7-1 Exploring Exponential Models

Quick Review

The general form of an exponential function is y = a b x , y equals , eh b to the x , comma  where x is a real number, a ≠ 0 , eh not equal to 0 comma   b > 0 , b greater than 0 comma  and b ≠ 1 . b not equal to 1 .  When b > 1 , b greater than 1 comma  the function models exponential growth, and b is the growth factor. When 0 < b < 1 , 0 less than b less than 1 comma  the function models exponential decay, and b is the decay factor. The y-intercept is (0, a).

Example

Determine whether y = 2 ( 1.4 ) x y equals 2 . open 1.4 close to the x  is an example of exponential growth or decay. Then, find the y-intercept.

Since b = 1.4 > 1 , b equals 1.4 greater than 1 comma  the function represents exponential growth.

Since a = 2 , eh equals 2 comma  the y-intercept is (0, 2).

Exercises

Determine whether each function is an example of exponential growth or decay. Then, find the y-intercept.

6. y = 5 x y equals , 5 to the x
7. y = 2 ( 4 ) x y equals 2 . open 4 close to the x
8. y = 0.2 ( 3.8 ) x y equals 0.2 . open 3.8 close to the x
9. y = 3 ( 0.25 ) x y equals 3 . open , 0.25 , close to the x
10. y = 25 7 ( 7 5 ) x y equals , 25 over 7 . open , 7 fifths , close to the x
11. y = 0.0015 ( 10 ) x y equals , 0.0015 . open 10 close to the x
12. y = 2.25 ( 1 3 ) x y equals , 2.25 . open , 1 third , close to the x
13. y = 0.5 ( 1 4 ) x y equals 0.5 . open , 1 fourth , close to the x

Write a function for each situation. Then find the value of each function after five years. Round to the nearest dollar.

14. A $12,500 car depreciates 9% each year.
15. A baseball card bought for $50 increases 3% in value each year.

7-2 Properties of Exponential Functions

Quick Review

Exponential functions can be translated, stretched, compressed, and reflected.

The graph of y = a b x − h + k y equals . eh b super x minus h plus end super . k  is the graph of the parent function y = b x y equals , b to the x  stretched or compressed by a factor | a |, reflected across the x-axis if a < 0 , eh less than 0 comma  and translated h units horizontally and k units vertically.

The continuously compounded interest formula is A = P e r t , eh equals . p e super r t end super . comma  where P is the principal, r is the annual interest rate, and t is time in years.

Example

How does the graph of y = − 3 x + 1 y equals , negative 3 to the x , plus 1  compare to the graph of the parent function?

The parent function is y = 3 x . y equals , 3 to the x , .

Since a = − 2 , eh equals negative 2 comma  the graph is reflected across the x-axis.

Since k = 1 , k equals 1 comma  it is translated up 1 unit.

Exercises

How does the graph of each function compare to the graph of the parent function?

16. y = 5 ( 2 ) x + 1 + 3 y equals 5 . open 2 close super x plus 1 end super . plus 3
17. y = − 2 ( 1 3 ) x − 2 y equals negative 2 . open , 1 third , close super x minus 2 end super

Find the amount in a continuously compounded account for the given conditions.

18. principal: $1000

    annual interest rate: 4.8%

    time: 2 years

19. principal: $250

    annual interest rate: 6.2%

    time: 2.5 years

Evaluate each expression to four decimal places.

20. e − 3 e super negative 3 end super
21. e − 1 e super negative 1 end super
22. e 5 e to the fifth
23. e − 1 2 e super negative , 1 half end super

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