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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