3.1. Exponential Functions. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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1 3.1 Exponential Functions Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2 Quick Review Evaluate the expression without using a calculator /3 Rewrite the expression using a single positive exponent. ( ) 2 4. a -3 Use a calculator to evaluate the expression Slide 3-2

3 Quick Review Solutions Evaluate the expression without using a calculator / Rewrite the expression using a single positive exponent ( a ) 6 a Use a calculator to evaluate the expression Slide 3-3

4 Today s Objectives Content Objective: n Exponential Functions and Their Graphs n The Natural Base e n Exponential functions model many growth patterns, including the growth of human and animal populations. Language Objective: n Use sentence frames to verbally determine a solution to all turn and talk questions. Report your conclusions about exponential functions to the teacher. Slide 3-4

5 Exponential Functions Let a and b be real number constants. An exponential function in x is a function that can be written in the form f (x) = a b x, where a is nonzero, b is positive, and b 1. The constant a is the initial value of f (the value at x = 0), and b is the base. The domain/input of exponential functions are exponents. The range/output is the base, b, multiplied by itself [insert exponent here] times and vertically stretched or shrunk by a factor of a. Slide 3-5

6 Finding an Exponential Function from its Table of Values Determine formulas for the exponential function g and h whose values are given in the table below. Slide 3-6

7 Because g is exponential, g(x) = a b x. Because g(0) = 4, a = 4. Because g(1) = 4 b 1 = 12, the base b = 3. So, g(x) = 4 3 x. Slide 3-7

8 Because h is exponential, h(x) = a b x. Because h(0) = 8, a = 8. Because h(1) = 8 b 1 = 2, the base b = 1/ 4. So, h(x) = x. Slide 3-8

9 Exponential Growth and Decay For any exponential function f (x) = a b x and any real number x, f (x +1) = b f (x). If a > 0 and b > 1, the function f is increasing and is an exponential growth function. The base b is its growth factor. If a > 0 and b < 1, the function f is decreasing and is an exponential decay function. The base b is its decay factor. Slide 3-9

10 Example Transforming Exponential Functions Describe how to transform the graph of f (x) = 2 x into the graph of g(x) = 2 x-2. Slide 3-10

11 Example Transforming Exponential Functions x x-2 Describe how to transform the graph of f( x) = 2 into the graph of g( x) = 2. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The graph of g(x) = 2 x-2 is obtained by translating the graph of f (x) = 2 x by 2 units to the right. Slide 3-11

12 Reflections: Transforming Exponential Functions Describe how to transform the graph of f (x) = 2 x into the graph of g(x) = 2 -x. Slide 3-12

13 Reflections: Transforming Exponential Functions Describe how to transform the graph of f (x) = 2 x into the graph of g(x) = 2 -x. Since the transformation is applied to x it will effect a horizontal property of the graph, therefore the graph of g(x) = 2 x is obtained by making a horizontal reflection of the graph. So flip f (x) = 2 x across the y-axis to produce g(x) = 2 x. Slide 3-13

14 The Natural Base e e = lim x 1+ 1 x Turn and Talk: When have we used limits? In your experience, what do limits tell us? Think about polynomial and rational functions what type of limit is indicated when x approaches infinity? Find the limit numerically, and make a conjecture about the value of Euler s constant also called the natural base e. Report to me: Each group must state there answer to the teacher. x Slide 3-14

15 Exponential Functions and the Base e Any exponential function f (x) = a b x can be rewritten as f (x) = a e kx, for any appropriately chosen real number constant k. If a > 0 and k > 0, f (x) = a e kx is an exponential growth function. If a > 0 and k < 0, f (x) = a e kx is an exponential decay function. Turn and Talk: What is the relationship between b, k, and e? Justify your response using the properties of exponents. Report to me: Each group must state there answer to the teacher. Slide 3-15

16 Exponential Functions and the Base e Turn and Talk: Classify these functions. What will they always have in common? Use the properties of exponents to explain why? Report to me: Each group must state there answer to the teacher. Slide 3-16

17 Example Transforming Exponential Functions Describe how to transform the graph of f (x) = e x into the graph of g(x) = e 3x. Since the transformation is applied to x it will effect a horizontal property of the graph, therefore the graph of g(x) = e 3x is obtained by making a horizontal reflection of the graph. So f (x) by g(x) = e 3x. Slide 3-17

18 AM: Graph exponential functions LO: I know by looking at the equation that I have an a = >, and a b = <. This means I have an example of exponential with a y-intercept of. Therefore, my answer is. 1 Slide 2-18

19 Graph Exponential Functions LO: I know by looking at the equation that a = >, and b = < value. This means I have an example of exponential with a y-intercept of. I also have a shift of units and a shift of units. Therefore, my answer is. Report to me: Each group must state the answer to the teacher. Pick up your AM Practice on Objectives #1 and #2 2 Slide 2-19

20 3.2 Exponential and Logistic Modeling Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

21 Quick Review Convert the percent to decimal form or the decimal into a percent % Show how to increase 25 by 8% using a single multiplication. Solve the equation algebraically b = 720 Solve the equation numerically b = Slide 3-21

22 Quick Review Solutions Convert the percent to decimal form or the decimal into a percent % % 3. Show how to increase 25 by 8% using a single multiplication. Solve the equation algebraically b = 720 ± 6 Solve the equation numerically b = Slide 3-22

23 What you ll learn about n n n n Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors. Slide 3-23

24 Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown. Time in years Population 0 P(0) = P = initial population 1 P(1) = P + Pr = P(1 + r) P(2) = P(1) (1 + r) = P(1 + r) 3 P(3) = P(2) (1 + r) = P(1 + r) M M t P( t) = P(1 + r) 0 t Slide 3-24

25 Exponential Population Model If a population P is changing at a constant percentage rate r each year, then t Pt ( ) = P(1 + r), where Pis the initial population, ris expressed as a decimal, 0 0 and t is time in years. Slide 3-25

26 Example Finding Growth and Decay Rates t Tell whether the population model Pt ( ) = 786, is an exponential growth function or exponential decay function, and find the constant percent rate of growth. Slide 3-26

27 Example Finding Growth and Decay Rates t Tell whether the population model Pt ( ) = 786, is an exponential growth function or exponential decay function, and find the constant percent rate of growth. Because 1+ r = 1.021, r =.021 > 0. So, P is an exponential growth function with a growth rate of 2.1%. Slide 3-27

28 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year. Slide 3-28

29 Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year. t Because P = 10 and r = 5% = 0.05, the function is P( t) = 10( ) or t Pt () = 10(1.05). 0 Slide 3-29

30 WP: Exponential Functions LO: We know that the initial value corresponds to a, so a =. Since the rate of growth is, the base will be equal to. This means our equation will be. Since time, x =, we can evaluate the function for x =. This give us a value of. 3 Slide 2-30

31 WP: Exponential Functions LO: We know that the initial value corresponds to a, so a =. Since the rate of is, the will be equal to. This means our equation will be. Since time, x =, we can the function for x =. This give us a value of. 3 Slide 2-31

32 WP: Exponential Functions LO: We know that the initial value corresponds to a, so a =. Since the rate of is, the will be equal to. This means our equation will be. Since time, x =, we can the function for x =. This give us a value of. 3 Slide 2-32

33 Example Modeling Bacteria Growth Suppose a culture of 200 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. Slide 3-33

34 Example Modeling Bacteria Growth Suppose a culture of 200 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350, = = M t Pt ( ) = represents the bacteria population thr after it is placed in the petri dish. To find out when the population will reach 350,000, solve t 350, 000 = for t using a calculator. t = or about 10 hours and 46 minutes. Slide 3-34

35 Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for Slide 3-35

36 Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for Let Pt ( ) be the population (in millions) of the U.S. tyears after t Using exponential regression, find a model Pt ( ) = To find the population in 2003 find P(103) = Slide 3-36

37 Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. Slide 3-37

38 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor, which spreads -0.9 t logistically so that St ( ) = 1500 /( e ) models the number of students who have heard the rumor by the end of t days, where t = 0 is the day the rumor begins to spread. (a) How many students have heard the rumor by the end of Day 0? (b) How long does it take for 1000 students to hear the rumor? Slide 3-38

39 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor, which spreads -0.9 t logistically so that St ( ) = 1500 /( e ) models the number of students who have heard the rumor by the end of t days, where t = 0 is the day the rumor begins to spread. (a) How many students have heard the rumor by the end of Day 0? (b) How long does it take for 1000 students to hear the rumor? -0.9 ( 0 ) (a) S(0) = 1500 /( e ) = 1500 /( ) = 1500 / 30 = 50. So 50 students have heard the rumor by the end of day t (b) Solve 1000 = 1500 /( e ) for t. t 4.5. So 1000 students have heard the rumor half way through the fifth day. Slide 3-39