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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. Algebra II (Bell work) • Do Problems # (Skip) (7.1) • Define exponential function and asymptote • Read student to student pg. 491

  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 Just Read 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 B > 0, exponential growth 0 < b < 1,exponential decay 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

  8. 7-1 The base , ,is less than 1. This is an exponential decay function. Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base. Step 2 Graph the function by using a table of values.

  9. 7-1 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

  10. 7-1 Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph. Step 1 Find the value of the base. The base , 1.2, is greater than 1. This is an exponential growth function. p(x) = 5(1.2x) Step 2 Graph the function by using a table of values.

  11. 7-1 Math Joke • Q: How did you know that your dentist studied algebra? • A: She said all that candy gave me exponential decay.

  12. 7-1 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.

  13. 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 f(t) = 5000(1 + 0.0625)t f(t) = 5000(1.0625)t 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**

  14. 7-1 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.

  15. 7-1 Graph the function. Use to find when the population will fall below 8000. 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 f(t) = 15,500(1 – 0.03)t f(t) = 15,500(0.97)t It will take about 22 years for the population to fall below 8000.

  16. 7-1 Graph the function. Use to find when the population will reach 20,000. 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 P(t) = 350(1 + 0.14)t P(t) = 350(1.14)t It will take about 31 years for the population to reach 20,000.

  17. 7-1 Graph the function. Use to find when the value will fall below 100. 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 f(t) = 1000(1 – 0.15)t f(t) = 1000(0.85)t It will take about 14.2 years for the value to fall below 100.

  18. 7-1 HW pg. 493 • 1-15, 21, 28, • B: 22 • On each graphing problem have a table of at least 5 values. Have points on either side of the y axis

  19. Algebra II (Bell work) • Turn in all of (7.1) • Define Inverse Relation and inverse function

  20. 7.2 Inverses of Relations and Functions Algebra II

  21. You can also find and apply inverses to relations and functions. 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.

  22. 7-2 Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Just Watch ● ● ● ● Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair.

  23. 7-2 Just Watch • 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

  24. 7-2 # 2 pg. 501 Graph the relation. Graph the Inverse. Identify the domain and range of each relation

  25. 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).

  26. 7-2 1 2 1 1 f(x) = x – 2 2 f–1(x) = x + Use inverse operations to write the inverse of f(x) = x – if possible.

  27. 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

  28. 7-2 Math Joke • Q: How did the chicken find the inverse? • A: It reflected the function across y = eggs

  29. 7-2 x 3 x 3 Use inverse operations to write the inverse of f(x) =. f(x) = f–1(x) = 3x

  30. 7-2 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

  31. 7-2 f(x) = x + 2 2 2 3 3 3 f–1(x) = x – Use inverse operations to write the inverse of f(x) = x + .

  32. 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

  33. 7-2 1 1 3 3 f–1(x) = x + 7 f–1(6) = (6) + 7= 2 + 7= 9  Use inverse operations to write the inverse of f(x) = 3(x – 7). f(x) = 3(x – 7) Check Use a sample input. f(9) = 3(9 – 7) = 3(2) = 6

  34. 7-2 f–1(x) = x +7 10 3 + 7 5 5 5  f(2) = 5(2) – 7 = 3 f–1(3) = = = 2 Use inverse operations to write the inverse of f(x) = 5x – 7. f(x) = 5x – 7. Check Use a sample input.

  35. Graph f(x) = – x– 5 . Then write the inverse and graph. y = – x – 5 x = – y – 5 1 1 1 1 2 x + 5= – y 2 2 2 –2x –10 = y y =–2(x + 5) f–1(x) = –2(x + 5) f–1(x) = –2x – 10

  36. 1 f(x) = – x – 5 2 f–1(x) = -2x - 10

  37. 7-2 Graph f(x) = x+ 2. Then write the inverse and graph. f–1(x) = y = x + 2 x = y + 2 2 2 3 3 2 2 3 3 3 3 2 2 x – 2 = y x – 3 x – 3 = y 3x – 6 = 2y

  38. 7-2 f–1(x) = 3 2 x – 3 Graph f(x) = x + 2

  39. 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.

  40. 7-2 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.

  41. 7-2 HW pg. 501 • 7.2- • 3-17 (Odd), 20-25, 41-46, • 38 (Good Bell work)

  42. Algebra I (Bell work) • Look over 8.4/Packet for any questions • Then we will cover quiz

  43. Quiz 8.1-8.3 • If you got an A on the last quiz, you automatically get 5 (QEC/TEC) points • For everyone else, improve your letter grade by at least one on the new quiz, you get 5 (QEC/TEC) • Must have passed the first quiz to be eligible to get extra credit • We will take the Quiz 8.1-8.3 Version B tomorrow • You will get a worksheet as an assignment tomorrow after quiz

  44. Algebra II (Bell work) • Turn in all of (7.2) • Have your books/notebooks open to 7.3

  45. 7.3 Logarithmic Functions Algebra II

  46. 7-3 Reading Math Read logba=x, as “the log base b of a is x.” Notice that the log is the exponent. You can write an exponential equation as a logarithmic equation and vice versa.

  47. 7-3 1 1 log6 = –1 6 6 1 1 2 log255 = 2 Write each exponential equation in logarithmic form. log3243 = 5 log1010,000 = 4 logac =b

  48. 7-3 1 1 3 3 8 = 2 1 4–2 = 16 Write each logarithmic form in exponential equation. 91 = 9 29 = 512 1 16 b0 = 1

  49. 7-3 A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105.

  50. 7-3 Math Joke • Q: Why are you drumming on your algebra book with two big sticks? • A: Because we’re studying log rhythms