In Lesson 8.1 You Studied Exponential Growth Functions

Section 8.2 Exponential Decay

In Lesson 8.1 you studied exponential growth functions.

Remember, if a > 0 and b > 1, the function y = abxis anexponential growth function.

In this lesson you will studyexponential decay functionswhich have the form ƒ(x) = abxwhere a > 0 and0 < b < 1.

Sample Problem #1

State whether ƒ(x) is an exponential growth or exponential decay function.

a. ƒ(x) =

b. ƒ(x) =

c. ƒ(x) = 10(3)-x

To see the basic shape of the graph of an exponential decay function, you can make atable of values and plot points, as shown below

Recall that in general the graph of an exponential function y = abxpasses throughthe point (0, a) and has the x-axis as an asymptote.

Example #1

Graph the function.

a. y =

SOLUTION

a. Plot (0, 3) and (1,)

Then, from right to left, draw a curve that begins just above the x-axis, passes through thetwo points, and moves up to the left.

Sample Problem #2

Graph the function

Remember that to graph a general exponential function, y = abx -h + k, begin bysketching the graph of y = abx. Then translate the graph horizontally by h units andvertically by k units.

Sample Problem #3

Graph the function

When a real-life quantity decreases by a fixed percent each year (or other timeperiod), the amount y of the quantity after t years can be modeled by the equationy = a(1 -r)twhere a is the initial amount and r is the percent decrease expressed as a decimal.

The quantity 1 -r is called thedecay factor.

Sample Problem #4

You buy a new car for $24,000. The value y of the car decreases by 16% each year.

a. Write an exponential decay model for the value of the car.

b.Use the model toestimate the value after 2 years.

c. Graph the model.

d. Use the graph to estimate when the car will have a value of $12,000.