1. 4.2 Exponential Functions
Additional Properties of Exponents
ax is a unique real number.
ab = ac if and only if b = c.
If a > 1 and m < n, then am < an.
If 0 < a < 1 and m < n, then am > an.
If a > 0, a  1, then
f (x) = ax
is the exponential function with base a.

2. 4.2 Graphs of Exponential Functions
Example Graph Determine the domain
and range of f.
Solution

3. 4.2 Graph of f (x) = ax, a > 1

4. 4.2 Graph of f (x) = ax, 0 < a < 1

5. 4.2 Comparing Graphs Example Explain why the graph of
is a reflection across the y-axis of the graph of
Analytic Solution
Show that g(x) = f (–x).

6. 4.2 Comparing Graphs Graphical Solution
The graph below indicates that g(x) is a reflection
across the y-axis of f (x).

7. 4.2 Using Translations to Graph an Exponential Function
Example Explain how the graph of
is obtained from the graph of
Solution

8. 4.2 Exponential Equations (Type I)
Example Solve
Solution
Write with the same base.
Set exponents equal and solve.

9. 4.2 The Natural Number e
Named after Swiss mathematician Leonhard Euler
e involves the expression
e is an irrational number
Since e is an important
base, calculators are
programmed to find
powers of e.

10. 4.2 Compound Interest Recall simple earned interest where
P is the principal (or initial investment),
r is the annual interest rate, and
t is the number of years.
If A is the final balance at the end of each year, then

11. 4.2 Compound Interest Formula
Example Suppose that $1000 is invested at an annual
rate of 8%, compounded quarterly. Find the total amount in
the account after 10 years if no withdrawals are made.
Solution .
Suppose that a principal of P dollars is invested at an annual interest rate r (as a decimal), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

12. 4.2 Continuous Compounding Formula
Example Suppose $5000 is deposited in an account
paying 8% compounded continuously for 5 years. Find the
total amount on deposit at the end of 5 years.
Solution
If P dollars is deposited at a rate of interest r compounded continuously for t years, the final amount A in dollars on deposit is

13. 4.2 Modeling the Risk of Alzheimer’s Disease
Example The chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by
where r is the risk (in decimal form) at age 55 and x
is the number of years greater than 55. What is the
risk at age 75?
Solution