a Check if the base of the exponential expression is less than one.
B
b Find the value of P(0).
A
aGrowth or Decay: Exponential Decay
Decay Factor: 0.9 Graph:
B
bExplanation: See solution.
Practice makes perfect
a
The function P(x)=2.28(0.9)^x can be classified as an exponential decay function because P(x) decreases by the same factor, 0.9, as x increases. For an exponential decay function, the base of the exponential expression is called the decay factor, which is 0.9 for P(x).
P(x)=2.28(0.9)^x
Now that we have classified the function, we can graph the function showing the decrease in the number of pay phones in millions x years since 1999. To do so, we will first make a table of values.
x
2.28(0.9^x)
P(x)=2.28(0.9^x)
1
2.28(0.9)^1
≈ 2.05
2
2.28(0.9)^2
≈ 1.85
3
2.28(0.9)^3
≈ 1.66
4
2.28(0.9)^4
≈ 1.50
5
2.28(0.9)^5
≈ 1.35
Note that P(x) gives the number of payphones in millions.
Let's now plot and connect the points ( 1, 2.05), ( 2, 1.85), ( 3, 1.66), ( 4, 1.50), and ( 5, 1.35) with a smooth curve.
b To find the y-intercept, we need to substitute 0 for x in the function. Let's do it!
The function intercepts the y-axis at 2.28, which tells us that the number of pay phones in 1999 is equal to 2.28 million. As x increases, y-values approach the x-axis and never equal to zero. Therefore, the asymptote is the x-axis. As a result, this model foresees that the number of pay phones will never be zero.
