Warm-Up1) A town with a current population of 701,500 has a growth rate of 1.4 percent. Find the multiplier for the rate of exponential growth.

5 minutes

2) The inflation rate of the U.S. is 2.9 percent. What this means is that every year, prices increase by 2.9 percent. If a pound of meat cost $2.55 four years ago, what does it cost now?

6.2.1 Exponential Functions6.2.1 Exponential Functions6.2.1 Exponential Functions6.2.1 Exponential FunctionsObjectives: •Classify an exponential function as representing exponential growth or exponential decay

Exponential Functionsy = x2

y = 2x

The function f(x) = bx is an exponential function with base b, where b is a positive real number other than 1 and x is any real number.

An asymptote is a line that a graph approaches (but does not reach) as its x- or y-values become very large or very small.

Exponential Functions

Graph y1 = 2x and y2 =

When b > 1, the function f(x) = bx represents exponential growth.

x12

When 0 < b < 1, the function f(x) = bx represents exponential decay.

Example 1

a) 4 f(x)

Graph f(x) = 2x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept.

b) 6 f( x)

y = 4(2x)exponential growth, since the base, 2, is > 1

y-intercept is 4 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 4

exponential decay, since the base, ½, is < 1

y-intercept is 6 because the graph of f(x) = 2x, which has a y-intercept of 1, is stretched by a factor of 6

xy 6 2

x1

62

Practice

11) f (x)

3

Graph f(x) = 2x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept.

12) f ( x)

4

Critical Thinking

y a f (x) ? What transformation of f occurs when a < 0 in

The graph is reflected across the x-axis.

Homework

p.367 #11-27 odds

Warm-UpTell whether each function represents exponential growth or decay.

4 minutes

1) f(x) = 12(2.5)x

2) f(x) = 24(0.5)x

3) f(x) = -3(8)x

4) f(x) = 2(4)-x

5) f(x) = 0.75(216)x

6.2.2 Exponential Functions6.2.2 Exponential Functions6.2.2 Exponential Functions6.2.2 Exponential FunctionsObjectives: •Calculate the growth of investments under various conditions

Compound InterestThe total amount of an investment, A, earning compound interest is

ntrA(t) P 1 ,

n

where P is the principal,

r is the annual interest rate, n is the number of times interest is compounded per

year, and t is the time in years.

Example 1Find the final amount of a $500 investment after 8 years at 7% interest compounded annually, quarterly, and monthly.

ntrA(t) P 1

n

180.07

A(t) 500 11

compounded annually:

= $859.09

4 80.07

A(t) 500 14

compounded quarterly:

= $871.11

12 80.07

A(t) 500 112

compounded monthly:

= $873.91

PracticeFind the final amount of a $2200 investment at 9% interest compounded monthly for 3 years.

Effective YieldThe effective yield is the annually compounded interest rate that yields the final amount of an investment.

Suppose you buy a motorcycle for $10,000 and sell it one year later for $13,000.

The effective yield would be 30% because you made 30% more ($3,000) than the original price you paid.

You can determine the effective yield by fitting an exponential regression equation to two points.

Example 2A collector buys an antique stove for $500 at the beginning of 1990 and sells it for $875 at the beginning of 1998. Find the effective yield.

Step 1: Find two points that represent the informationafter 0 years the stove was worth $500

after 8 years the stove was worth $875

(0,500)

(8,875)

Step 2: Enter the two points on a list and find the exponential regression equation that fits the points.

The multiplier is about 1.07251.0725 – 1 =

0.0725

= 7.25%

PracticeFind the effective yield for a painting bought for $100,000 at the end of 1994 and sold for $200,000 at the end of 2004.

Homework

p.367 #29-35 odds,47,51