3. Exponential Growth

4. Exponential Decay

5. Graph the exponential function given by
Example
Graph the exponential function given by
Solution x
y, or f(x) 1 –1 2 –2 3
1/3 9
1/9 27

6. Graph the exponential function given by
Example
Graph the exponential function given by
Solution x
y, or f(x) 1 –1 2 –2 –3
1/3 3
1/9 9 27

8. 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.

9. C. All graphs of f (x ) = ax go through the y-intercept (0, 1).
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.

10. Example
Zero x =
Horizontal asymptote x = -5

11. 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 .

15. 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

16. Example
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.
Solution
= $60000( )t
= $60000(1.05)t
b) A(6) = $60000(1.05)6

18. Linear vs Polynomial vs Exponential