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Chapter 7 Exponential and Logarithmic Functions

Chapter 7 Exponential and Logarithmic Functions

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Chapter 7 Exponential and Logarithmic Functions

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  1. Chapter 7 Exponential and Logarithmic Functions Algebra II

  2. Table of Contents • 7.1 – Exponential Functions, Growth, and Decay • 7.2- Inverses of Relations and Functions • 7.3 – Logarithmic Functions • 7.4- Properties of Logarithms • 7.5 – Exponential and Logarithmic Equations and Inequalities • 7.6 – The Natural Base, e • 7.7 – Transforming Exponential and Logarithmic Functions

  3. 7-1 Algebra II (Bell work) Define Exponential Function, base, asymptote, exponential growth, exponential decay Book Extra Credit (Multi Step Test Prep, Due Tomorrow) Test Corrections (Due Wednesday) Solve For Skips (At your table) Evaluate. 1. 100(1.08)20 2. 100(0.95)25 3.100(1 – 0.02)10 4. 100(1 + 0.08)–10 ≈ 466.1 ≈ 27.74 ≈ 81.71 ≈ 46.32

  4. 7.1 Exponential Functions, Growth, and Decay Algebra II

  5. 7-1 Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = bx, where the baseb is a constant and the exponent x is the independent variable.

  6. 7-1 The graph of the parent function f(x) = 2xis shown. The domain is all real numbers and the range is {y|y > 0}.

  7. 7-1 Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2xcannot be zero. In this case, the x-axis is an asymptote. An asymptoteis a line that a graphed function approaches as the value of x gets very large or very small. as·ymp·tote B > 0, exponential growth 0 < b < 1,exponential decay

  8. 7-1 Math Joke • Q: How did you know that your dentist studied algebra? • A: She said all that candy gave me exponential decay.

  9. 7-1 The base , ,is less than 1. This is an exponential decay function. Example 1A: Graphing Exponential Functions Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base.

  10. Example 1A Continued 7-1 Step 2 Graph the function by using a table of values.

  11. 7-1 Example 1B: Graphing Exponential Functions Tell whether the function shows growth or decay. Then graph. g(x) = 100(1.05)x Step 1 Find the value of the base. The base, 1.05, is greater than 1. This is an exponential growth function. g(x) = 100(1.05)x

  12. 7-1 Check It Out! Example 1 Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph. Step 1 Find the value of the base. p(x) = 5(1.2x) The base , 1.2, is greater than 1. This is an exponential growth function.

  13. Check It Out! Example 1 Continued 7-1 Step 2 Graph the function by using a table of values.

  14. 7-1 Algebra II (Bell work) Copy down formula below Questions 7-1 Day 1? Read student to student pg. 491 You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r,is called the growth factor. Similarly, 1 – ris the decay factor.

  15. Example 2: Economics Application 7-1 Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000? Step 1 Write a function to model the growth in value of her investment. f(t) = a(1 + r)t Exponential growth function. f(t) = 5000(1 + 0.0625)t Substitute 5000 for a and 0.0625 for r. f(t) = 5000(1.0625)t Simplify.

  16. 7-1 Example 2 Continued Step 2 When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points of the function. Step 3 Use the graph to predict when the value of the investment will reach $10,000. Use the feature to find the t-value where f(t) ≈ 10,000. ** Can also use 2nd , Table to find it**

  17. 7-1 Example 2 Continued Table Feature The function value is approximately 10,000 when t ≈ 11.43 The investment will be worth $10,000 about 11.43 years after it was purchased.

  18. Example 3: Depreciation Application 7-1 A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8000. f(t) = a(1 – r)t Exponential decay function. f(t) = 15,500(1 – 0.03)t Substitute 15,500 for a and 0.03 for r. f(t) = 15,500(0.97)t Simplify.

  19. Example 3 Continued 7-1 Graph the function. Use to find when the population will fall below 8000. It will take about 22 years for the population to fall below 8000.

  20. 7-1 Check It Out! Example 2 In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000. P(t) = a(1 + r)t Exponential growth function. P(t) = 350(1 + 0.14)t Substitute 350 for a and 0.14 for r. P(t) = 350(1.14)t Simplify.

  21. 7-1 Graph the function. Use to find when the population will reach 20,000. Check It Out! Example 2 Continued It will take about 31 years for the population to reach 20,000.

  22. Check It Out! Example 3 7-1 A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100. f(t) = a(1 – r)t Exponential decay function. f(t) = 1000(1 – 0.15)t Substitute 1,000 for a and 0.15 for r. f(t) = 1000(0.85)t Simplify.

  23. 7-1 Graph the function. Use to find when the value will fall below 100. Check It Out! Example 3 Continued It will take about 14.2 years for the value to fall below 100.

  24. 7-1 HW • Day 1: 1-4, 7-9, 12-14, 47-49 • Minimum 5 points (Show on a table) • Graph enough so that your on both sides of the y axis • Choose whatever scale you want, large graphs can be roughly sketched • Round to 2 decimal places • Pg. 472 Multi Step Test Prep (Due Tuesday) • Day 2: 5, 6, 11, 15, 18, 19, 21, Bonus 20 • (Due End of Class Wednesday)

  25. Pre Algebra (Bell work) • Grab a study guide off of the side table • Work on your review assignment/cheat sheet • That extra credit page is due today • Test 8.1-8.5 Tomorrow • Wednesday Drink/Drive • Thursday (SDAP), bring a book or other HW To class

  26. 7-2 y = y + 5 x + 7 8 3 y = ± x Algebra II (Bell work) Copy down the following vocabulary To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation. Solve For Skips Solve for y. 1.x = 3y –7 2.x = y = 8x – 5 y = 4 – x 3.x = 4 – y 4.x = y2

  27. 7.2 Inverses of Relations and Functions Algebra II

  28. 7-2 Example 1: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. ● ● ● ● Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair.

  29. Example 1 Continued 7-2 • Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • • • • Domain:{x|0 ≤ x ≤ 8} Range :{y|2 ≤ x ≤ 9} • • Domain:{x|2 ≤ x ≤ 9} Range :{y|0 ≤ x ≤ 8} Blue Line Red Line

  30. 7-2 Math Joke • Q: How did the chicken find the inverse? • A: It reflected the function across y = eggs

  31. 7-2 When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal. Functions that undo each other are inverse functions. To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).

  32. 7-2 1 2 Example 2: Writing Inverses of by Using Inverse Functions Use inverse operations to write the inverse of f(x) = x – if possible.

  33. Example 2 Continued 7-2 1 1 1 = 2 2 1 1 2 1 1 1 f(x) = x – f(1) = 1 – 2 2 2 2 2 f–1(x) = x + Substitute for x. f–1( ) = + The inverse function does undo the original function.  Check Use the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1

  34. 7-2 x 3 Check It Out! Example 2a Use inverse operations to write the inverse of f(x) =.

  35. 7-2 Check It Out! Example 2a Continued 1 x 3 3 1 1 1 = 1 3 3 3 3 f–1( ) = 3( ) Substitute for x. The inverse function does undo the original function.  CheckUse the input x = 1 in f(x). f(x) = f(1) = Substitute 1 for x. Substitute the result into f–1(x) f–1(x) = 3x = 1

  36. 7-2 2 3 Check It Out! Example 2b Use inverse operations to write the inverse of f(x) = x + .

  37. Check It Out! Example 2b Continued 7-2 f(1) = 1 + 5 5 5 = 3 5 3 3 2 2 2 2 f(x) = x + 3 3 3 3 3 Substitute for x. f–1(x) = x – f–1( ) = – The inverse function does undo the original function.  CheckUse the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1

  38. 7-2 Helpful Hint The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses Algebra II (Bell work) Questions 7.2? (Day 1) Copy down the Helpful hint and info below Undo operations in the opposite order of the order of operations.

  39. 7-2 Example 3: Writing Inverses of Multi-Step Functions Use inverse operations to write the inverse of f(x) = 3(x – 7).

  40. Check It Out! Example 3 7-2 Use inverse operations to write the inverse of f(x) = 5x – 7.

  41. Example 4: Writing and Graphing Inverse Functions Graph f(x) = – x– 5 . Then write the inverse and graph. 1 2

  42. Example 4 Continued 1 f(x) = – x – 5 2 f–1(x) = -2x - 10

  43. Check It Out! Example 4 7-2 Graph f(x) = x+ 2. Then write the inverse and graph. 2 3

  44. Check It Out! Example 4 7-2 f–1(x) = 3 2 f –1 f x – 3 Set y = f(x). Then graph f–1.

  45. 7-2 Remember! In a real-world situation, don’t switch the variables, because they are named for specific quantities. Anytime you need to undo an operation or work backward from a result to the original input, you can apply inverse functions.

  46. Example 5: Retailing Applications 7-2 L = c – 2.50 = L 13.70 – 2.50 0.80 0.80 Juan buys a CD online for 20% off the list price. He has to pay $2.50 for shipping. The total charge is $13.70. What is the list price of the CD? c = 0.80L + 2.50 Step 1 Write an equation for the total charge as a function of the list price. c–2.50 = 0.80L Step 2 Find the inverse function that models list price as a function of the change. Step 3 Evaluate the inverse function for c = $13.70. = 14 The list price of the CD is $14.

  47. 7-2 Check It Out! Example 5 To make tea, use teaspoon of tea per ounce of water plus a teaspoon for the pot. Use the inverse to find the number of ounces of water needed if 7 teaspoons of tea are used. 1 1 1 6 6 6 t = z + 1 Step 1 Write an equation for the number of ounces of water needed. t– 1 = z Step 2 Find the inverse function that models ounces as a function of tea. 6t – 6 = z Step 3 Evaluate the inverse function for t = 7. z = 6(7) – 6 = 36 36 ounces of water should be added.

  48. 7-2 HW, 7-2 • Day 1: 2-7, 18-22, 32, 33, Bonus 39 • Day 2: 8-17, 26-28, 31a • If needed 61-68

  49. 7-3 Algebra II (Bell work) 1) Define: Logarithm, Common Logarithm, Logarithmic Function 2) Copy the know it note on pg. 506 Solve for Skips 1. 4–3 2. 4. 2

  50. 7.3 Logarithmic Functions Algebra II