Reteach 7-1

Name

Date

Class

Reteach

LESSON

7-1

Exponential Functions, Growth, and Decay

The base of an exponential function indicates whether the function shows growth or decay. Exponential function: f  x   ab x •

a is a constant

•

b is the base. The base is a constant. If 0  b  1, the function shows decay. If b  1, the function shows growth.

•

x is an exponent.

f  x   1.2 x a1 b  1.2 b  1, so the function shows exponential growth. 

g  x   10  0.6  x a  10 b  0.6 0  b  1, so the function shows exponential decay.

Y

































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Tell whether each function shows growth or decay. Then graph. x 1. h  x   0.8  1.6 

a

2. p  x   12  0.7  x

0.8

b

1.6

a

h  x  shows exponential growth. 

12

p  x  shows exponential decay.

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b

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Copyright © by Holt, Rinehart and Winston. All rights reserved.

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Holt Algebra 2

5/15/06 1:31:11 PM Process Black

Name LESSON

7-1

Date

Class

Reteach Exponential Functions, Growth, and Decay (continued)

When an initial amount, a, increases or decreases by a constant rate, r, over a number of time periods, t, this formula shows the final amount, A  t . t At   a 1  r 

A  t , the final amount, is a function of time, t.

Time, t, is measured in years.

The rate, r, usually is a percent. a is the initial amount.

An initial amount of $15,000 increases by 12% per year. In how many years will the amount reach $25,000? Step 1 Identify values for a and r. a  $15,000

Remember: On the graph, x corresponds to t and y corresponds to f  t .

r  12%  0.12

Step 2 Substitute values for a and r into the formula. f t   a 1  r  t f  t   15,000  1  0.12  t f  t   15,000  1.12  t

Simplify.

Step 3 Graph the function using a graphing calculator. Modify the scales: [ 0, 10 ] and [ 0, 30,000 ]. Step 4 Use the graph and the [TRACE] feature on the calculator to find f  t   25,000. Step 5 Use the graph to approximate the value of t when f  t   25,000. t  4.5 when f  t   25,000 The amount will reach $25,000 in about 4.5 years. Write an exponential function and graph the function to solve. 3. An initial amount of $40,000 increases by 8% per year. In how many years will the amount reach $60,000? a. a  b. r  c. f  t  

 

40,000 0.08 f  t  ⫽ 40,000  1.08  t

       

d. Approximate t when f  t   60,000 t

Y

 

5.25 yr

    

Copyright © by Holt, Rinehart and Winston. All rights reserved.

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