duration': 2366.453, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU5800014.jpg', 'highlights': ['Interval notation is crucial for representing inequalities and emphasizes the use of soft and hard brackets.', 'Understanding the concept of distance on a number line is crucial in solving absolute value equations.', 'Reversing the inequality when multiplying or dividing by a negative number is emphasized in solving linear inequalities.', 'Solving absolute value inequalities involves thinking in terms of distance on the number line and rewriting the inequality without absolute value signs.', "Emphasis is placed on using 'and' and 'or' conjunctions in finding overlapping or inclusive solutions on the number line.", 'Converting absolute value inequalities to interval notation demonstrates the equivalence of different formulations and their representation on the number line.', 'Solving associated equation first is crucial in solving inequalities involving x squared or higher power terms.', 'Using test values to determine where the expression is positive or negative is an effective approach in finding solutions.', 'Examples illustrate the process of solving compound inequalities by addressing each part separately and then combining the solutions at the end.', 'Emphasizing the importance of the direction of inequalities and the practical implications on the number line.']}, {'end': 10143.671, 'segs': [{'end': 8226.946, 'src': 'embed', 'start': 8196.499, 'weight': 0, 'content': [{'end': 8207.962, 'text': 'then the distance between them is given by the formula the square root of x two, minus, x one squared, plus, y two, minus y one squared.', 'start': 8196.499, 'duration': 11.463}, {'end': 8212.263, 'text': 'This formula actually comes from the Pythagorean theorem.', 'start': 8209.202, 'duration': 3.061}, {'end': 8216.825, 'text': 'Let me draw a right triangle with these two points.', 'start': 8213.004, 'duration': 3.821}, {'end': 8226.946, 'text': 'as two of its vertices, then the length of this side is the difference between the two x coordinates.', 'start': 8217.358, 'duration': 9.588}], 'summary': 'The distance between two points is calculated using the pythagorean theorem formula.', 'duration': 30.447, 'max_score': 8196.499, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU8196499.jpg'}, {'end': 8662.13, 'src': 'embed', 'start': 8600.325, 'weight': 1, 'content': [{'end': 8601.485, 'text': 'In this video,', 'start': 8600.325, 'duration': 1.16}, {'end': 8609.248, 'text': 'we use the midpoint formula to find the midpoint of a line segment just by taking the average of the x coordinates and the average of the y coordinates.', 'start': 8601.485, 'duration': 7.763}, {'end': 8613.249, 'text': 'This video is about graphs and equations of circles.', 'start': 8610.228, 'duration': 3.021}, {'end': 8624.552, 'text': 'Suppose we want to find the equation of a circle of radius five centered at the point three to look something like this.', 'start': 8614.409, 'duration': 10.143}, {'end': 8639.364, 'text': 'For any point x y on the circle, we know that the distance of that point x y from the center is equal to the radius that is five.', 'start': 8627.453, 'duration': 11.911}, {'end': 8648.654, 'text': 'From the distance formula, that distance of five is equal to the square root of the difference of the x coordinates.', 'start': 8640.605, 'duration': 8.049}, {'end': 8651.697, 'text': "That's x minus three.", 'start': 8649.795, 'duration': 1.902}, {'end': 8662.13, 'text': "squared plus the difference of the y coordinates, that's y minus two squared.", 'start': 8653.326, 'duration': 8.804}], 'summary': 'Using midpoint formula to find center of circle with radius five at point (3,2)', 'duration': 61.805, 'max_score': 8600.325, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU8600325.jpg'}, {'end': 8782.938, 'src': 'embed', 'start': 8759.356, 'weight': 3, 'content': [{'end': 8766.798, 'text': "That's the general formula for a circle with radius r and center h k.", 'start': 8759.356, 'duration': 7.442}, {'end': 8770.159, 'text': 'Notice that the coordinates h and k are subtracted here.', 'start': 8766.798, 'duration': 3.361}, {'end': 8777.175, 'text': 'the two squares are added, because they are in the distance formula, and the radius is squared on the other side.', 'start': 8771.052, 'duration': 6.123}, {'end': 8782.938, 'text': 'If you remember this general formula, that makes it easy to write down the equation for a circle.', 'start': 8778.236, 'duration': 4.702}], 'summary': 'General circle formula: (x-h)^2 + (y-k)^2 = r^2', 'duration': 23.582, 'max_score': 8759.356, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU8759356.jpg'}, {'end': 9201.806, 'src': 'embed', 'start': 9172.115, 'weight': 4, 'content': [{'end': 9175.077, 'text': 'We also showed a method of completing the square.', 'start': 9172.115, 'duration': 2.962}, {'end': 9183.96, 'text': 'When you have an equation for a circle in disguise, completing the square will help you rewrite it into the standard form.', 'start': 9176.537, 'duration': 7.423}, {'end': 9189.219, 'text': 'This video is about graphs and equations of lines.', 'start': 9186.257, 'duration': 2.962}, {'end': 9194.462, 'text': "Here we're given the graph of a line, we want to find the equation.", 'start': 9190.84, 'duration': 3.622}, {'end': 9201.806, 'text': 'One standard format for the equation of a line is y equals mx plus b.', 'start': 9195.482, 'duration': 6.324}], 'summary': 'Demonstrated method of completing the square for rewriting circle equations. explained relationship between graphs and equations of lines.', 'duration': 29.691, 'max_score': 9172.115, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9172115.jpg'}, {'end': 9421.673, 'src': 'embed', 'start': 9394.014, 'weight': 5, 'content': [{'end': 9405.543, 'text': 'So now I can write out my final equation for my line, y equals negative 3 fourths x plus 11 fourths by plugging in for m and b.', 'start': 9394.014, 'duration': 11.529}, {'end': 9408.586, 'text': "Next, let's find the equation for this horizontal line.", 'start': 9405.543, 'duration': 3.043}, {'end': 9412.611, 'text': 'A horizontal line has slope zero.', 'start': 9410.41, 'duration': 2.201}, {'end': 9418.052, 'text': 'So if we think of it as y equals mx plus b, m is going to be zero.', 'start': 9412.931, 'duration': 5.121}, {'end': 9420.572, 'text': "In other words, it's just y equals b.", 'start': 9418.092, 'duration': 2.48}, {'end': 9421.673, 'text': 'y is some constant.', 'start': 9420.572, 'duration': 1.101}], 'summary': 'The equation of a horizontal line is y equals a constant, with slope zero.', 'duration': 27.659, 'max_score': 9394.014, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9394014.jpg'}, {'end': 9668.912, 'src': 'embed', 'start': 9641.707, 'weight': 6, 'content': [{'end': 9646.768, 'text': 'In this video, we saw that you can find the equation for a line if you know the slope.', 'start': 9641.707, 'duration': 5.061}, {'end': 9652.089, 'text': 'And you know one point.', 'start': 9648.808, 'duration': 3.281}, {'end': 9656.49, 'text': 'you can also find the equation for the line if you know two points,', 'start': 9652.089, 'duration': 4.401}, {'end': 9664.371, 'text': 'because you can use the two points to get the slope and then plug in one of those points to figure out the rest of the equation.', 'start': 9656.49, 'duration': 7.881}, {'end': 9668.912, 'text': 'We saw two standard forms for the equation of a line.', 'start': 9666.371, 'duration': 2.541}], 'summary': 'Finding equation of a line using slope and one/two points, two standard forms.', 'duration': 27.205, 'max_score': 9641.707, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9641707.jpg'}, {'end': 9737.575, 'src': 'embed', 'start': 9704.086, 'weight': 8, 'content': [{'end': 9719.173, 'text': "In other words, the rise over the run is three over four, then any other line that's parallel to this line will have the same slope,", 'start': 9704.086, 'duration': 15.087}, {'end': 9721.734, 'text': 'the same rise over run.', 'start': 9719.173, 'duration': 2.561}, {'end': 9728.592, 'text': "So that's our first fact to keep in mind, parallel lines have the same slopes.", 'start': 9723.851, 'duration': 4.741}, {'end': 9737.575, 'text': 'Suppose on the other hand, we want to find a line perpendicular to our original line with its original slope of three fourths.', 'start': 9729.533, 'duration': 8.042}], 'summary': 'Parallel lines have the same slope; perpendicular line has original slope of three fourths.', 'duration': 33.489, 'max_score': 9704.086, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9704086.jpg'}, {'end': 9791.842, 'src': 'embed', 'start': 9762.382, 'weight': 7, 'content': [{'end': 9769.048, 'text': "So I'll write this as a principle that perpendicular lines have opposite reciprocal slopes.", 'start': 9762.382, 'duration': 6.666}, {'end': 9774.608, 'text': "To get the hang of what it means to be an opposite reciprocal, let's look at a few examples.", 'start': 9770.063, 'duration': 4.545}, {'end': 9783.156, 'text': "So here's the original slope, and this will be the opposite reciprocal, which would represent the slope of our perpendicular line.", 'start': 9775.168, 'duration': 7.988}, {'end': 9791.842, 'text': 'So for example, if m1 was 2, then the opposite reciprocal, reciprocal of two is one half opposite means I changed the sign.', 'start': 9783.637, 'duration': 8.205}], 'summary': 'Perpendicular lines have opposite reciprocal slopes, e.g. m1=2, m2=-1/2', 'duration': 29.46, 'max_score': 9762.382, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9762382.jpg'}, {'end': 10006.994, 'src': 'embed', 'start': 9981.835, 'weight': 9, 'content': [{'end': 9989.462, 'text': 'So on my original slope of my original line is negative two, which means my perpendicular slope is the opposite reciprocal.', 'start': 9981.835, 'duration': 7.627}, {'end': 9995.328, 'text': "So I take the reciprocal of of negative two, that's negative one half and I change the sign.", 'start': 9989.983, 'duration': 5.345}, {'end': 9998.992, 'text': 'So that gives me one half as the slope of my perpendicular line.', 'start': 9995.408, 'duration': 3.584}, {'end': 10006.994, 'text': "Now my new line I know it's going to be y equals one half x plus B for some B.", 'start': 9999.792, 'duration': 7.202}], 'summary': 'The original slope is -2, the perpendicular slope is 1/2 for the new line y = 1/2x + b.', 'duration': 25.159, 'max_score': 9981.835, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU9981835.jpg'}, {'end': 10071.989, 'src': 'embed', 'start': 10042.068, 'weight': 10, 'content': [{'end': 10050.332, 'text': 'So the y coordinate is always three, which means that my line will be a horizontal line through that height at height y equals three.', 'start': 10042.068, 'duration': 8.264}, {'end': 10056.702, 'text': 'If I want something parallel to this line, it will also be a horizontal line.', 'start': 10050.799, 'duration': 5.903}, {'end': 10063.745, 'text': "Since it goes through the point negative two, one, let's see negative two, one has to go through that point there.", 'start': 10057.222, 'duration': 6.523}, {'end': 10071.989, 'text': "It's going to have always have a y coordinate of one the same as the y coordinate of the point it goes through.", 'start': 10063.765, 'duration': 8.224}], 'summary': 'The line is horizontal at y=3, parallel line goes through (-2,1)', 'duration': 29.921, 'max_score': 10042.068, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10042068.jpg'}, {'end': 10125.376, 'src': 'embed', 'start': 10092.2, 'weight': 11, 'content': [{'end': 10097.503, 'text': 'So I need a vertical line that goes through the point three, four.', 'start': 10092.2, 'duration': 5.303}, {'end': 10103.506, 'text': "Okay, and so I'm going to draw a vertical line there.", 'start': 10097.523, 'duration': 5.983}, {'end': 10110.107, 'text': 'Now vertical lines have the form x equals something for their equation.', 'start': 10103.526, 'duration': 6.581}, {'end': 10116.751, 'text': "And to find what x equals, I just need to look at the x coordinate of the point I'm going through this point three, four.", 'start': 10110.628, 'duration': 6.123}, {'end': 10125.376, 'text': 'So that x coordinate is three and all the points on this, this perpendicular and vertical line have x coordinate three.', 'start': 10117.151, 'duration': 8.225}], 'summary': 'Draw a vertical line through the point (3,4) with x=3.', 'duration': 33.176, 'max_score': 10092.2, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10092200.jpg'}], 'start': 8168.484, 'title': 'Equations of lines and circles', 'summary': 'Covers distance and midpoint formulas, circle equations, transforming equations, and finding parallel and perpendicular lines. it includes examples and principles for finding slopes, transforming equations, and finding equations of lines.', 'chapters': [{'end': 8692.982, 'start': 8168.484, 'title': 'Distance, midpoint, and circle equations', 'summary': 'Explains the distance formula using the pythagorean theorem, the midpoint formula for finding the midpoint of a line segment, and the equation of a circle of radius 5 centered at the point (3, 2) using the distance formula, illustrating through examples.', 'duration': 524.498, 'highlights': ['The distance formula is derived using the Pythagorean theorem, where the distance between two points (x1, y1) and (x2, y2) is given by the square root of (x2 - x1)^2 + (y2 - y1)^2, demonstrated in an example with coordinates (-1, 5) and (4, 2) resulting in a distance of √34.', 'The midpoint formula is utilized to find the midpoint of a line segment by taking the average of the x and y coordinates, exemplified with the segment between the points (-1, 5) and (4, 2) yielding the midpoint coordinates of (3/2, 7/2).', 'The equation of a circle with a radius of 5 centered at the point (3, 2) is derived using the distance formula, resulting in the equation x - 3)^2 + (y - 2)^2 = 25.']}, {'end': 9201.806, 'start': 8693.462, 'title': 'Equation of a circle: general formula and transformations', 'summary': 'Discusses the general formula for a circle with radius r centered at point h k, the method of transforming an equation into the standard form x minus h squared plus y minus k squared equals r squared, and the application of completing the square to rewrite equations in the standard form. it also covers finding the equation of a line in the standard format y equals mx plus b.', 'duration': 508.344, 'highlights': ['The general formula for a circle with radius r and center h k is x minus h squared plus y minus k squared equals r squared.', 'The method of completing the square is used to transform an equation into the standard form x minus h squared plus y minus k squared equals r squared.', 'The standard format for the equation of a line is y equals mx plus b.']}, {'end': 9840.793, 'start': 9201.806, 'title': 'Equation of a line and slope', 'summary': 'Explains how to find the equation of a line using the slope intercept form and the point slope form, and also discusses the properties of parallel and perpendicular lines, providing examples and principles for finding their slopes.', 'duration': 638.987, 'highlights': ['The chapter explains how to find the equation of a line using the slope intercept form and the point slope form.', 'Parallel lines have the same slopes, while perpendicular lines have opposite reciprocal slopes.', 'Principles for finding the slopes of parallel and perpendicular lines are demonstrated with examples.']}, {'end': 10143.671, 'start': 9840.793, 'title': 'Finding parallel and perpendicular lines', 'summary': 'Explains how to find the equations of parallel and perpendicular lines, using the slope and given points, with examples involving standard form and horizontal/vertical lines.', 'duration': 302.878, 'highlights': ['The slope of the original line is 4/3, resulting in the slope of the parallel line also being 4/3.', 'The equation of a line perpendicular to a given line through a given point is y = 1/2x - 1.', 'A horizontal line parallel to y = 3 will also be y = 3.', 'A vertical line perpendicular to y = 4 through the point (3, 4) is x = 3.']}], 'duration': 1975.187, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU8168484.jpg', 'highlights': ['The distance formula is derived using the Pythagorean theorem, resulting in a distance of √34.', 'The midpoint formula finds the midpoint of a line segment, yielding the midpoint coordinates of (3/2, 7/2).', 'The equation of a circle with a radius of 5 centered at the point (3, 2) is derived using the distance formula.', 'The general formula for a circle with radius r and center h k is x minus h squared plus y minus k squared equals r squared.', 'The method of completing the square is used to transform an equation into the standard form.', 'The standard format for the equation of a line is y equals mx plus b.', 'The chapter explains how to find the equation of a line using the slope intercept form and the point slope form.', 'Parallel lines have the same slopes, while perpendicular lines have opposite reciprocal slopes.', 'The slope of the original line is 4/3, resulting in the slope of the parallel line also being 4/3.', 'The equation of a line perpendicular to a given line through a given point is y = 1/2x - 1.', 'A horizontal line parallel to y = 3 will also be y = 3.', 'A vertical line perpendicular to y = 4 through the point (3, 4) is x = 3.']}, {'end': 11938.06, 'segs': [{'end': 10176.083, 'src': 'embed', 'start': 10145.512, 'weight': 0, 'content': [{'end': 10149.114, 'text': 'This video introduces functions and their domains and ranges.', 'start': 10145.512, 'duration': 3.602}, {'end': 10159.637, 'text': 'function is a correspondence between input numbers, usually the x values, and output numbers, usually the y values,', 'start': 10150.834, 'duration': 8.803}, {'end': 10163.578, 'text': 'that sends each input number to exactly one output number.', 'start': 10159.637, 'duration': 3.941}, {'end': 10176.083, 'text': 'Sometimes a function is thought of as a rule or machine in which you can feed in x values as input and get out y values as output.', 'start': 10164.638, 'duration': 11.445}], 'summary': 'Introduction to functions, domains, and ranges.', 'duration': 30.571, 'max_score': 10145.512, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10145512.jpg'}, {'end': 10362.819, 'src': 'embed', 'start': 10329.329, 'weight': 4, 'content': [{'end': 10339.534, 'text': 'we could rewrite that as a squared plus six a plus nine plus one, or a squared plus six a plus 10.', 'start': 10329.329, 'duration': 10.205}, {'end': 10347.038, 'text': "When evaluating a function on a complex expression, it's important to keep the parentheses when you plug in for x.", 'start': 10339.534, 'duration': 7.504}, {'end': 10350.52, 'text': 'That way, you evaluate the function on the whole expression.', 'start': 10347.038, 'duration': 3.482}, {'end': 10362.819, 'text': 'For example, it would be wrong to write f of a plus three equals a plus three, squared plus one, without the parentheses,', 'start': 10351.561, 'duration': 11.258}], 'summary': 'Importance of keeping parentheses when evaluating complex expressions.', 'duration': 33.49, 'max_score': 10329.329, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10329329.jpg'}, {'end': 10586.604, 'src': 'embed', 'start': 10562.7, 'weight': 1, 'content': [{'end': 10570.344, 'text': "If we meet a function that's described as an equation instead of a graph, one way to find the domain and range are to graph the function.", 'start': 10562.7, 'duration': 7.644}, {'end': 10576.601, 'text': "but it's often possible to find the domain at least more quickly by using algebraic considerations.", 'start': 10571.24, 'duration': 5.361}, {'end': 10586.604, 'text': 'We think about what x values it makes sense to plug into this expression and what x values need to be excluded because they make the algebraic expression impossible to evaluate.', 'start': 10577.422, 'duration': 9.182}], 'summary': 'To find domain and range, graph function or use algebraic considerations.', 'duration': 23.904, 'max_score': 10562.7, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10562700.jpg'}, {'end': 10942.061, 'src': 'embed', 'start': 10912.256, 'weight': 3, 'content': [{'end': 10915.237, 'text': 'Notice this this function is an even function.', 'start': 10912.256, 'duration': 2.981}, {'end': 10924.701, 'text': 'That means it has mirror symmetry across the y axis, the left side looks like exactly like the mirror image of the right side.', 'start': 10916.678, 'duration': 8.023}, {'end': 10929.823, 'text': 'That happens because when you square a positive number like two,', 'start': 10925.641, 'duration': 4.182}, {'end': 10936.125, 'text': 'you get the exact same y value as when you square its mirror image x value of negative two.', 'start': 10929.823, 'duration': 6.302}, {'end': 10942.061, 'text': "The next function y equals x cubed, I'll call that a cubic.", 'start': 10937.659, 'duration': 4.402}], 'summary': 'Functions: even function with mirror symmetry, y=x^3 is a cubic.', 'duration': 29.805, 'max_score': 10912.256, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10912256.jpg'}, {'end': 11848.33, 'src': 'embed', 'start': 11821.641, 'weight': 2, 'content': [{'end': 11826.384, 'text': 'Pause the video for a moment and see if you could describe what happens in these four transformations.', 'start': 11821.641, 'duration': 4.743}, {'end': 11831.467, 'text': "In the first example, we're subtracting four on the outside of the function.", 'start': 11827.745, 'duration': 3.722}, {'end': 11835.215, 'text': 'Adding and subtracting means a translation or shift.', 'start': 11832.532, 'duration': 2.683}, {'end': 11840.902, 'text': "And since we're on the outside of the function affects the y values that's moving us vertically.", 'start': 11836.236, 'duration': 4.666}, {'end': 11848.33, 'text': 'So this transformation should take the square root of graph and move it down by four units.', 'start': 11841.823, 'duration': 6.507}], 'summary': 'Describing four transformations, including subtracting four on the outside of the function, resulting in a vertical shift down by four units.', 'duration': 26.689, 'max_score': 11821.641, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU11821641.jpg'}], 'start': 10145.512, 'title': 'Understanding functions and their properties', 'summary': 'Covers functions and their domains, evaluation of functions, toolkit functions, and function transformations, with examples illustrating key concepts and properties, helping in understanding the correspondence between input and output numbers and the effects of transformations on functions.', 'chapters': [{'end': 10227.853, 'start': 10145.512, 'title': 'Functions and domains', 'summary': 'Introduces functions and their domains and ranges, explaining that a function is a correspondence between input and output numbers, with examples including the biological mother function, which satisfies the condition of sending each input to exactly one output, and the distinction between a function and a non-function based on the number of output possibilities.', 'duration': 82.341, 'highlights': ['The chapter emphasizes that a function is a correspondence between input and output numbers, with each input number being sent to exactly one output number.', 'The example of the biological mother function is used to illustrate a function that satisfies the condition of sending each input to exactly one output.', 'The distinction between a function and a non-function is explained based on the number of output possibilities, with examples including individuals having more than one mother.']}, {'end': 10851.89, 'start': 10228.254, 'title': 'Understanding functions and evaluating domains and ranges', 'summary': 'Explains the concept of functions, using equations and graphs, and provides examples of evaluating functions and finding their domains and ranges, emphasizing the importance of excluding values that make the denominator zero or the expression inside a square root sign negative.', 'duration': 623.636, 'highlights': ['The domain and range of a function are crucial, as they determine the set of possible input and output values, as exemplified by finding the domain and range of a given function graphically and algebraically.', 'The concept of function evaluation is demonstrated through examples, including the use of function notation, and the importance of correctly evaluating complex expressions by preserving parentheses to ensure accurate results.', 'The distinction between a graph representing a function and violating the vertical line test, along with the significance of the vertical line test in determining whether a graph represents a function, is explained with clear examples.', 'The process of evaluating a function on a graph is demonstrated, emphasizing the significance of finding corresponding y values for given x values, and the concept of undefined values in the context of functions is illustrated with a specific example.', 'The importance of considering exclusion of values that make the denominator zero or the expression inside a square root sign negative in determining the domain of a function is highlighted, along with the implications of these exclusions in finding the legitimate domain of a function involving both a square root and a denominator.']}, {'end': 11268.229, 'start': 10853.203, 'title': 'Toolkit functions and their graphs', 'summary': 'Introduces the toolkit functions, including y=x, y=x squared, y=x cubed, y=square root of x, absolute value function, y=2 to the x, y=1 over x, and y=1 over x squared, highlighting their key properties and symmetries.', 'duration': 415.026, 'highlights': ['Toolkit functions include y=x, y=x squared, and y=x cubed, each exhibiting different symmetries and shapes.', 'The function y=square root of x has a domain of x values greater than or equal to 0.', 'The absolute value function exhibits mirror symmetry and a V-shaped graph.', 'The exponential function y=2 to the x represents exponential growth, doubling the y-coordinate with each increase in the x-coordinate by 1.', 'The function y=1 over x is an example of a hyperbola, exhibiting odd symmetry and a shape in the first and third quadrants.']}, {'end': 11938.06, 'start': 11269.69, 'title': 'Function transformations & graphing', 'summary': 'Explains function transformations with examples and rules, including the effects of numbers inside and outside of the function, such as translations, stretches, shrinks, and reflections, using the square root function to illustrate these concepts.', 'duration': 668.37, 'highlights': ['The chapter explains function transformations with examples and rules, including the effects of numbers inside and outside of the function, such as translations, stretches, shrinks, and reflections.', 'The video gives some rules and examples for transformations of functions, particularly focusing on toolkit functions like y equals square root of x, y equals x squared, and y equals the absolute value of x.', 'The chapter emphasizes the impact of numbers on the inside and outside of the function, explaining that numbers on the outside affect the y values, resulting in vertical motions, while numbers on the inside affect the x values, leading to horizontal motions.', 'The chapter illustrates the impact of multiplication and negative signs on function transformations, showcasing how multiplication results in stretches or shrinks, while a negative sign leads to reflection in the vertical direction.', 'The chapter provides graphical representations for each function transformation, visually depicting the shifts and changes in the graphs.']}], 'duration': 1792.548, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU10145512.jpg', 'highlights': ['The chapter emphasizes that a function is a correspondence between input and output numbers, with each input number being sent to exactly one output number.', 'The domain and range of a function are crucial, as they determine the set of possible input and output values, as exemplified by finding the domain and range of a given function graphically and algebraically.', 'The chapter explains function transformations with examples and rules, including the effects of numbers inside and outside of the function, such as translations, stretches, shrinks, and reflections.', 'Toolkit functions include y=x, y=x squared, and y=x cubed, each exhibiting different symmetries and shapes.', 'The concept of function evaluation is demonstrated through examples, including the use of function notation, and the importance of correctly evaluating complex expressions by preserving parentheses to ensure accurate results.']}, {'end': 14656.786, 'segs': [{'end': 12024.058, 'src': 'embed', 'start': 11938.06, 'weight': 0, 'content': [{'end': 11944.341, 'text': 'because the square root of one fourth x is the same thing as the square root of one fourth times the square root of x,', 'start': 11938.06, 'duration': 6.281}, {'end': 11948.182, 'text': 'which is the same thing as one half times the square root of x.', 'start': 11944.341, 'duration': 3.841}, {'end': 11958.464, 'text': 'And So now we can see algebraically that a vertical shrink by a factor of one half is the same as a horizontal stretch by a factor of four,', 'start': 11948.182, 'duration': 10.282}, {'end': 11960.565, 'text': 'at least for this function, the square root function.', 'start': 11958.464, 'duration': 2.101}, {'end': 11966.527, 'text': "This video gave some rules for transformations of functions, which I'll repeat below.", 'start': 11962.305, 'duration': 4.222}, {'end': 11975.431, 'text': 'Numbers on the outside correspond to changes in the y values or vertical motions.', 'start': 11967.688, 'duration': 7.743}, {'end': 11982.92, 'text': 'numbers on the inside of the function affect the x values and result in horizontal motions.', 'start': 11976.838, 'duration': 6.082}, {'end': 11989.703, 'text': 'Adding and subtracting corresponds to translations or shifts.', 'start': 11984.261, 'duration': 5.442}, {'end': 11996.266, 'text': 'Multiplying and dividing by numbers corresponds to stretches and shrinks.', 'start': 11991.524, 'duration': 4.742}, {'end': 12002.448, 'text': 'And putting in a negative sign corresponds to a reflection.', 'start': 11997.806, 'duration': 4.642}, {'end': 12011.67, 'text': 'horizontal reflection, if the negative sign is on the inside, and a vertical reflection, if the negative sign is on the outside.', 'start': 12004.265, 'duration': 7.405}, {'end': 12024.058, 'text': 'Knowing these basic rules about transformations empowers you to be able to sketch graphs of much more complicated functions like y equals three times the square root of x plus two,', 'start': 12012.931, 'duration': 11.127}], 'summary': 'Algebraic rules for function transformations include vertical/horizontal motions, shifts, stretches, shrinks, and reflections.', 'duration': 85.998, 'max_score': 11938.06, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU11938060.jpg'}, {'end': 12084.62, 'src': 'embed', 'start': 12060.274, 'weight': 13, 'content': [{'end': 12067.499, 'text': "So by making sure that a is not zero, we make sure there's really an x squared term, which is the hallmark of a quadratic function.", 'start': 12060.274, 'duration': 7.225}, {'end': 12074.405, 'text': 'Please pause the video for a moment and decide which of these equations represent quadratic functions.', 'start': 12068.801, 'duration': 5.604}, {'end': 12084.62, 'text': 'The first function can definitely be written in the form g of x equals a x squared plus b x plus c.', 'start': 12075.834, 'duration': 8.786}], 'summary': 'Making a non-zero ensures presence of x squared term, hallmark of quadratic function.', 'duration': 24.346, 'max_score': 12060.274, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU12060274.jpg'}, {'end': 12206.042, 'src': 'embed', 'start': 12175.052, 'weight': 12, 'content': [{'end': 12179.856, 'text': 'So in fact, our function can be written in the right form, and it is a quadratic function.', 'start': 12175.052, 'duration': 4.804}, {'end': 12190.625, 'text': 'A function that is already written in the form y equals ax squared plus bx plus C is said to be in standard form.', 'start': 12182.558, 'duration': 8.067}, {'end': 12193.798, 'text': 'So our first example.', 'start': 12192.438, 'duration': 1.36}, {'end': 12206.042, 'text': "g of x is, in standard form, a function that's written in the format of the last function, that is, in the form of y, equals a times x minus h,", 'start': 12193.798, 'duration': 12.244}], 'summary': 'The function is a quadratic function written in standard form.', 'duration': 30.99, 'max_score': 12175.052, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU12175052.jpg'}, {'end': 12380.769, 'src': 'embed', 'start': 12356.256, 'weight': 5, 'content': [{'end': 12364.661, 'text': "a negative coefficient to the x squared term means the parabola will be pointing down, whereas a positive coefficient, like here the coefficient's 1,", 'start': 12356.256, 'duration': 8.405}, {'end': 12365.881, 'text': "means the parabola's pointing up.", 'start': 12364.661, 'duration': 1.22}, {'end': 12368.543, 'text': "I'll write that rule over here.", 'start': 12367.402, 'duration': 1.141}, {'end': 12374.286, 'text': 'So if a is bigger than 0, the parabola opens up.', 'start': 12369.864, 'duration': 4.422}, {'end': 12380.769, 'text': 'And if the value of the coefficient a is less than zero, then the parabola opens down.', 'start': 12375.966, 'duration': 4.803}], 'summary': 'Parabola opens up if a > 0, down if a < 0.', 'duration': 24.513, 'max_score': 12356.256, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU12356256.jpg'}, {'end': 12479.621, 'src': 'embed', 'start': 12447.526, 'weight': 14, 'content': [{'end': 12449.888, 'text': 'So our parabola will look something like this.', 'start': 12447.526, 'duration': 2.362}, {'end': 12457.393, 'text': 'Notice how easy it was to just read off the vertex, when our quadratic function is written in this form.', 'start': 12452.049, 'duration': 5.344}, {'end': 12470.374, 'text': 'In any parabola, any quadratic function written in the form a times x minus h squared plus k has a vertex at h k.', 'start': 12458.094, 'duration': 12.28}, {'end': 12479.621, 'text': "By the same reasoning, we're moving the parabola with a vertex at the origin to the right by h and by k.", 'start': 12470.374, 'duration': 9.247}], 'summary': 'Quadratic function in form a(x-h)^2+k allows for easy vertex extraction.', 'duration': 32.095, 'max_score': 12447.526, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU12447526.jpg'}, {'end': 13264.675, 'src': 'embed', 'start': 13240.687, 'weight': 10, 'content': [{'end': 13250.553, 'text': 'But the x coordinate of the vertex, which is exactly halfway in between the two x intercepts will be at negative b over to a.', 'start': 13240.687, 'duration': 9.866}, {'end': 13252.354, 'text': "That's where the vertex formula comes from.", 'start': 13250.553, 'duration': 1.801}, {'end': 13257.213, 'text': 'And it turns out that this formula works even when there are no x intercepts.', 'start': 13253.632, 'duration': 3.581}, {'end': 13264.675, 'text': 'Even when the quadratic formula gives us no solutions, the vertex still has the x coordinate negative b over 2a.', 'start': 13257.333, 'duration': 7.342}], 'summary': 'The x-coordinate of the vertex is at -b/2a, even when there are no x-intercepts.', 'duration': 23.988, 'max_score': 13240.687, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU13240687.jpg'}, {'end': 13486.305, 'src': 'embed', 'start': 13449.864, 'weight': 6, 'content': [{'end': 13455.527, 'text': 'In this last example, the degree is four, but the number of turning points is one, not three.', 'start': 13449.864, 'duration': 5.663}, {'end': 13462.73, 'text': 'In fact, it turns out that, while the number of turning points is not always equal to the degree minus one,', 'start': 13456.847, 'duration': 5.883}, {'end': 13465.912, 'text': 'it is always less than or equal to the degree minus one.', 'start': 13462.73, 'duration': 3.182}, {'end': 13472.155, 'text': "That's a useful fact to remember when you're sketching graphs or recognizing graphs of polynomials.", 'start': 13466.952, 'duration': 5.203}, {'end': 13486.305, 'text': 'The end behavior of a function is how the ends of the function look as x gets bigger and bigger heads towards infinity or x goes through larger and larger negative numbers towards negative infinity.', 'start': 13473.615, 'duration': 12.69}], 'summary': 'Degree 4 has 1 turning point, not 3. turning points ≤ degree - 1. important for graph sketching.', 'duration': 36.441, 'max_score': 13449.864, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU13449864.jpg'}, {'end': 13552.345, 'src': 'embed', 'start': 13527.253, 'weight': 7, 'content': [{'end': 13534.837, 'text': "If you study these examples and others, you might notice there's a relationship between the degree of the polynomial,", 'start': 13527.253, 'duration': 7.584}, {'end': 13538.638, 'text': 'the leading coefficients of the polynomials, and the end behavior.', 'start': 13534.837, 'duration': 3.801}, {'end': 13552.345, 'text': 'Specifically, these four types of end behavior are determined by whether the degree is even or odd and by whether the leading coefficient is positive or negative.', 'start': 13539.499, 'duration': 12.846}], 'summary': 'Studying polynomials reveals relationship between degree, leading coefficients, and end behavior.', 'duration': 25.092, 'max_score': 13527.253, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU13527253.jpg'}, {'end': 13909.18, 'src': 'embed', 'start': 13882.079, 'weight': 15, 'content': [{'end': 13890.488, 'text': 'The phrase initial value comes from the fact that if we plug in x equals 0, we get a times b to the 0.', 'start': 13882.079, 'duration': 8.409}, {'end': 13894.79, 'text': 'Well, anything to the 0 is just 1, so this is equal to a.', 'start': 13890.488, 'duration': 4.302}, {'end': 13897.352, 'text': 'In other words, f of 0 equals a.', 'start': 13894.79, 'duration': 2.562}, {'end': 13907.359, 'text': 'So if we think of starting out when x equals 0, we get the y value of a.', 'start': 13897.352, 'duration': 10.007}, {'end': 13909.18, 'text': "That's why it's called the initial value.", 'start': 13907.359, 'duration': 1.821}], 'summary': 'Initial value is a, when x equals 0.', 'duration': 27.101, 'max_score': 13882.079, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU13882079.jpg'}, {'end': 14079.763, 'src': 'embed', 'start': 14049.199, 'weight': 8, 'content': [{'end': 14053.302, 'text': 'The parameter B tells us how the graph is increasing or decreasing.', 'start': 14049.199, 'duration': 4.103}, {'end': 14058.246, 'text': 'Specifically, if B is greater than one, the graph is increasing.', 'start': 14053.663, 'duration': 4.583}, {'end': 14063.21, 'text': 'And if B is less than one, the graph is decreasing.', 'start': 14059.787, 'duration': 3.423}, {'end': 14068.734, 'text': 'The closer B is to the number one, the flatter the graph.', 'start': 14065.131, 'duration': 3.603}, {'end': 14079.763, 'text': 'So, for example, if I were to graph, y equals point two, five to the x and y equals 0.4 to the x.', 'start': 14070.135, 'duration': 9.628}], 'summary': 'Parameter b determines graph trend: b>1 increases, b<1 decreases, closer to 1 flattens.', 'duration': 30.564, 'max_score': 14049.199, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU14049199.jpg'}, {'end': 14438.633, 'src': 'embed', 'start': 14392.478, 'weight': 9, 'content': [{'end': 14407.447, 'text': 'In other words, your salary after two years is your original salary multiplied by the growth factor of 1.03 taken to the T power.', 'start': 14392.478, 'duration': 14.969}, {'end': 14417.013, 'text': 'Let me write this as a formula s of t, where s of t is your salary is equal to 40, 000 times 1.03 to the T.', 'start': 14409.008, 'duration': 8.005}, {'end': 14431.786, 'text': 'This is an exponential function, that is a function of the form A times B to the T, where your initial value A is 40, 000.', 'start': 14421.738, 'duration': 10.048}, {'end': 14438.633, 'text': 'And your base B is 1.03.', 'start': 14431.787, 'duration': 6.846}], 'summary': 'Salary growth formula: s(t) = 40,000 * 1.03^t', 'duration': 46.155, 'max_score': 14392.478, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU14392478.jpg'}], 'start': 11938.06, 'title': 'Function and graph transformations', 'summary': 'Covers rules for function transformations, including vertical and horizontal motions, adding, subtracting, multiplying, and dividing functions. it also discusses identifying, graphing, and applying quadratic functions, as well as polynomials, their graphs, and exponential functions with real-world applications.', 'chapters': [{'end': 12024.058, 'start': 11938.06, 'title': 'Function transformations and rules', 'summary': 'Discusses the rules for transformations of functions, including the relationship between vertical and horizontal motions and the effects of adding, subtracting, multiplying, dividing, and using negative signs, empowering the ability to sketch graphs of complex functions.', 'duration': 85.998, 'highlights': ['Knowing the basic rules about transformations empowers you to be able to sketch graphs of much more complicated functions', 'A vertical shrink by a factor of one half is the same as a horizontal stretch by a factor of four', 'Numbers on the outside correspond to changes in the y values or vertical motions, while numbers on the inside affect the x values and result in horizontal motions', 'Adding and subtracting correspond to translations or shifts, while multiplying and dividing by numbers corresponds to stretches and shrinks', 'Putting in a negative sign corresponds to a reflection, with a horizontal reflection for the negative sign on the inside and a vertical reflection for the negative sign on the outside']}, {'end': 12564.331, 'start': 12024.058, 'title': 'Identifying quadratic functions', 'summary': 'Discusses quadratic functions, identifying them in standard and vertex forms, and graphing them. it also explains the characteristics of parabolas, including their vertex, x-intercepts, and orientation, with examples demonstrating the application of these concepts.', 'duration': 540.273, 'highlights': ['Quadratic functions and their standard form', 'Identifying quadratic functions', 'Parabola characteristics']}, {'end': 13264.675, 'start': 12564.331, 'title': 'Graphing quadratic functions', 'summary': 'Explains how to graph quadratic functions, find the vertex using the vertex formula, convert quadratic functions from standard form to vertex form and vice versa, and justifies the vertex formula with examples, providing a clear understanding of the topic.', 'duration': 700.344, 'highlights': ['The parabola opens up if the coefficient of x squared is greater than zero, and down if a is less than zero.', 'The x coordinate of the vertex is given by negative b over 2a.', 'The vertex formula works even when there are no x intercepts, giving the vertex the x coordinate negative b over 2a.']}, {'end': 13719.89, 'start': 13266.556, 'title': 'Polynomials and graphs', 'summary': 'Explains the key concepts of polynomials and their graphs, including definitions of polynomial terms, the relationship between degree and turning points, end behavior based on degree and leading coefficient, and using turning points and end behavior to determine the degree and leading coefficient of a polynomial.', 'duration': 453.334, 'highlights': ['The end behavior of a function is influenced by whether the degree is even or odd and by whether the leading coefficient is positive or negative.', 'The number of turning points in a polynomial is always less than or equal to the degree minus one.', 'The degree and leading coefficient of a polynomial can be determined by analyzing its end behavior and the number of turning points on its graph.']}, {'end': 14656.786, 'start': 13721.131, 'title': 'Exponential functions and their applications', 'summary': 'Discusses the definition and properties of exponential functions, including their graphical representation, initial value, base, and applications in modeling real-world scenarios such as salary growth and population growth. it also covers the concept of exponential decay in the context of drug metabolism.', 'duration': 935.655, 'highlights': ['The number a in the expression f of x equals a times b to the x is called the initial value.', 'The parameter B tells us how the graph is increasing or decreasing. Specifically, if B is greater than one, the graph is increasing. And if B is less than one, the graph is decreasing.', 'Your salary after two years is your original salary multiplied by the growth factor of 1.03 squared.']}], 'duration': 2718.726, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU11938060.jpg', 'highlights': ['Knowing basic rules about transformations empowers to sketch graphs of more complicated functions', 'A vertical shrink by a factor of one half is the same as a horizontal stretch by a factor of four', 'Numbers on the outside correspond to changes in the y values or vertical motions, while numbers on the inside affect the x values and result in horizontal motions', 'Adding and subtracting correspond to translations or shifts, while multiplying and dividing by numbers corresponds to stretches and shrinks', 'Putting in a negative sign corresponds to a reflection, with a horizontal reflection for the negative sign on the inside and a vertical reflection for the negative sign on the outside', 'The parabola opens up if the coefficient of x squared is greater than zero, and down if a is less than zero', 'The number of turning points in a polynomial is always less than or equal to the degree minus one', 'The degree and leading coefficient of a polynomial can be determined by analyzing its end behavior and the number of turning points on its graph', 'The parameter B tells us how the graph is increasing or decreasing. Specifically, if B is greater than one, the graph is increasing. And if B is less than one, the graph is decreasing', 'Your salary after two years is your original salary multiplied by the growth factor of 1.03 squared', 'The x coordinate of the vertex is given by negative b over 2a', 'The vertex formula works even when there are no x intercepts, giving the vertex the x coordinate negative b over 2a', 'Quadratic functions and their standard form', 'Identifying quadratic functions', 'Parabola characteristics', 'The number a in the expression f of x equals a times b to the x is called the initial value', 'The end behavior of a function is influenced by whether the degree is even or odd and by whether the leading coefficient is positive or negative']}, {'end': 16157.813, 'segs': [{'end': 14753.522, 'src': 'embed', 'start': 14719.907, 'weight': 6, 'content': [{'end': 14730.313, 'text': "And in general, after t hours I'll have 400 multiplied by this growth or shrinkage factor of 0.89 raised to the t power.", 'start': 14719.907, 'duration': 10.406}, {'end': 14736.072, 'text': 'since each hour the amount of Seroquel gets multiplied by point eight nine.', 'start': 14730.313, 'duration': 5.759}, {'end': 14737.213, 'text': 'that number less than one.', 'start': 14736.072, 'duration': 1.141}, {'end': 14748.099, 'text': "I'll write my exponential decay function as f of t equals 400 times 0.89 to the t,", 'start': 14738.353, 'duration': 9.746}, {'end': 14753.522, 'text': 'where f of t represents the number of milligrams of Seroquel in the body.', 'start': 14748.099, 'duration': 5.423}], 'summary': 'Exponential decay model for seroquel dosage: f(t) = 400 * 0.89^t', 'duration': 33.615, 'max_score': 14719.907, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU14719907.jpg'}, {'end': 14959.662, 'src': 'embed', 'start': 14922.908, 'weight': 5, 'content': [{'end': 14925.93, 'text': 'An antique car is worth $50, 000 now, and its value increases by 7% each year.', 'start': 14922.908, 'duration': 3.022}, {'end': 14932.902, 'text': "Let's write an equation to model its value x years from now.", 'start': 14929.38, 'duration': 3.522}, {'end': 14939.005, 'text': 'After one year, its value is the 50, 000 plus 0.07 times the 50, 000.', 'start': 14934.383, 'duration': 4.622}, {'end': 14942.507, 'text': "That's because its value has grown by 7%, or 0.07 times 50, 000.", 'start': 14939.005, 'duration': 3.502}, {'end': 14959.662, 'text': 'This can be written as 50, 000 times 1 plus 0.07.', 'start': 14942.507, 'duration': 17.155}], 'summary': 'An antique car worth $50,000 increases by 7% annually.', 'duration': 36.754, 'max_score': 14922.908, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU14922908.jpg'}, {'end': 15277.998, 'src': 'embed', 'start': 15248.931, 'weight': 4, 'content': [{'end': 15256.114, 'text': 'We can see from the equation that the number of bacteria is increasing and not decreasing, because the base of the exponential function 1.45,', 'start': 15248.931, 'duration': 7.183}, {'end': 15260.109, 'text': 'is bigger than 1..', 'start': 15256.114, 'duration': 3.995}, {'end': 15269.674, 'text': 'Notice that our equation f of x equals 12 times 1.45 to the x, has the form of a times b to the x,', 'start': 15260.109, 'duration': 9.565}, {'end': 15274.736, 'text': 'or we can think of it as a times one plus r to the x.', 'start': 15269.674, 'duration': 5.062}, {'end': 15277.998, 'text': 'Here, a is 12, b is 1.45, and r is 0.45.', 'start': 15274.736, 'duration': 3.262}], 'summary': 'Bacteria is increasing exponentially with a base of 1.45, a=12, b=1.45, r=0.45.', 'duration': 29.067, 'max_score': 15248.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU15248931.jpg'}, {'end': 15507.563, 'src': 'embed', 'start': 15469.244, 'weight': 3, 'content': [{'end': 15486.308, 'text': 'For example, here, we have an initial value of 100 and a 15% increase.', 'start': 15469.244, 'duration': 17.064}, {'end': 15495.594, 'text': 'And here, we have an initial value of 50 and a 40% decrease.', 'start': 15487.729, 'duration': 7.865}, {'end': 15502.519, 'text': 'Exponential functions can be used to model compound interest for loans and bank accounts.', 'start': 15497.756, 'duration': 4.763}, {'end': 15507.563, 'text': 'Suppose you invest $200 in a bank account that earns 3% interest every year.', 'start': 15503.82, 'duration': 3.743}], 'summary': 'Exponential functions model financial changes, e.g. 15% increase and 40% decrease; $200 investment at 3% interest.', 'duration': 38.319, 'max_score': 15469.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU15469244.jpg'}, {'end': 15647.337, 'src': 'embed', 'start': 15610.058, 'weight': 2, 'content': [{'end': 15617.041, 'text': "But in the next few examples, we'll see what happens when the interest rate accumulates more frequently, twice a year, or every month, for example.", 'start': 15610.058, 'duration': 6.983}, {'end': 15626.085, 'text': "Let's deposit $300 in an account that earns 4.5% annual interest compounded semi-annually.", 'start': 15618.762, 'duration': 7.323}, {'end': 15629.587, 'text': 'This means two times a year, or every six months.', 'start': 15626.626, 'duration': 2.961}, {'end': 15647.337, 'text': "A 4.5% annual interest rate compounded two times a year means that we're actually getting 4.5 over 2% interest every time the interest is compounded.", 'start': 15631.986, 'duration': 15.351}], 'summary': 'Exploring the impact of different compounding frequencies on a $300 deposit with 4.5% annual interest.', 'duration': 37.279, 'max_score': 15610.058, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU15610058.jpg'}, {'end': 15916.295, 'src': 'embed', 'start': 15874.904, 'weight': 1, 'content': [{'end': 15881.33, 'text': 'Similarly, after three years or 36 months, your loan amount will be 1200 times 1.005 to the 36th power.', 'start': 15874.904, 'duration': 6.426}, {'end': 15891.057, 'text': "And in general, after t years, that's 12 t months.", 'start': 15886.374, 'duration': 4.683}, {'end': 15896.661, 'text': 'So the interest will be compounded 12 t times.', 'start': 15891.838, 'duration': 4.823}, {'end': 15901.625, 'text': 'And so we have to raise the 1.005 to the 12 t power.', 'start': 15897.322, 'duration': 4.303}, {'end': 15916.295, 'text': 'This gives us the general formula for the money owed is p of t equals 1200 times 1.005 to the 12 t, where t is the number of years.', 'start': 15902.525, 'duration': 13.77}], 'summary': 'Loan amount after 3 years: 1200 * 1.005^36', 'duration': 41.391, 'max_score': 15874.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU15874904.jpg'}, {'end': 16080.7, 'src': 'embed', 'start': 16036.813, 'weight': 0, 'content': [{'end': 16046.757, 'text': 'The formula for continuous compounding is P of t equals A times e to the RT, where P of t is the amount of money.', 'start': 16036.813, 'duration': 9.944}, {'end': 16050.638, 'text': 'T is the time in years.', 'start': 16048.037, 'duration': 2.601}, {'end': 16054.96, 'text': 'A is the initial amount of money.', 'start': 16052.839, 'duration': 2.121}, {'end': 16065.086, 'text': 'And R is the annual interest rate, written as a decimal.', 'start': 16054.98, 'duration': 10.106}, {'end': 16080.7, 'text': "So 0.025 in this problem from the 2.5% annual interest rate, he represents the famous constant Euler's constant, which is about 2.718.", 'start': 16065.726, 'duration': 14.974}], 'summary': 'Continuous compounding formula: p(t) = a * e^(rt), where p(t) is the amount of money, t is time, a is initial amount, and r is annual interest rate.', 'duration': 43.887, 'max_score': 16036.813, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU16036813.jpg'}], 'start': 14658.286, 'title': 'Exponential functions and applications', 'summary': 'Covers exponential decay and growth functions, with specific examples including medication dosage, car value depreciation, bacterial growth, population decrease, and compound interest modeling.', 'chapters': [{'end': 14921.788, 'start': 14658.286, 'title': 'Exponential functions and decay', 'summary': 'Explains the exponential decay function f(t) = 400 * 0.89^t, where t represents the number of hours since the dose was given, and how to find the growth factor b for modeling exponential decay, with a specific example of seroquel dosage resulting in 24.4 milligrams after 24 hours.', 'duration': 263.502, 'highlights': ['The exponential decay function f(t) = 400 * 0.89^t models the number of milligrams of Seroquel in the body after t hours, with 24.4 milligrams after 24 hours.', 'The process of finding the growth factor B involves converting the percent increase or decrease to a decimal and adding or subtracting it from one, depending on whether the quantity is increasing or decreasing.', 'The common form for exponential decay functions is f(t) = a * b^t, where a represents the initial amount and b represents the growth factor, with b being less than one for exponential decay.']}, {'end': 15322.57, 'start': 14922.908, 'title': 'Exponential growth and decay equations', 'summary': "Explains exponential growth and decay using the examples of an antique car's value increasing by 7% annually, a toyota prius's value decreasing by 5% annually, and the number of bacteria in a petri dish increasing by 45% every hour.", 'duration': 399.662, 'highlights': ["The antique car's value increases by 7% annually, modeled by the equation V(x) = 50,000 * 1.07^x.", "The Toyota Prius's value decreases by 5% annually, modeled by the equation V(x) = 3000 * 0.95^x.", 'The number of bacteria in the petri dish increases by 45% every hour, modeled by the equation f(x) = 12 * 1.45^x.']}, {'end': 16157.813, 'start': 15322.571, 'title': 'Exponential functions and compound interest', 'summary': 'Explains the modeling of exponential functions with an example of salamander population decrease and demonstrates the application of exponential functions in modeling compound interest for loans and bank accounts with specific examples, formulas, and calculations.', 'duration': 835.242, 'highlights': ['The exponential function for modeling salamander population decrease is explained with an initial value of 3000, a growth factor of 0.78, and a 22% decrease annually.', 'The application of exponential functions in modeling compound interest for a bank account with an initial investment of $200 at 3% interest annually is demonstrated, resulting in a total of $268.78 after 10 years.', 'The concept of compound interest compounded semi-annually is illustrated with an initial deposit of $300 at 4.5% interest, resulting in a total of $409.65 after 7 years.', 'The application of compound interest in a loan scenario with an initial amount of $1200 at 6% interest compounded monthly is showcased, resulting in a total of $1436.02 after 3 years.', 'The formula for compound interest compounded continuously is presented as P(t) = A * e^(RT), with a specific example resulting in $4532.59 after 5 years.']}], 'duration': 1499.527, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU14658286.jpg', 'highlights': ['The formula for compound interest compounded continuously is presented as P(t) = A * e^(RT), with a specific example resulting in $4532.59 after 5 years.', 'The application of compound interest in a loan scenario with an initial amount of $1200 at 6% interest compounded monthly is showcased, resulting in a total of $1436.02 after 3 years.', 'The concept of compound interest compounded semi-annually is illustrated with an initial deposit of $300 at 4.5% interest, resulting in a total of $409.65 after 7 years.', 'The application of exponential functions in modeling compound interest for a bank account with an initial investment of $200 at 3% interest annually is demonstrated, resulting in a total of $268.78 after 10 years.', 'The number of bacteria in the petri dish increases by 45% every hour, modeled by the equation f(x) = 12 * 1.45^x.', "The antique car's value increases by 7% annually, modeled by the equation V(x) = 50,000 * 1.07^x.", 'The exponential decay function f(t) = 400 * 0.89^t models the number of milligrams of Seroquel in the body after t hours, with 24.4 milligrams after 24 hours.']}, {'end': 17255.069, 'segs': [{'end': 16239.388, 'src': 'embed', 'start': 16199.207, 'weight': 0, 'content': [{'end': 16202.289, 'text': 'The number A is called the base of the logarithm.', 'start': 16199.207, 'duration': 3.082}, {'end': 16206.151, 'text': "It's also called the base when we write it in this exponential form.", 'start': 16202.709, 'duration': 3.442}, {'end': 16217.893, 'text': 'Some students find it helpful to remember this relationship log base A of B equals C means A to the C equals B by drawing arrows.', 'start': 16207.665, 'duration': 10.228}, {'end': 16222.137, 'text': 'A to the C equals B.', 'start': 16218.614, 'duration': 3.523}, {'end': 16224.839, 'text': 'Other students like to think of it in terms of asking a question.', 'start': 16222.137, 'duration': 2.702}, {'end': 16239.388, 'text': "Log base A of B asks, what power do you raise A to in order to get B? Let's look at some examples.", 'start': 16225.699, 'duration': 13.689}], 'summary': 'Logarithm base a is defined as a raised to the power of c equals b.', 'duration': 40.181, 'max_score': 16199.207, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU16199207.jpg'}, {'end': 16512.056, 'src': 'embed', 'start': 16431.7, 'weight': 1, 'content': [{'end': 16432.759, 'text': "We're never going to get 0.", 'start': 16431.7, 'duration': 1.059}, {'end': 16437.802, 'text': "Even if we raise 10 to the 0 power, we'll just get 1.", 'start': 16432.759, 'duration': 5.043}, {'end': 16441.562, 'text': "So there's no way to get 0, and the log base 10 of 0 does not exist.", 'start': 16437.802, 'duration': 3.76}, {'end': 16447.386, 'text': "If you try it on your calculator using the log base 10 button, you'll get an error message.", 'start': 16442.543, 'duration': 4.843}, {'end': 16451.71, 'text': 'Same thing happens when we do log base 10 of negative 100.', 'start': 16448.507, 'duration': 3.203}, {'end': 16456.613, 'text': "We're asking 10 to what power equals negative 100, and there's no exponent that will work.", 'start': 16451.71, 'duration': 4.903}, {'end': 16469.301, 'text': "And more generally, it's possible to take the log of numbers that are greater than 0, but not for numbers that are less than or equal to 0.", 'start': 16458.715, 'duration': 10.586}, {'end': 16478.469, 'text': "In other words, the domain of the function log base a of x, no matter what base you're using for a, the domain is going to be all positive numbers.", 'start': 16469.301, 'duration': 9.168}, {'end': 16481.671, 'text': 'A few notes on notation.', 'start': 16480.391, 'duration': 1.28}, {'end': 16494.346, 'text': "When you see ln of x, that's called natural log, and it means the log base e of x, where e is that famous number that's about 2.718.", 'start': 16482.393, 'duration': 11.953}, {'end': 16501.63, 'text': "When you see log of x with no base at all, by convention, that means log base 10 of x, and it's called the common log.", 'start': 16494.346, 'duration': 7.284}, {'end': 16507.674, 'text': 'Most scientific calculators have buttons for natural log and for common log.', 'start': 16503.451, 'duration': 4.223}, {'end': 16512.056, 'text': "Let's practice rewriting expressions with logs in them.", 'start': 16509.175, 'duration': 2.881}], 'summary': 'Logarithms have domain of all positive numbers, log base 10 of 0 and negative numbers are undefined.', 'duration': 80.356, 'max_score': 16431.7, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU16431700.jpg'}, {'end': 17092.274, 'src': 'embed', 'start': 17062.816, 'weight': 6, 'content': [{'end': 17068.318, 'text': "it'll still have the same vertical asymptote, since a vertical line shifted up by five units is still a vertical line.", 'start': 17062.816, 'duration': 5.502}, {'end': 17073.12, 'text': "But instead of going through one zero, it'll go through the point one five.", 'start': 17068.779, 'duration': 4.341}, {'end': 17076.322, 'text': "So I'll draw a rough sketch here.", 'start': 17074.181, 'duration': 2.141}, {'end': 17081.025, 'text': "Let's compare our starting function.", 'start': 17079.284, 'duration': 1.741}, {'end': 17084.488, 'text': 'y equals ln x and the transformed version.', 'start': 17081.025, 'duration': 3.463}, {'end': 17092.274, 'text': 'y equals ln x, plus five in terms of the domain, the range and the vertical asymptote.', 'start': 17084.488, 'duration': 7.786}], 'summary': 'Shifting y=ln x up by 5 units changes its zero and asymptote.', 'duration': 29.458, 'max_score': 17062.816, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17062816.jpg'}, {'end': 17168.828, 'src': 'embed', 'start': 17140.36, 'weight': 3, 'content': [{'end': 17147.942, 'text': 'And finally, we already saw that the original vertical asymptote of the y-axis x equals 0, when we shift that up by 5 units,', 'start': 17140.36, 'duration': 7.582}, {'end': 17152.375, 'text': "it's still the vertical line x equals 0..", 'start': 17147.942, 'duration': 4.433}, {'end': 17155.7, 'text': "In this next example, we're starting with a log base 10 function.", 'start': 17152.375, 'duration': 3.325}, {'end': 17162.249, 'text': 'And since the plus two is on the inside, that means we shift that graph left by two.', 'start': 17157.102, 'duration': 5.147}, {'end': 17166.048, 'text': "So I'll draw our basic log function.", 'start': 17163.567, 'duration': 2.481}, {'end': 17168.828, 'text': "Here's our basic log function.", 'start': 17166.968, 'duration': 1.86}], 'summary': 'Shifting the log function left by 2 units.', 'duration': 28.468, 'max_score': 17140.36, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17140360.jpg'}], 'start': 16159.815, 'title': 'Logarithms and log functions', 'summary': 'Introduces logarithms, including writing exponents as logarithms, examples of logarithmic calculations, and domain of logarithmic functions, along with discussing natural log and common log, converting between exponential equations and logs, and graphing log functions by hand, including transformations and specific examples like ln(x+5) and log(x+2).', 'chapters': [{'end': 16481.671, 'start': 16159.815, 'title': 'Introduction to logarithms', 'summary': 'Introduces logarithms, including the concept of writing exponents as logarithms, examples of logarithmic calculations, and the domain of logarithmic functions, emphasizing that logs of numbers greater than 0 exist but not for numbers less than or equal to 0.', 'duration': 321.856, 'highlights': ['Logarithms are a way of writing exponents, where log base A of B equals C means that A to the C equals B, providing a method for expressing exponents in a different form.', 'Examples of logarithmic calculations include log base 2 of 8 equals 3, log base 2 of 1 is 0, and log base 10 of 0 does not exist, demonstrating the application of logarithms in determining exponents and the non-existence of logarithms for certain numbers.', 'The domain of the function log base a of x is all positive numbers, indicating the restriction on the range of numbers for which logarithms exist, with logs not existing for numbers less than or equal to 0.']}, {'end': 16838.028, 'start': 16482.393, 'title': 'Understanding logarithms and log functions', 'summary': 'Discusses the concept of natural log and common log, rewriting expressions with logs, converting between exponential equations and logs, and graphing a log function by hand in the form y = log base 2 of x.', 'duration': 355.635, 'highlights': ['The chapter discusses the concept of natural log and common log, rewriting expressions with logs, converting between exponential equations and logs, and graphing a log function by hand in the form y = log base 2 of x.', 'Log base 3 of 1 9th is negative 2 can be rewritten as the expression 3 to the negative 2 equals 1 9th, and log of 13 can be rewritten as 10 to the 1.11394 equals 13.', 'The video introduced the idea of logs and the fact that log base A of B equals C means the same thing as A to the C equals B.', 'Graphing a log function by hand in the form y = log base 2 of x by plotting specific points and working with fractional values for x.']}, {'end': 17255.069, 'start': 16838.088, 'title': 'Graphing log functions', 'summary': 'Explains the graph of y = log base 2 of x, clarifies its domain, range, and vertical asymptote, and demonstrates the transformations of basic log functions, including shifting and scaling, with examples of ln(x+5) and log(x+2).', 'duration': 416.981, 'highlights': ['The domain of y = log base 2 of x is x values greater than zero, with a range of all real numbers and a vertical asymptote at the y axis (x=0).', 'The transformation y = ln(x+5) shifts the graph of y = ln(x) five units upwards, maintaining the original domain, range, and vertical asymptote.', 'The transformation y = log(x+2) shifts the graph of y = log(x) two units to the left, with the domain, range, and vertical asymptote remaining unaltered.']}], 'duration': 1095.254, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU16159815.jpg', 'highlights': ['Logarithms express exponents differently, e.g., log base A of B equals C means A to the C equals B.', 'Domain of log base a of x is all positive numbers, logs do not exist for numbers less than or equal to 0.', 'Examples: log base 2 of 8 equals 3, log base 2 of 1 is 0, log base 10 of 0 does not exist.', 'Graphing a log function by hand in the form y = log base 2 of x by plotting specific points.', 'Natural log and common log, converting between exponential equations and logs.', 'Domain of y = log base 2 of x is x values greater than zero, range of all real numbers.', 'Transformation y = ln(x+5) shifts the graph of y = ln(x) five units upwards.', 'Transformation y = log(x+2) shifts the graph of y = log(x) two units to the left.']}, {'end': 18586.17, 'segs': [{'end': 17437.261, 'src': 'embed', 'start': 17409.377, 'weight': 1, 'content': [{'end': 17412.38, 'text': 'For example, when we take 10 to the power of log base 10 of 1000,', 'start': 17409.377, 'duration': 3.003}, {'end': 17418.485, 'text': "the 10 to the power and the log base 10 undo each other and we're left with the 1000..", 'start': 17412.38, 'duration': 6.105}, {'end': 17425.691, 'text': 'This happens for any base a, a to the log base a of x is equal to x.', 'start': 17418.485, 'duration': 7.206}, {'end': 17437.261, 'text': 'We can describe this by saying that an exponential function and a log function with the same base undo each other.', 'start': 17425.691, 'duration': 11.57}], 'summary': 'Exponential and log functions with the same base undo each other', 'duration': 27.884, 'max_score': 17409.377, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17409377.jpg'}, {'end': 17754.017, 'src': 'embed', 'start': 17716.212, 'weight': 0, 'content': [{'end': 17720.517, 'text': "I'll make these base two to agree with my base that I'm using for my exponent rules.", 'start': 17716.212, 'duration': 4.305}, {'end': 17727.059, 'text': 'In words, this says the log of the product is the sum of the logs.', 'start': 17721.932, 'duration': 5.127}, {'end': 17730.764, 'text': 'Since logs really represent exponents.', 'start': 17728.12, 'duration': 2.644}, {'end': 17739.035, 'text': 'this is saying that when you multiply two numbers together, you add their exponents, which is just what we said for the exponent version.', 'start': 17730.764, 'duration': 8.271}, {'end': 17754.017, 'text': 'The quotient rule for exponents can be rewritten in terms of logs by saying the log of x divided by y is equal to the log of x minus the log of y.', 'start': 17740.612, 'duration': 13.405}], 'summary': 'Logarithm rules state that the log of a product is the sum of the logs, and the log of a quotient is the difference of the logs.', 'duration': 37.805, 'max_score': 17716.212, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17716212.jpg'}, {'end': 17805.46, 'src': 'embed', 'start': 17771.588, 'weight': 2, 'content': [{'end': 17774.129, 'text': "That's how we described the exponent rule above.", 'start': 17771.588, 'duration': 2.541}, {'end': 17788.193, 'text': 'Finally, the power rule for exponents can be rewritten in terms of logs by saying the log of x to the n is equal to n times log of x.', 'start': 17775.689, 'duration': 12.504}, {'end': 17797.657, 'text': 'Sometimes people describe this rule by saying when you take the log of an expression with an exponent, you can bring down the exponent and multiply.', 'start': 17790.615, 'duration': 7.042}, {'end': 17805.46, 'text': 'If we think of x as being some power of 2, this is really saying when we take a power to a power, we multiply their exponents.', 'start': 17798.678, 'duration': 6.782}], 'summary': 'The power rule for exponents can be rewritten in terms of logs, stating that log of x to the n is equal to n times log of x.', 'duration': 33.872, 'max_score': 17771.588, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17771588.jpg'}, {'end': 18155.976, 'src': 'embed', 'start': 18129.857, 'weight': 3, 'content': [{'end': 18135.883, 'text': 'Anytime you have an equation like this, one that has variables in the exponent.', 'start': 18129.857, 'duration': 6.026}, {'end': 18140.667, 'text': 'logs are the tool of choice for getting those variables down where you can solve for them.', 'start': 18135.883, 'duration': 4.784}, {'end': 18147.033, 'text': "In this video, I'll do a few examples of solving equations with variables in the exponents.", 'start': 18142.008, 'duration': 5.025}, {'end': 18155.976, 'text': "For our first example, let's solve for x the equation 5 times 2 to the x plus 1 equals 17.", 'start': 18148.291, 'duration': 7.685}], 'summary': 'Solving equations with variables in exponents using logs.', 'duration': 26.119, 'max_score': 18129.857, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU18129857.jpg'}], 'start': 17255.069, 'title': 'Log functions and rules', 'summary': 'Covers computing the domain of a log function, the relationship between log and exponential functions, log rules and their relation to exponent rules, and solving equations with logs, with examples resulting in solutions of x=0.765 and x=5.106.', 'chapters': [{'end': 17342.806, 'start': 17255.069, 'title': 'Log functions and domain', 'summary': 'Discusses how to compute the domain of a log function using algebra, where the domain for the given log function is x values from negative infinity to 2 thirds, not including 2 thirds, and highlights the importance of understanding the basic shape of the graph of a log function and the relationship between logs and exponents.', 'duration': 87.737, 'highlights': ['The domain for the given log function is x values from negative infinity to 2 thirds, not including 2 thirds.', 'Understanding the basic shape of the graph of a log function is essential.', 'The importance of the relationship between logs and exponents is emphasized.']}, {'end': 17603.299, 'start': 17344.147, 'title': 'Log and exponential functions', 'summary': 'Explains the relationship between log and exponential functions, demonstrating that they undo each other with the same base, and provides examples to illustrate these rules.', 'duration': 259.152, 'highlights': ['The log and the exponential function with the same base undo each other, as shown by log base 10 of 10 cubed equaling 3 and e to the log base e of 9.6 equaling 9.6.', 'The rules are explained by demonstrating that the log base a of a to the x equals x, and a to the log base a of x equals x for any base a.', 'Examples are provided to illustrate the rules, such as 3 to the log base 3 of 1.4 equaling 1.4, and ln of e to the x equaling x.', 'The statement ln of 10 to the x equaling x is refuted with an example, exhibiting that the bases must be the same for logs and exponents to undo each other.']}, {'end': 17818.705, 'start': 17604.66, 'title': 'Log rules and exponent rules', 'summary': 'Discusses the properties of log rules and their relation to exponent rules, including the product, quotient, and power rules, and provides examples with base 2.', 'duration': 214.045, 'highlights': ['The log rules are closely related to the exponent rules', 'Product rule for exponents: 2 to the m times 2 to the n is equal to 2 to the m plus n', 'Quotient rule for exponents: 2 to the m divided by 2 to the n is equal to 2 to the m minus n', 'Power rule for exponents: log of x to the n is equal to n times log of x']}, {'end': 18586.17, 'start': 17820.049, 'title': 'Log rules and solving equations', 'summary': 'Covers log rules including product, quotient, and power rules, and demonstrates solving equations with variables in the exponents using logs, with examples resulting in solutions of x=0.765 and x=5.106.', 'duration': 766.121, 'highlights': ['The log rules include product, quotient, and power rules, allowing the rewriting of logs as sums or differences and bringing down exponents to multiply.', 'Solving equations with variables in the exponents involves isolating the terms with exponents, taking the log of both sides, using log rules to bring down exponents, and then isolating the variable by factoring and dividing.', 'Examples result in solutions of x=0.765 and x=5.106.']}], 'duration': 1331.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU17255069.jpg', 'highlights': ['The importance of the relationship between logs and exponents is emphasized.', 'The log and the exponential function with the same base undo each other.', 'The log rules are closely related to the exponent rules.', 'Solving equations with variables in the exponents involves isolating the terms with exponents.']}, {'end': 19890.367, 'segs': [{'end': 18648.064, 'src': 'embed', 'start': 18591.28, 'weight': 0, 'content': [{'end': 18600.586, 'text': 'And now I can divide using my calculator, I can get a decimal answer of 2.0433.', 'start': 18591.28, 'duration': 9.306}, {'end': 18606.75, 'text': 'This video gave some examples of solving equations with variables in the exponent.', 'start': 18600.586, 'duration': 6.164}, {'end': 18616.677, 'text': 'And the key idea was to take the log of both sides and use the log properties to bring the exponents down.', 'start': 18607.411, 'duration': 9.266}, {'end': 18624.155, 'text': 'This video gives some examples of equations with logs in them, like this one.', 'start': 18620.029, 'duration': 4.126}, {'end': 18632.607, 'text': "In order to solve equations like this one, we have to free the variable from the log, and we'll do that using exponential functions.", 'start': 18625.457, 'duration': 7.15}, {'end': 18640.14, 'text': 'My first step in solving pretty much any kind of equation is to simplify it and isolate the tricky part.', 'start': 18634.457, 'duration': 5.683}, {'end': 18643.621, 'text': 'In this case, the tricky part is the part with the log in it.', 'start': 18640.88, 'duration': 2.741}, {'end': 18648.064, 'text': 'So I can isolate it by first adding 3 to both sides.', 'start': 18644.262, 'duration': 3.802}], 'summary': 'Solving equations with logs and exponents, yielding a decimal answer of 2.0433.', 'duration': 56.784, 'max_score': 18591.28, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU18591280.jpg'}, {'end': 18762.766, 'src': 'embed', 'start': 18735.592, 'weight': 4, 'content': [{'end': 18746.038, 'text': "There's one last step we need to do when solving equations with logs in them, and that's to check our answers because we may get extraneous solutions.", 'start': 18735.592, 'duration': 10.446}, {'end': 18753.883, 'text': "An extraneous solution is a solution that comes out of the solving process but doesn't actually satisfy the original equation.", 'start': 18746.719, 'duration': 7.164}, {'end': 18762.766, 'text': 'And those can happen for equations with logs in them, because we might get a solution that makes the argument of the log negative or zero,', 'start': 18754.543, 'duration': 8.223}], 'summary': "When solving equations with logs, it's important to check for extraneous solutions to ensure they satisfy the original equation.", 'duration': 27.174, 'max_score': 18735.592, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU18735592.jpg'}, {'end': 19102.243, 'src': 'embed', 'start': 19075.851, 'weight': 5, 'content': [{'end': 19080.213, 'text': 'Before I leave this problem, I do want to mention that some people have an alternative approach.', 'start': 19075.851, 'duration': 4.362}, {'end': 19088.737, 'text': 'Some people like to start with the original equation and then use log rules to combine everything into one log expression.', 'start': 19080.873, 'duration': 7.864}, {'end': 19094.959, 'text': "So, since we have the sum of two logs, we know that's the same as the log of a product, right?", 'start': 19089.877, 'duration': 5.082}, {'end': 19099.962, 'text': 'So we can rewrite the left side as log of x plus 3 times x.', 'start': 19094.999, 'duration': 4.963}, {'end': 19102.243, 'text': 'That equals 1.', 'start': 19100.822, 'duration': 1.421}], 'summary': 'Some people use log rules to combine equations into one log expression.', 'duration': 26.392, 'max_score': 19075.851, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19075851.jpg'}, {'end': 19345.333, 'src': 'embed', 'start': 19314.887, 'weight': 1, 'content': [{'end': 19325.049, 'text': "Now it's easy to isolate t just by dividing both sides by ln of 1.065.", 'start': 19314.887, 'duration': 10.162}, {'end': 19331.35, 'text': 'Typing that into my calculator, I get that t is approximately 3.54 years.', 'start': 19325.049, 'duration': 6.301}, {'end': 19340.072, 'text': 'In the next example, we have a population of bacteria that initially contains 1.5 million bacteria.', 'start': 19334.071, 'duration': 6.001}, {'end': 19342.793, 'text': "And it's growing by 12% per day.", 'start': 19340.572, 'duration': 2.221}, {'end': 19345.333, 'text': 'We want to find the doubling time.', 'start': 19343.933, 'duration': 1.4}], 'summary': 'Isolate t by dividing both sides by ln 1.065. t is approximately 3.54 years. a population of 1.5 million bacteria is growing by 12% per day. seeking the doubling time.', 'duration': 30.446, 'max_score': 19314.887, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19314887.jpg'}, {'end': 19682.132, 'src': 'embed', 'start': 19654.042, 'weight': 6, 'content': [{'end': 19657.125, 'text': 'But of course, the most accurate thing is just to leave B as it is.', 'start': 19654.042, 'duration': 3.083}, {'end': 19664.271, 'text': "And I'll do that and rewrite my equation is y equals 350 times two to the one 15th to the t.", 'start': 19657.605, 'duration': 6.666}, {'end': 19671.963, 'text': "Now, I'd like to work this problem one more time.", 'start': 19669.639, 'duration': 2.324}, {'end': 19678.814, 'text': "And this time, I'm going to use the form of the equation y equals a times e to the rt.", 'start': 19672.424, 'duration': 6.39}, {'end': 19682.132, 'text': 'This is called a continuous growth model.', 'start': 19679.69, 'duration': 2.442}], 'summary': 'The equation for continuous growth model is y = 350 * 2^(1/15)t.', 'duration': 28.09, 'max_score': 19654.042, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19654042.jpg'}, {'end': 19863.149, 'src': 'embed', 'start': 19833.535, 'weight': 7, 'content': [{'end': 19835.758, 'text': 'one a times e to the RT.', 'start': 19833.535, 'duration': 2.223}, {'end': 19840.629, 'text': "In this last example, we're going to work with half life.", 'start': 19837.287, 'duration': 3.342}, {'end': 19843.291, 'text': 'half life is pretty much like doubling time.', 'start': 19840.629, 'duration': 2.662}, {'end': 19853.417, 'text': 'it just means the amount of time that it takes for a quantity to decrease to half as much.', 'start': 19843.291, 'duration': 10.126}, {'end': 19857.687, 'text': 'as originally started with.', 'start': 19856.047, 'duration': 1.64}, {'end': 19863.149, 'text': "We're told that the half-life of radioactive carbon-14 is 5, 750 years.", 'start': 19858.968, 'duration': 4.181}], 'summary': 'Half-life of radioactive carbon-14 is 5,750 years.', 'duration': 29.614, 'max_score': 19833.535, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19833535.jpg'}], 'start': 18591.28, 'title': 'Logarithmic equations and exponential functions', 'summary': 'Covers solving equations with logarithms and logs, highlighting the process, importance of checking for extraneous solutions, and real-life applications, resulting in specific examples and leading to a real-life application involving exponential equations, with examples resulting in x=2.0433, and includes exploring exponential growth and decay, finding doubling time, working with half-life, and demonstrating how to solve for t and b and convert between different exponential equation forms.', 'chapters': [{'end': 18753.883, 'start': 18591.28, 'title': 'Solving equations with logs', 'summary': 'Demonstrated the process of solving equations with logs, such as isolating the variable, applying exponential functions to free the variable, and checking for extraneous solutions, with an example resulting in x=2.0433.', 'duration': 162.603, 'highlights': ['The video illustrated solving equations with logs and provided an example resulting in x=2.0433.', 'The key idea was to take the log of both sides and use the log properties to bring the exponents down.', 'The process included isolating the tricky part with log, using exponential functions to free the variable, and checking for extraneous solutions.']}, {'end': 18943.648, 'start': 18754.543, 'title': 'Solving logarithmic equations', 'summary': 'Discusses solving equations with logarithms, demonstrating the process with specific examples and emphasizing the importance of checking for extraneous solutions.', 'duration': 189.105, 'highlights': ['The process of solving equations with logarithms is demonstrated with specific examples.', 'Emphasis is placed on the importance of checking for extraneous solutions.']}, {'end': 19312.625, 'start': 18944.129, 'title': 'Solving quadratic equation with logs', 'summary': 'Covers the process of solving a quadratic equation involving logarithms using factoring, checking for extraneous solutions, and applying log rules to solve for the variable, with an alternative method to combine logs into one expression, leading to a real-life application involving exponential equations.', 'duration': 368.496, 'highlights': ['The process of solving a quadratic equation involving logarithms using factoring, checking for extraneous solutions, and applying log rules to solve for the variable.', 'The alternative method to combine logs into one expression, leading to a real-life application involving exponential equations.']}, {'end': 19890.367, 'start': 19314.887, 'title': 'Exponential growth and decay', 'summary': 'Explores exponential growth and decay, including finding doubling time and working with half-life, and demonstrates how to solve for t and b and convert between different exponential equation forms.', 'duration': 575.48, 'highlights': ['The chapter explains how to find doubling time in exponential growth using an equation for the amount of bacteria, demonstrating that it only depends on the growth rate, not the initial population, and showing the process of solving for the time it takes for the population to double, with an example showing the calculation to be approximately 6.12 days.', 'It then delves into solving for the growth factor in exponential growth using an equation y = a * b^t, demonstrating the process of solving for b and calculating it as approximately 1.047294, and then converting the equation to y = a * e^(rt) form and solving for r, showing the equivalence between the two forms of the exponential equation.', 'Lastly, it explores the concept of half-life, explaining that it means the amount of time that it takes for a quantity to decrease to half as much as originally started with, and provides an example of using carbon dating to find the age of a sample containing radioactive carbon-14 with a half-life of 5,750 years.']}], 'duration': 1299.087, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU18591280.jpg', 'highlights': ['The video illustrated solving equations with logs and provided an example resulting in x=2.0433.', 'The chapter explains how to find doubling time in exponential growth, with an example showing the calculation to be approximately 6.12 days.', 'The process of solving a quadratic equation involving logarithms using factoring, checking for extraneous solutions, and applying log rules to solve for the variable.', 'The process included isolating the tricky part with log, using exponential functions to free the variable, and checking for extraneous solutions.', 'Emphasis is placed on the importance of checking for extraneous solutions.', 'The alternative method to combine logs into one expression, leading to a real-life application involving exponential equations.', 'It then delves into solving for the growth factor in exponential growth using an equation y = a * b^t, demonstrating the process of solving for b and calculating it as approximately 1.047294.', 'Lastly, it explores the concept of half-life, explaining that it means the amount of time that it takes for a quantity to decrease to half as much as originally started with.']}, {'end': 21567.467, 'segs': [{'end': 19922.125, 'src': 'embed', 'start': 19891.287, 'weight': 0, 'content': [{'end': 19893.969, 'text': "Let's use the continuous growth model this time.", 'start': 19891.287, 'duration': 2.682}, {'end': 19896.89, 'text': 'So our final amount.', 'start': 19894.029, 'duration': 2.861}, {'end': 19901.532, 'text': 'so this is our amount of radioactive C.', 'start': 19896.89, 'duration': 4.642}, {'end': 19911.023, 'text': '14 is going to be the initial amount, times e to the RT.', 'start': 19901.532, 'duration': 9.491}, {'end': 19912.784, 'text': 'we could have used the other model too.', 'start': 19911.023, 'duration': 1.761}, {'end': 19917.485, 'text': 'we could have used f of t equals a times b to the t, but I just want to use a continuous model for practice.', 'start': 19912.784, 'duration': 4.701}, {'end': 19922.125, 'text': 'So we know that our half life is 5750.', 'start': 19918.365, 'duration': 3.76}], 'summary': 'Using continuous growth model for radioactive c; half life is 5750.', 'duration': 30.838, 'max_score': 19891.287, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19891287.jpg'}, {'end': 20860.491, 'src': 'embed', 'start': 20832.179, 'weight': 1, 'content': [{'end': 20840.067, 'text': "We've seen that systems of linear equations can have one solution when the lines that the equations represent intersect in one point.", 'start': 20832.179, 'duration': 7.888}, {'end': 20847.505, 'text': 'They can be inconsistent and have no solutions that corresponds to parallel lines,', 'start': 20841.462, 'duration': 6.043}, {'end': 20854.188, 'text': 'or they can be dependent and have infinitely many solutions that corresponds to the lines lying on top of each other.', 'start': 20847.505, 'duration': 6.683}, {'end': 20860.491, 'text': "In this video, I'll work through a problem involving distance rate and time.", 'start': 20856.769, 'duration': 3.722}], 'summary': 'Systems of linear equations can have one solution, be inconsistent, or be dependent. this video will cover a problem on distance, rate, and time.', 'duration': 28.312, 'max_score': 20832.179, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU20832179.jpg'}, {'end': 20926.473, 'src': 'embed', 'start': 20899.249, 'weight': 2, 'content': [{'end': 20909.438, 'text': "For example, if you normally walk at 3 miles per hour but you're walking on a moving sidewalk that's going at a rate of 2 miles per hour,", 'start': 20899.249, 'duration': 10.189}, {'end': 20918.366, 'text': 'then your total speed of travel with respect to something stationary is going to be 3 plus 2, or 5 miles per hour.', 'start': 20909.438, 'duration': 8.928}, {'end': 20926.473, 'text': "I'll write that as a formula as r1, or the first rate plus the second rate, is equal to the total rate.", 'start': 20920.348, 'duration': 6.125}], 'summary': 'Total speed on moving sidewalk: 3 + 2 = 5 mph', 'duration': 27.224, 'max_score': 20899.249, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU20899249.jpg'}, {'end': 21243.731, 'src': 'embed', 'start': 21217.915, 'weight': 3, 'content': [{'end': 21234.105, 'text': "how much household bleach should be combined with 70 liters of a weaker 1% hypochlorite solution in order to form a solution that's 2.5% sodium hypochlorite?", 'start': 21217.915, 'duration': 16.19}, {'end': 21238.908, 'text': 'I want to turn this problem into a system of equations.', 'start': 21236.326, 'duration': 2.582}, {'end': 21243.731, 'text': "So I'm asking myself what quantities are going to be equal to each other?", 'start': 21239.268, 'duration': 4.463}], 'summary': 'To form 2.5% sodium hypochlorite, how much household bleach to add to 70 liters of 1% solution?', 'duration': 25.816, 'max_score': 21217.915, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU21217915.jpg'}], 'start': 19891.287, 'title': 'Mathematical problem solving', 'summary': 'Covers topics such as continuous growth model for radioactive decay with an example yielding a decay time of approximately 13,351 years, solving systems of linear equations, distance-rate-time problems, and solving mixture problems resulting in a solution of 70 liters of the weaker solution combined with 30 liters of household bleach.', 'chapters': [{'end': 20141.247, 'start': 19891.287, 'title': 'Continuous growth model and half-life', 'summary': 'Introduces the continuous growth model for radioactive decay, demonstrating the calculation of the continuous growth rate and solving for time using natural logarithms, with an example yielding a decay time of approximately 13,351 years.', 'duration': 249.96, 'highlights': ['The continuous growth model for radioactive decay is introduced, with the calculation of the continuous growth rate using natural logarithms and the equation f(t) = a * e^(ln(1/2)/5750 * t).', 'Solving for the decay time using the continuous growth model yields approximately 13,351 years, aligning with the concept of half-life and demonstrating the application of the model in practical scenarios.', 'Introduction of the ideas of doubling time and half-life in an exponential growth model, providing context for understanding the decay process and its relationship to time and quantity changes.']}, {'end': 20854.188, 'start': 20142.608, 'title': 'Solving linear equations', 'summary': 'Discusses the concept of solving systems of linear equations, demonstrating two main methods of substitution and elimination, and illustrating different outcomes such as having one solution, no solution, or infinitely many solutions, through graphical and algebraic representations.', 'duration': 711.58, 'highlights': ['The chapter discusses the concept of solving systems of linear equations', 'Illustrating different outcomes such as having one solution, no solution, or infinitely many solutions', 'Demonstrating two main methods of substitution and elimination', 'Graphical and algebraic representations of the solutions']}, {'end': 21188.916, 'start': 20856.769, 'title': 'Distance rate and time problem', 'summary': 'Discusses the relationship between distance, rate, and time, utilizing the formula distance equals rate times time and showcasing the concept that rates add. it then demonstrates the application of these principles in solving a problem to find the rate of the river current, resulting in the discovery that the speed of the current is three miles per hour.', 'duration': 332.147, 'highlights': ['The key relationship to remember is that the rate of travel is the distance traveled divided by the time it takes to travel it, demonstrated by the formula r times t is equal to d, which states that distance is equal to rate times time.', 'The concept that rates add is illustrated using the example of walking on a moving sidewalk, where the total speed of travel with respect to something stationary is the sum of the individual speeds, such as 3 plus 2, resulting in a total speed of 5 miles per hour.', 'The problem of finding the rate of the river current is solved by organizing the information into a chart, charting the distance traveled, the rate of travel, and the time taken for both upstream and downstream travel, leading to the discovery that the speed of the current is three miles per hour.', 'The system of equations representing the distance, rate, and time problem is solved by isolating t in each equation, then equating the two expressions for t and solving for the variable r, resulting in the determination that the speed of the current is three miles per hour.', 'Solving for the speed of the current is accomplished by multiplying both sides of the equation by the least common denominator to clear the denominator, leading to the conclusion that the speed of the current is three miles per hour.']}, {'end': 21567.467, 'start': 21190.077, 'title': 'Mixture problem solution', 'summary': 'Discusses solving a mixture problem by creating a system of equations to determine the quantity of household bleach needed to form a 2.5% sodium hypochlorite solution by mixing it with a 1% hypochlorite solution, resulting in a solution of 70 liters of the weaker solution combined with 30 liters of household bleach.', 'duration': 377.39, 'highlights': ['The volume of household bleach needed to form a 2.5% sodium hypochlorite solution by mixing with a 1% hypochlorite solution is 30 liters.', 'The technique used to solve this equation can be applied to solve various other equations involving mixtures of items.', 'The information about the quantity of water before and after mixing is redundant and not necessary for solving the problem.']}], 'duration': 1676.18, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU19891287.jpg', 'highlights': ['The continuous growth model for radioactive decay yields a decay time of approximately 13,351 years', 'Solving systems of linear equations involves different outcomes such as one solution, no solution, or infinitely many solutions', 'The concept that rates add is illustrated using the example of walking on a moving sidewalk', 'The volume of household bleach needed to form a 2.5% sodium hypochlorite solution by mixing with a 1% hypochlorite solution is 30 liters']}, {'end': 22674.259, 'segs': [{'end': 21633.152, 'src': 'embed', 'start': 21603.595, 'weight': 0, 'content': [{'end': 21608.739, 'text': 'The simpler function f of x equals one over x is also considered a rational function.', 'start': 21603.595, 'duration': 5.144}, {'end': 21612.741, 'text': 'You can think of one and x as very simple polynomials.', 'start': 21609.239, 'duration': 3.502}, {'end': 21617.064, 'text': 'The graph of this rational function is shown here.', 'start': 21614.202, 'duration': 2.862}, {'end': 21621.048, 'text': 'this graph looks different from the graph of a polynomial.', 'start': 21618.427, 'duration': 2.621}, {'end': 21624.249, 'text': 'For one thing, its end behavior is different.', 'start': 21621.948, 'duration': 2.301}, {'end': 21633.152, 'text': 'The end behavior of a function is the way the graph looks when x goes through really large positive or really large negative numbers.', 'start': 21625.369, 'duration': 7.783}], 'summary': 'The function f(x) = 1/x is a rational function with unique end behavior.', 'duration': 29.557, 'max_score': 21603.595, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU21603595.jpg'}, {'end': 21726.705, 'src': 'embed', 'start': 21698.371, 'weight': 1, 'content': [{'end': 21704.198, 'text': 'And as we approach the x value of negative 5 from the left, our y values are going up towards positive infinity.', 'start': 21698.371, 'duration': 5.827}, {'end': 21712.328, 'text': 'We say that this graph has a vertical asymptote at x equals negative 5.', 'start': 21705.18, 'duration': 7.148}, {'end': 21716.674, 'text': 'A vertical asymptote is a vertical line that the graph gets closer and closer to.', 'start': 21712.328, 'duration': 4.346}, {'end': 21721.981, 'text': "Finally, there's something really weird going on at x equals 2.", 'start': 21717.998, 'duration': 3.983}, {'end': 21726.705, 'text': "There's a little open circle there, like the value at x equals 2 is dug out.", 'start': 21721.981, 'duration': 4.724}], 'summary': 'Graph has a vertical asymptote at x = -5 and an open circle at x = 2.', 'duration': 28.334, 'max_score': 21698.371, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU21698371.jpg'}, {'end': 22131.9, 'src': 'embed', 'start': 22106.405, 'weight': 5, 'content': [{'end': 22112.409, 'text': 'In the second case, where the degree of the numerator and the degree of the denominator are equal, things cancel out.', 'start': 22106.405, 'duration': 6.004}, {'end': 22120.313, 'text': "And so we get a horizontal asymptote at the y value that's equal to the ratio of the leading coefficients.", 'start': 22113.069, 'duration': 7.244}, {'end': 22128.218, 'text': 'Finally, in the third case, when the degree of the numerator is bigger than the degree of the denominator,', 'start': 22121.614, 'duration': 6.604}, {'end': 22131.9, 'text': 'then the numerator is getting really big compared to the denominator.', 'start': 22128.218, 'duration': 3.682}], 'summary': 'Degree equality results in horizontal asymptote at leading coefficient ratio, while numerator dominance leads to significant increase.', 'duration': 25.495, 'max_score': 22106.405, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22106405.jpg'}, {'end': 22422.526, 'src': 'embed', 'start': 22366.634, 'weight': 3, 'content': [{'end': 22372.84, 'text': 'We learned to find the vertical asymptotes and wholes by looking at the factored version of the functions.', 'start': 22366.634, 'duration': 6.206}, {'end': 22380.867, 'text': 'The wholes correspond to the x values that make the numerator and denominator zero, whose corresponding factors cancel.', 'start': 22373.8, 'duration': 7.067}, {'end': 22387.616, 'text': 'The vertical asymptotes correspond to the x values that make the denominator 0,', 'start': 22382.335, 'duration': 5.281}, {'end': 22392.317, 'text': 'even after factoring any common factors in the numerator and denominator.', 'start': 22387.616, 'duration': 4.701}, {'end': 22400.119, 'text': 'This video is about combining functions by adding them, subtracting them, multiplying them, and dividing them.', 'start': 22393.917, 'duration': 6.202}, {'end': 22406.72, 'text': 'Suppose we have two functions, f of x equals x plus 1, and g of x equals x squared.', 'start': 22401.519, 'duration': 5.201}, {'end': 22410.476, 'text': 'One way to combine them is by adding them together.', 'start': 22408.254, 'duration': 2.222}, {'end': 22422.526, 'text': 'This notation f plus g of x means the function defined by taking f of x and adding it to g of x.', 'start': 22411.337, 'duration': 11.189}], 'summary': 'Learned to find vertical asymptotes, wholes, and combine functions by adding, subtracting, multiplying, and dividing.', 'duration': 55.892, 'max_score': 22366.634, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22366634.jpg'}], 'start': 21569.068, 'title': 'Rational functions and their graphs', 'summary': 'Discusses rational functions, their graphs, and asymptotes, with a focus on horizontal and vertical asymptotes. it explains finding asymptotes and combining functions, demonstrating with examples and practical applications.', 'chapters': [{'end': 21745.98, 'start': 21569.068, 'title': 'Rational functions and their graphs', 'summary': 'Discusses rational functions, with a focus on their graphs, including the concept of horizontal and vertical asymptotes and holes, revealing how the behavior of the graph can be predicted by examining the equation.', 'duration': 176.912, 'highlights': ['Rational functions are functions that can be written as a ratio or quotient of two polynomials.', 'The graph of the rational function f(x) = 1/x has a horizontal asymptote at y = 3 and a vertical asymptote at x = -5, with a hole at x = 2.', 'The end behavior of the rational function f(x) = 1/x is different from that of a polynomial, as it levels off at y = 3 for large x values.']}, {'end': 21972.102, 'start': 21747.202, 'title': 'Horizontal and vertical asymptotes', 'summary': 'Explains how to find horizontal and vertical asymptotes by analyzing the highest exponent terms in the function, resulting in a horizontal asymptote at y=3 and vertical asymptotes at x=-5 and x=2, with a hole at (2, 4/3).', 'duration': 224.9, 'highlights': ["The function's y values are approximately 3x squared over x squared, resulting in a horizontal asymptote at y=3.", 'The vertical asymptotes occur where the denominator of the function is 0, at x=-5 and x=2.', 'The function has a hole at x=2 with a y value of 4 thirds.']}, {'end': 22674.259, 'start': 21973.843, 'title': 'Finding asymptotes and combining functions', 'summary': 'Explains how to find vertical and horizontal asymptotes of rational functions and demonstrates the process through three examples. it also discusses combining functions by adding, subtracting, multiplying, and dividing them and provides practical applications of these operations.', 'duration': 700.416, 'highlights': ['The process of finding vertical asymptotes and holes involves identifying where the denominator is zero, while horizontal asymptotes are determined by considering the highest power terms in the numerator and denominator.', 'In the first example, 5x over 3x squared simplifies to 5 over 3x, resulting in a horizontal asymptote at y equals 0, demonstrating the concept of horizontal asymptotes through numerical evaluation.', 'The concept of horizontal asymptotes is further illustrated in the second example, where the highest power terms simplify to 2 thirds, leading to a horizontal asymptote at y equals 2 thirds as x approaches infinity.', 'The third example showcases a scenario where no horizontal asymptote exists, as the highest power terms result in x over 2, which grows without bound, demonstrating cases where the degree of the numerator is larger than the degree of the denominator.', 'The relationship between the degrees of the numerator and denominator determines the presence or absence of horizontal asymptotes, with smaller degree leading to the numerator approaching zero, equal degrees resulting in cancellation, and larger degree leading to the numerator growing without bound.', 'The process of finding vertical asymptotes and wholes involves factoring the numerator and denominator, with common factors indicating the presence of a hole instead of a vertical asymptote, exemplified through a detailed example.', 'The practical application of horizontal asymptotes, vertical asymptotes, and holes is demonstrated through the analysis and graphing of a rational function, providing insights into the behavior of the function at different points.', 'The video also delves into combining functions by adding, subtracting, multiplying, and dividing them, showcasing examples and practical applications of these operations in graphical and numerical contexts.']}], 'duration': 1105.191, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU21569068.jpg', 'highlights': ['Rational functions are functions that can be written as a ratio or quotient of two polynomials.', 'The graph of the rational function f(x) = 1/x has a horizontal asymptote at y = 3 and a vertical asymptote at x = -5, with a hole at x = 2.', 'The end behavior of the rational function f(x) = 1/x is different from that of a polynomial, as it levels off at y = 3 for large x values.', 'The process of finding vertical asymptotes and holes involves identifying where the denominator is zero, while horizontal asymptotes are determined by considering the highest power terms in the numerator and denominator.', 'The practical application of horizontal asymptotes, vertical asymptotes, and holes is demonstrated through the analysis and graphing of a rational function, providing insights into the behavior of the function at different points.', 'The relationship between the degrees of the numerator and denominator determines the presence or absence of horizontal asymptotes, with smaller degree leading to the numerator approaching zero, equal degrees resulting in cancellation, and larger degree leading to the numerator growing without bound.', 'The video also delves into combining functions by adding, subtracting, multiplying, and dividing them, showcasing examples and practical applications of these operations in graphical and numerical contexts.']}, {'end': 24225.197, 'segs': [{'end': 22735.733, 'src': 'embed', 'start': 22705.596, 'weight': 9, 'content': [{'end': 22713.783, 'text': 'we can think of it schematically in this diagram f acts on a number x and produces a number f of x.', 'start': 22705.596, 'duration': 8.187}, {'end': 22722.827, 'text': 'then G takes that output, f of x and produces a new number g of f of x.', 'start': 22713.783, 'duration': 9.044}, {'end': 22731.311, 'text': 'Our composition of functions g composed with f is the function that goes all the way from x to g of f of x.', 'start': 22722.827, 'duration': 8.484}, {'end': 22735.733, 'text': "Let's work out some examples where our functions are defined by tables of values.", 'start': 22731.311, 'duration': 4.422}], 'summary': 'Composition of functions explained using a diagram and examples.', 'duration': 30.137, 'max_score': 22705.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22705596.jpg'}, {'end': 22806.821, 'src': 'embed', 'start': 22773.168, 'weight': 6, 'content': [{'end': 22777.249, 'text': '7 becomes our new x value in our table of values for g.', 'start': 22773.168, 'duration': 4.081}, {'end': 22781.39, 'text': 'The x value of 7 corresponds to the g of x value of 10.', 'start': 22777.249, 'duration': 4.141}, {'end': 22786.465, 'text': 'So g of 7 is equal to 10.', 'start': 22781.39, 'duration': 5.075}, {'end': 22791.309, 'text': 'we found that g composed with f of four is equal to 10.', 'start': 22786.465, 'duration': 4.844}, {'end': 22799.715, 'text': 'If instead, we want to find f composed with g of four, well, we can rewrite that as f of g of four.', 'start': 22791.309, 'duration': 8.406}, {'end': 22802.257, 'text': 'And again, work from the inside out.', 'start': 22800.396, 'duration': 1.861}, {'end': 22805.52, 'text': "Now we're trying to find g of four.", 'start': 22803.118, 'duration': 2.402}, {'end': 22806.821, 'text': 'So four is our x value.', 'start': 22805.54, 'duration': 1.281}], 'summary': 'G(7) = 10, g(f(4)) = 10, f(g(4)) = f(g(4))', 'duration': 33.653, 'max_score': 22773.168, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22773168.jpg'}, {'end': 22970.419, 'src': 'embed', 'start': 22876.49, 'weight': 3, 'content': [{'end': 22884.331, 'text': 'But f of eight, eight is not on the table as an x value for the for the f function.', 'start': 22876.49, 'duration': 7.841}, {'end': 22887.852, 'text': 'And so there is no f of eight, this does not exist.', 'start': 22884.711, 'duration': 3.141}, {'end': 22898.104, 'text': 'we can say that six is not in the domain for F composed with G.', 'start': 22889.178, 'duration': 8.926}, {'end': 22906.93, 'text': "Even though it was in the domain of G, we couldn't follow all the way through and get a value for F composed with G of six.", 'start': 22898.104, 'duration': 8.826}, {'end': 22913.185, 'text': "Next, let's turn our attention to the composition of functions that are given by equations.", 'start': 22908.522, 'duration': 4.663}, {'end': 22921.751, 'text': 'p of x is x squared plus x and q of x is negative two x, we want to find q composed with p of one.', 'start': 22914.326, 'duration': 7.425}, {'end': 22929.175, 'text': 'As usual, I can rewrite this as q of p of one and work from the inside out.', 'start': 22923.051, 'duration': 6.124}, {'end': 22936.238, 'text': "p of one is one squared plus one, so that's two.", 'start': 22930.736, 'duration': 5.502}, {'end': 22939.76, 'text': 'So this is the same thing as q of two.', 'start': 22937.159, 'duration': 2.601}, {'end': 22945.142, 'text': 'But q of two is negative two times two or negative four.', 'start': 22940.8, 'duration': 4.342}, {'end': 22947.724, 'text': 'So this evaluates to negative four.', 'start': 22945.723, 'duration': 2.001}, {'end': 22954.727, 'text': 'In this next example, we want to find q composed of p of some arbitrary x.', 'start': 22948.724, 'duration': 6.003}, {'end': 22960.13, 'text': "I'll rewrite it as usual as q of p of x and work from the inside out.", 'start': 22954.727, 'duration': 5.403}, {'end': 22967.056, 'text': "Well, p we know the formula for that, that's x squared plus x.", 'start': 22961.11, 'duration': 5.946}, {'end': 22970.419, 'text': 'So I can replace my p with that expression.', 'start': 22967.056, 'duration': 3.363}], 'summary': 'F of 8 does not exist; q composed with p of 1 equals -4.', 'duration': 93.929, 'max_score': 22876.49, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22876490.jpg'}, {'end': 23123.215, 'src': 'embed', 'start': 23088.866, 'weight': 5, 'content': [{'end': 23096.691, 'text': 'Once again, we see that q composed with p is not necessarily equal to p composed with q.', 'start': 23088.866, 'duration': 7.825}, {'end': 23099.834, 'text': 'Please pause the video and try this last example yourself.', 'start': 23096.691, 'duration': 3.143}, {'end': 23112.145, 'text': 'And working from the inside out, we can replace P of x with its expression x squared plus x.', 'start': 23105.359, 'duration': 6.786}, {'end': 23116.709, 'text': 'And then we need to evaluate P on x squared plus x.', 'start': 23112.145, 'duration': 4.564}, {'end': 23123.215, 'text': 'That means we plug in x squared plus x everywhere we see an x in this formula.', 'start': 23116.709, 'duration': 6.506}], 'summary': 'Composition of functions not always commutative. evaluate p(x) = x^2 + x.', 'duration': 34.349, 'max_score': 23088.866, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU23088866.jpg'}, {'end': 23376.382, 'src': 'embed', 'start': 23341.944, 'weight': 11, 'content': [{'end': 23356.196, 'text': 'We also learned to break apart a complicated function into a composition of two functions by boxing one piece of the function and letting the first function applied in the composition.', 'start': 23341.944, 'duration': 14.252}, {'end': 23362.681, 'text': 'Let that be the inside of the box and the second function applied in the composition be whatever happens to the box.', 'start': 23356.856, 'duration': 5.825}, {'end': 23376.382, 'text': 'the inverse of a function undoes what the function does.', 'start': 23373.479, 'duration': 2.903}], 'summary': 'Learned to break apart functions, apply compositions, and understand function inverses.', 'duration': 34.438, 'max_score': 23341.944, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU23341944.jpg'}, {'end': 23536.671, 'src': 'embed', 'start': 23479.285, 'weight': 0, 'content': [{'end': 23481.887, 'text': 'Inverse functions reverse the roles of y and x.', 'start': 23479.285, 'duration': 2.602}, {'end': 23486.962, 'text': "I'm going to plot the points for y equals f of x in blue.", 'start': 23483.7, 'duration': 3.262}, {'end': 23492.945, 'text': "Next, I'll plot the points for y equals f inverse of x in red.", 'start': 23488.743, 'duration': 4.202}, {'end': 23497.948, 'text': 'Pause the video for a moment and see what kind of symmetry you observe in this graph.', 'start': 23494.066, 'duration': 3.882}, {'end': 23500.73, 'text': 'How are the blue points related to the red points?', 'start': 23498.429, 'duration': 2.301}, {'end': 23506.633, 'text': 'You might have noticed that the blue points and the red points are mirror images.', 'start': 23502.071, 'duration': 4.562}, {'end': 23508.975, 'text': 'over the mirror line y equals x.', 'start': 23506.633, 'duration': 2.342}, {'end': 23515.925, 'text': 'So our second key fact is that the graph of y equals f, inverse of x,', 'start': 23510.523, 'duration': 5.402}, {'end': 23523.269, 'text': 'can be obtained from the graph of y equals f of x by reflecting over the line y equals x.', 'start': 23515.925, 'duration': 7.344}, {'end': 23527.851, 'text': 'This makes sense because inverses reverse the roles of y and x.', 'start': 23523.269, 'duration': 4.582}, {'end': 23534.55, 'text': "In this same example, let's compute f inverse of f of 2.", 'start': 23529.727, 'duration': 4.823}, {'end': 23536.671, 'text': 'This open circle means composition.', 'start': 23534.55, 'duration': 2.121}], 'summary': 'Inverse functions mirror each other over y=x line.', 'duration': 57.386, 'max_score': 23479.285, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU23479285.jpg'}, {'end': 24166.503, 'src': 'embed', 'start': 24132.455, 'weight': 10, 'content': [{'end': 24142.36, 'text': 'The range of f inverse is the y values for f inverse which correspond to the x values or the domain of f.', 'start': 24132.455, 'duration': 9.905}, {'end': 24146.382, 'text': 'In this video, we discussed five key properties of inverse functions.', 'start': 24142.36, 'duration': 4.022}, {'end': 24153.606, 'text': 'Inverse functions reverse the roles of y and x.', 'start': 24148.303, 'duration': 5.303}, {'end': 24161.41, 'text': 'The graph of y equals f inverse of x is the graph of y equals f of x.', 'start': 24153.606, 'duration': 7.804}, {'end': 24166.503, 'text': 'reflected over the line y equals x.', 'start': 24162.662, 'duration': 3.841}], 'summary': 'Inverse functions reverse roles of y and x, with 5 key properties discussed.', 'duration': 34.048, 'max_score': 24132.455, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU24132455.jpg'}, {'end': 24225.197, 'src': 'embed', 'start': 24210.945, 'weight': 1, 'content': [{'end': 24215.289, 'text': 'And the range of f is the domain of f inverse.', 'start': 24210.945, 'duration': 4.344}, {'end': 24225.197, 'text': 'These properties of inverse functions will be important when we study exponential functions and their inverses, logarithmic functions.', 'start': 24216.718, 'duration': 8.479}], 'summary': 'The range of f is the domain of f inverse, important for studying exponential and logarithmic functions.', 'duration': 14.252, 'max_score': 24210.945, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU24210945.jpg'}], 'start': 22675.801, 'title': 'Functions composition and inverse', 'summary': 'Delves into the composition of functions, demonstrating non-commutativity, and explores the properties and graphical representation of inverse functions, highlighting their relationship with original functions. it discusses finding the inverse function, its properties, and the graphical representation.', 'chapters': [{'end': 22906.93, 'start': 22675.801, 'title': 'Composition of functions', 'summary': 'Explains the composition of functions, where g composed with f is not the same as f composed with g, illustrated through examples and tables of values.', 'duration': 231.129, 'highlights': ['G composed with F is not the same thing as F composed with G.', 'G composed with f of four is equal to 10.', 'F composed with G of six does not exist as six is not in the domain for F composed with G.']}, {'end': 23149.907, 'start': 22908.522, 'title': 'Composition of functions', 'summary': 'Explains the composition of functions using equations, demonstrating how to find q composed with p of one, q composed with p of some arbitrary x, and p composed with q of x, highlighting the distinction between q composed with p and p composed with q.', 'duration': 241.385, 'highlights': ['The chapter explains the composition of functions using equations, demonstrating how to find q composed with p of one, q composed with p of some arbitrary x, and p composed with q of x.', 'q composed with p of one evaluates to negative four, and q composed with p of x is -2x squared - 2x.', 'The distinction between q composed with p and p composed with q is emphasized, showcasing that q composed with p is not necessarily equal to p composed with q.']}, {'end': 23965.528, 'start': 23150.711, 'title': 'Composition and inverse functions', 'summary': 'Explores the concept of composing functions, breaking down a complex function into a composition of two functions, and the properties and graphical representation of inverse functions, demonstrating their symmetry and the relationship between a function and its inverse.', 'duration': 814.817, 'highlights': ['The chapter explores the concept of composing functions', 'The properties and graphical representation of inverse functions are demonstrated', 'The chapter discusses breaking down a complex function into a composition of two functions']}, {'end': 24225.197, 'start': 23967.029, 'title': 'Inverse functions and their properties', 'summary': 'Discusses finding the inverse function and its properties, such as domain, range, and the relationship between the original function and its inverse. it also highlights the key properties of inverse functions, including their graphical representation and the relationship between their domains and ranges.', 'duration': 258.168, 'highlights': ['The graph of y equals f inverse of x is the graph of y equals f of x reflected over the line y equals x.', 'The domain of f is the range of f inverse, and the range of f is the domain of f inverse.', 'Inverse functions reverse the roles of y and x.']}], 'duration': 1549.396, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LwCRRUa8yTU/pics/LwCRRUa8yTU22675801.jpg', 'highlights': ['The graph of y equals f inverse of x is the graph of y equals f of x reflected over the line y equals x.', 'The domain of f is the range of f inverse, and the range of f is the domain of f inverse.', 'Inverse functions reverse the roles of y and x.', 'The chapter explains the composition of functions using equations, demonstrating how to find q composed with p of one, q composed with p of some arbitrary x, and p composed with q of x.', 'q composed with p of one evaluates to negative four, and q composed with p of x is -2x squared - 2x.', 'The distinction between q composed with p and p composed with q is emphasized, showcasing that q composed with p is not necessarily equal to p composed with q.', 'G composed with f of four is equal to 10.', 'G composed with F is not the same thing as F composed with G.', 'F composed with G of six does not exist as six is not in the domain for F composed with G.', 'The chapter explores the concept of composing functions', 'The properties and graphical representation of inverse functions are demonstrated', 'The chapter discusses breaking down a complex function into a composition of two functions']}], 'highlights': ['The formula for compound interest compounded continuously is presented as P(t) = A * e^(RT), with a specific example resulting in $4532.59 after 5 years.', 'The process of solving a quadratic equation involving logarithms using factoring, checking for extraneous solutions, and applying log rules to solve for the variable.', 'The graph of the rational function f(x) = 1/x has a horizontal asymptote at y = 3 and a vertical asymptote at x = -5, with a hole at x = 2.', 'The product rule states x^n * x^m = x^(n+m), demonstrated by 2^3 * 2^4 = 2^7', 'The importance of considering plus or minus solutions for even roots or even number powers is emphasized, exemplified by the equation x squared equals four and the cube root of negative 8, highlighting the need to include both positive and negative answers.']}