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Name. .Class. Date.

Exponential Functions, Growth and Decay

Essential question: What are the characteristics of an exponential junction?

In an exponential function, the variable is an exponent. The parentfunction isf(x) — If, where b is any real number greater than 0, except 1.

CC.9-12,F.IE7e

E X A M P L E - Graphing f(x) = If for b > 1

Graph f(x) = 2*.

A Complete the table of values below.

B Plot the points on the graph and connect the points with a smooth curve.

-3

-2

-1

f (x) = 2"

I j.

4

1 0

—

-f

-t-M-4-

^REFLECT \. What happens to/(x) as x increases without bound? What happens to/(X) as x

decreases without bound?

j. Does the graph intersect the x-axis? Explain how you know.

1 c. What are the domain and range of

Chapter 4 191 Lesson 1

CC.9~12.F.IF.7e

E 1 A M P I , ii . Grapmrag ffcr) =

Graph f(x) = (1)*

A Complete the table of values below.

B Plot the points on the graph and connect the points with a smooth curve.

-3

-2

-1

•y| I i

i D I--ff'- ' 4—

*

( REFLECT ̂ \. What happens to/(^) as x increases without bound? What happens to/(x) as x

decreases without bound?

f i \. How do the domain and range off(x) — 1^1 compare to the domain and range ofW = 2'?

2c, What do you notice about the y-intercepts of the graphs off(x) = I ̂ J and f(x) = 2X1Why does this make sense?

. What transformation can you use to obtain the graph off(x) = f ̂ 1 from the graphof/(.x) - 2*?

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Chapter 4 192 Lesson 1

CC9-12.F.IR7e

E N G A G E Recognizing Types of Exponential Functions

A function of the form/(x) = V is an exponential growth function if b > 1 and anexponential decay function if 0 < b < I.

Exponential Growthf(x) = b* f or b > 1

Exponential Decayf(x) = b* for 0 <b<

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Q.

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I f"REFLECT'";\. Describe the end behavior of an exponential growth function.

3b. Describe the end behavior of an exponential decay function.

3e. Explain why the point (1, b} is always on the graph off(x) = b*.

3d. Explain why the point (0, 1) is always on the graph of/(x) = bx.

-. Aief(x) = 3X and g(x) = 5"x both exponential growth functions or both exponentialdecay functions? Although they have the same end behavior, how you do thinktheir graphs differ? Explain your reasoning.

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31. Are/(jc) = fi] andg(x) = f|) both exponential growth functions or both

exponential decay functions? Although they have the same end behavior, how doyou think their graphs differ? Explain your reasoning.

Chapter 4 193 Lesson 1

o CC.9-12.FLE.3

E JC F L 0 11 Comparing Onea?, Cuibk, aaid

Compare each of the functions f(x) = x + 3 and g(x) = x3 to the exponentialfunction h(x) = 3* for x > 0.

A Complete the table of values for the three functions.

X

0

1

2

3

4

5

f (x) = x + 3

3

g(x) = x3

0

Ai(x) = 3'

1

B The graph of h(x) — 3X is shown onthe coordinate grid below. Graph/(X) = x + 3 on the same grid.

2 4 6 10

C The graph of h(x) = 3* is shown onthe coordinate grid below. Graphg(x) — x? on the same grid.

8 10

[ REFLECTJ\. How do the values of h(x) compare to those of/(x) and g(x) as x increases

without bound?

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Chapter 4 194 Lesson 1

PRACTICE

Tell whether the function describes an exponential growth function or anexponential decay function. Explain how you know without graphing.

2.

3. 4. = gf

5. In an exponential function, f(x) = If, b is not allowed to be 1. Explain why thisrestriction exists.

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6. Complete the table for/(jc) = 4*. Then sketch the graph of the function.

-1

f(x)

7. Complete the table for/(x) = f|] . Then sketch the graph of the function.

-3

-2

-1

f(x)

Chapter 4 195 Lesson 1

8. Compare the graph off(x) = 2X to the graph of g(x) = x2.

9. Enter the functions/(jc) = 10* and g(x) = f^rl into your graphing calculator.

a. Look at a table of values for the two functions. For a given x-value, how do thecorresponding function values compare?

b. Look at graphs of the two functions. How are the two graphs related to each other?

le graph of an exponential function f(x) — b* is shown.

a. Which of the labeled points, (0, 1) or (1, 5), allows you todetermine the value of b! Why doesn't the other point help?

-b. What is the value of bl Explain how you know.

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11. Given an exponential function y = V, when you double the value of x, how doesthe value of y change? Explain.

12. Given an exponential function y = b*, when you add 2 to the value of x, how doesthe value of y change? Explain.

13. Error Analysis A student says that the function/(:c) = f~J is an exponentialdecay function. Explain the student's error.

14. One method of cutting a long piece of string into smaller pieces is to make individualcuts, so that 1 cut results in 2 pieces, 2 cuts result in 3 pieces, and so on. Anothermethod of cutting the string is to fold it onto itself and cut the folded end, then foldthe pieces onto themselves and cut their folded ends at the same time, and continueto fold and cut, so that 1 cut results in 2 pieces, 2 cuts result in 4 pieces, and so on.For each method, write a function that gives the number p of pieces in terms of thenumber c of cuts. Which function grows faster? Why?

10no3

Chapter 4 196 Lesson 1

Name, . Class. Date.

/mm^m^^^^E^^^K^m^^^^^g^m^^a,^^^^^^

Tell whether the function shows growth or decay. Then graph.

1. g(x) = -(2)x 2. Mx) = -0.5(0.2)x

.10

-5 -4 -3 -2 -i °

I i 1 4io

3.y(x) = -2(0.5)x

4-20-

4-30-

-50

1 2 3 4 5X

-5 -4 -3 -2 -1

4. p(x) = 4(1.4)x

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4-30-

--40-

-50

1 2 3 4 5

Eoura

y

-5 -4 -3 -2 -1

-1-0-

4-2Q-

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-40-

-50

1 2 3 4 5

Solve.

5. A certain car's value depreciates about 15%each year. This is modeled by the function

V(t) = 20,000(0.85)'

where $20,000 is the value of a brand-new model.

a. Graph the function.

b. Suppose the car was worth $20,000 in 2005.What is the first year that the value of this carwill be worth less than half of that value?

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1 2 3 4 5' ; ; i !

30,000

25,000

20,000

15,000

10,000

5,000

y

1 2 3 4 5 6 7' >» X9 10

Chapter 4 197 Lesson 1

Problem Solving

Justin drove his pickup truck about 22,000 miles in 2004. He read thatin 1988 the average residential vehicle traveled about 10,200 miles,which increased by about 2.9% per year through 2004.

1. Write a function for the average mileage, m(t), asa function of t, the time in years since 1988.

2. Assume that the 2.9% increase is valid through 2008 and use your functionto complete the table to show the average annual miles driven.

Year

t

m(t)

1988

0

10,200

1992

4

1996 2000 2004 2008

3. Did Justin drive more or fewer miles than the average residential vehicledriver in 2004? by how much (to the nearest 100 miles)?

4. Later Justin read that the annual mileage for light trucks increased by 7.8%per year from 1988 to 2004.

a. Write a function for the average miles driven fora light truck, n(t), as a function of t, the time inyears since 1988. He assumes that the averagenumber of miles driven in 1988 was 10,200.

b. Graph the function. Then use your graph toestimate the average number of miles driven(to the nearest 1000) for a light truckin 2004.

c. Did Justin drive more or fewer miles than theaverage light truck driver in 2004? by howmuch?

}400 00̂

35000

30000

20000

0

f

2 4 6 8 10 12 14 16 18 2093

Justin bought his truck new for $32,000. Its value decreases 9.0%each year. Choose the letter for the best answer.

5. Which function represents the yearlyvalue of Justin's truck?

A f(t) = 32,000(1 +0.9)'

B f(t) = 32,000(1 -0.9)f

C f(t) = 32,000(1 + 0.09)f

D f(t) = 32,000(1 -0.09)'

6. When will the value of Justin's truck fallbelow half of what he paid for it?

F In 6 years

G In 8 years

H In 10 years

J In 12 years

Chapter 4 198 Lesson 1