Algebra 1 Common Core Student Edition, Grade 8-9

When can you use an
exponential growth
function?
You can use an
exponential growth
function when an initial
a m o u n t increases by a
fixed percent each tim e
period.

Pr o b l em 1 Modeling Exponential Growth

Economics Since 2005, the amount of money spent at restaurants in the

United States has increased about 7% each year. In 2005, about $360 billion

was spent at restaurants.

0 If the trend continues, about how much will be spent at restaurants in 2015?

Relate y = a • bx Use an exponential function.

Define Let x = the num ber of years since 2005.

L ety = the annual am ount spent at restaurants (in billions of dollars).

Let a = the initial am ount spent (in billions of dollars), 360.

Let b = the growth factor, which is 1 + 0.07 = 1.07.

Write y = 360 • 1.07x

Use th e equation to p red ict th e an n u a l sp en d in g in 2015.

y = 360 • 1.07*

= 360 * 1.0710 2015 is 10 yr a fte r 2005, so su b stitu te 10 fo r x.

~ 708 Round to the nearest billion dollars.

About $708 billion will be sp en t at restau ran ts in the U nited States in 2015 if the

tren d continues.

0 What is an expression that represents the equivalent monthly increase of spending at

U.S. restaurants in 2005?

You will n ee d to find an expression of th e form rm, w here r is approxim ately the

m onthly growth factor an d m is the n u m b e r of m onths. You know th a t 1.07x represents

the yearly increase w here x is th e n u m b e r of years.

12x

1.07x = 1. 07 tY There are 12xm onths in xyears.

= (l.07^)12x Power raised to a power

= 1.005712x Simplify.

= 1.0057m Let 12x = m, the number of months.

The expression 1.0057m represents th e equivalent m onthly increase of spending at

restaurants.

Go t I t? 1. Suppose that in 1985, there were 285 cell p h o n e subscribers in a small town.

The n u m b er of subscribers increased by 75% each year after 1985. How m any

cell p hone subscribers were in the small tow n in 1994? Write an expression to

represent the equivalent m onthly cell p h o n e subscription increase.

W hen a b an k pays interest on b o th th e principal and th e interest an account has already

earned, th e b an k is paying compound interest. C om p o u n d interest is an exam ple of

exponential growth.

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Lesson 7-7 Exponential G row th and Decay _i 461