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Exponential Growth
Exponential Decay
ExampleGraph the exponential function given by ( ) 3 .xf x
Solution
x y, or f(x)
01
–12
–23
13
1/39
1/927 4 3 2 1 0 1 2 3 4
2
1
1
2
3
4
5
6
7
8
9
10
Example
Graph the exponential function given by 1
( ) .3
xf x
Solution
x y, or f(x)
01–12–2–3
11/33
1/99
27 4 3 2 1 0 1 2 3 4
2
1
1
2
3
4
5
6
7
8
9
10
4 3 2 1 0 1 2 3 4
2
1
1
2
3
4
5
6
7
8
9
10
( ) 3xy f x 1
( )3
xy f x
From the previous two examples, we can make the following observations.A. For a > 1, the graph of f (x ) = ax increases from left to right. The greater the value of a, the steeper the curve, this is called exponential growth.B. For 0 < a < 1, the graph of f (x ) = ax decreases from left to right. For smaller values of a, the graph becomes steeper, this is called exponential decay.
D. All graphs of f (x) = ax have the x-axis as the asymptote.
E. If f (x ) = ax, with a > 0, a not 1, the domain of f is all real numbers, and the range of f is all positive real numbers.
, 1,
exponential growth
xy a a , 0 1,
exponential decay
xy a a
intercept (0,1)y
0
horizontial asymptote
y
C. All graphs of f (x ) = ax go through the y-intercept (0, 1).
3Consider the exponential function ( ) 4 5.
(a) Graph ( ) in the window [ 7,2] [ 7,7].
(b) Find the zero of ( ).
(c) Find the equation of the horizontal asymptote.
xf x
f x
f x
Example
Horizontal asymptote x = -5
Zero x = -1.839
Equations with x and y Interchanged
It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged x = ay .
Applications of Exponential Functions
Example Interest compounded annually.
The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula
(1 ) .tA P i
Example
Solution
Suppose that $60,000 is invested at 5% interest,
compounded annually.
a) Find a function for the amount in the account after t years.
b) Find the amount of money in the account at t = 6.
a) ( ) (1 )tA t P i = $60000(1 + 0.05)t
= $60000(1.05)t
b) A(6) = $60000(1.05)6 $80,405.74.
Linear vs Polynomial vs Exponential
