866 Chapter 12 Graphing Exponential and Logarithmic Functions

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1 7. Determine the amount of mone in Helen s account at the end of 3 ears if it is compounded: a. twice a ear. b. monthl. c. dail.. What effect does the frequenc of compounding have on the amount of mone in her savings account? Chapter Graphing Eponential and Logarithmic Functions

2 Problem 3 Eas e Recall that in Problem the variable n represented the number of compound periods per ear. Let s eamine what happens as the interest becomes compounded more frequentl. 1. Imagine that Helen finds a different bank that offers her 1% interest. Complete the table to calculate how much Helen would accrue in 1 ear for each period of compounding if she starts with $1. Even though we are working with mone, in order to see the pattern, it will be helpful to compute the compound interest to at least si decimal places. Period of Compounding n 5 Formula Amount Yearl 1 1 ( ) 1? 1. Semi-Annuall 1 ( )? 1.5 Quarterl 1 ( )? 1 Monthl Weekl Dail Hourl Ever Minute Ever Second. Make an observation about the frequenc of compounding and the amount that Helen earns. What is it approaching?. Properties of Eponential Graphs 7

3 The amount that Helen s earnings approach is actuall an irrational number called e. e It is often referred to as the natural base e. In geometr, ou worked with p, an irrational number that was approimated as and so on. Pi is an incredibl important part of man geometric formulas and occurs so frequentl that, rather than write out each time, we use the smbol p. Similarl, the smbol e is used to represent the constant It is often used in models of population changes as well as radioactive deca of substances, and it is vital in phsics and calculus. The number e goes on forever with no repetition. There are even competitions to see who can memorize the most digits of e. In 7, Bhaskar Karmakar from India set a World Record for memorizing 5 digits! The smbol for the natural base e was first used b Swiss mathematician Leonhard Euler in 177 as part of a research manuscript he wrote at age 1. In fact, he used it so much, e became known as Euler s number. The constant e represents continuous growth and has man other mathematical properties that make it unique, which ou will stud further in calculus. 3. The following graphs are sketched on the coordinate plane shown. f() 5, g() 5 3, h() 5 1, j() 5 ( 3 5 ), k() a. Label each function b. Consider the function m() 5 e. Use our knowledge of the approimate value of e to sketch its graph. Eplain our reasoning. Chapter Graphing Eponential and Logarithmic Functions

4 c. Using the functions f() 5, g() 5 3, m() 5 e, approimate the values of f(), g(), and m() on the number line. Eplain our reasoning. 1 Problem It Keeps Growing and Growing and Growing 1. The formula for population growth is N(t) 5 N e rt. Complete the table to identif the contetual meaning of each quantit. Quantit Contetual Meaning N r t N(t). Wh is e used as the base? 3. How could this formula be used to represent a decline in population?. Properties of Eponential Graphs 9

5 . The population of the cit of Fredericksburg, Virginia, was approimatel 19,3 in and has been continuousl growing at a rate of.9% each ear. a. Use the formula for population growth to write a function to model this growth. b. Use our function model to predict the population of Fredericksburg in 13. c. What value does our function model give for the population of Fredericksburg in the ear 19? 5. Use a graphing calculator to estimate the number of ears it would take Fredericksburg to grow to, people, assuming that the population trend continues. Be prepared to share our solutions and methods. 7 Chapter Graphing Eponential and Logarithmic Functions

6 I Like to Move It Transformations of Eponential Functions.3 Learning Goals In this lesson, ou will: Dilate, reflect, and translate eponential functions using reference points and transformational function form. Investigate graphs of eponential functions through intercepts, asmptotes, intervals of increase and decrease, and end behavior. Describe how transformations of eponential functions affect their ke characteristics. And Warhol was an American pop artist whose work eplored the relationship between artistic epression, celebrit culture, and advertisement. A recurring theme throughout Warhol s art is the transformation of the mundane and commonplace into art. His most renowned images are silk-screened reproductions of Campbell s soup cans and publicit photographs of pop culture icons like Mariln Monroe and Elvis Presle. Have ou ever seen an of And Warhol s work? 71

7 problem 1 It s the Same... But Different! 1. The two tables show four eponential functions and four eponential graphs. a. Match the eponential function to its corresponding graph, and write the function under the graph it represents. b. Eplain the method(s) ou used to match the functions with their graphs. Eponential Functions f() 5 1 h() 5 1 g() 5 1 j() 5 1 Eponential Graphs 7 Chapter Graphing Eponential and Logarithmic Functions

8 . Analze the graphs. a. Write an equation for h() in terms of f(). Describe the transformation on f(). b. Write an equation for g() in terms of f(). Describe the transformation on f(). c. Write an equation for j() in terms of f(). Describe the transformation on f(). 3. Determine the asmptotes, intervals of increase and decrease, and end behavior for each eponential function. Function Asmptotes Intervals of Increase and Decrease End Behavior f() 5 1 g() 5 1 h() 5 1 j() 5 1. How would the graph of k() 5 ( 1 1 ) compare to the graph of g() 5 1?.3 Transformations of Eponential Functions 73

9 5. How do the transformations on f() affect the asmptotes, intervals of increase and decrease, and end behavior? Problem Keep On Moving Consider the functions 5 f() and g() 5 Af(B( C)) 1 D. Recall that the D-value translates f() verticall, the C-value translates f() horizontall, the A-value verticall stretches or compresses f(), and the B-value horizontall stretches or compresses f(). Eponential functions are transformed in the same manner. The function f() 5 3 is shown. Recall the ke characteristics of basic eponential functions, including a domain of all real numbers, a range of positive numbers, and a horizontal asmptote at 5. f() Suppose that a() 5 f( 1 1). a. Describe the transformation on the graph of f() that produces a(). b. Complete the table to determine the corresponding points on a(), given reference points on f(). Then, graph and label a(). f() 5 3 Reference Points on f() ( 1, 1 3 ) (, 1) Corresponding Points on a() 1 (1, 3) 7 Chapter Graphing Eponential and Logarithmic Functions

10 c. Determine the domain, range, and asmptotes of a().. Suppose that b() 5 f() 1 1. a. Describe the transformation on the graph of f() that produces b(). b. Complete the table to determine the corresponding points on b(), given reference points on f(). Then, graph and label b(). f() 5 3 Reference Points on f() ( 1, 1 3 ) (, 1) Corresponding Points on b() 1 (1, 3) c. Determine the domain, range, and asmptotes of b()..3 Transformations of Eponential Functions 75

11 3. Suppose that c() 5 f() 5. a. Describe the transformation on the graph of f() that produces c(). b. Complete the table to determine the corresponding points on c(), given reference points on f(). Then, graph and label c(). f() 5 3 Reference Points on f() ( 1, 1 3 ) (, 1) Corresponding Points on c() 1 (1, 3) c. Determine the domain, range, and asmptotes of c(). 7 Chapter Graphing Eponential and Logarithmic Functions

12 . Suppose that d() 5 f(). a. Describe the transformation on the graph of f() that produces c(). b. Complete the table to determine the corresponding points on d(), given reference points on f(). Then, graph and label d(). f() 5 3 Reference Points on f() ( 1, 1 3 ) (, 1) Corresponding Points on d() 1 (1, 3) c. Determine the domain, range, and asmptotes of d(). 5. Analze the transformations performed on f() in Questions 1 through. a. Which, if an, of these transformations affected the domain, range, and asmptotes? b. What generalizations can ou make about the effects of transformations on the domain, range, and asmptotes of eponential functions?.3 Transformations of Eponential Functions 77

13 . Andres and Tomas each described the effects of transforming the graph of f() 5 3, such that p() 5 3f(). Andres p() = 3f() The A-value is 3 so the graph is stretched verticall b a scale factor of 3. Tomas p() = 3f() p() = 3?3 p() = p() = f( + 1) The C-value is 1 so the graph is horizontall translated 1 unit to the left. a. Eplain Andres and Thomas reasoning. b. Determine the domain, range, and asmptotes of p(). c. What generalizations can ou make about the effects of vertical dilations on the domain, range, and asmptotes of eponential functions? 7 Chapter Graphing Eponential and Logarithmic Functions

14 Problem 3 Multiple Transformations 1. Analze the graphs of f() and g(). Describe the transformations performed on f() to create g(). Then, write an equation for g() in terms of f(). For each set of points shown on f(), the corresponding points are shown on g(). a. g() 5 f() (1, 5) 1 (1, ) 5 (, 1) 1 (1, ) 5 (, 1) (3, 5) g() b. g() f() (1, 1) (1, 1) g() 1 1 (1, ) 1 (, 1) (, 5) (1, ) Transformations of Eponential Functions 79

15 c. g() 5 1 f() 1 g() 1 1 (1, ) (1, 5) 1 (1, ) (, 1) 1 (, 1) (1, ). The equation for an eponential function m() is given. The equation for the transformed function t() in terms of m() is also given. Describe the graphical transformation(s) on m() that produce(s) t(). Then, write an eponential equation for t(). a. m() 5 t() 5.5m( 1 3) b. m() 5 e t() 5 m() 1 c. m() 5 t() 5 m() Be prepared to share our solutions and methods. Chapter Graphing Eponential and Logarithmic Functions