1. Precalculus Essentials
Fifth Edition
Chapter 3
Exponential and Logarithmic Functions
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2. 3.1 Exponential Functions

3. Objectives Evaluate exponential functions.
Graph exponential functions.
Evaluate functions with base e.
Use compound interest formulas.

4. Definition of the Exponential Function
The exponential function f with base b is defined by
where b is a positive constant other than 1 (b > 0 and b ≠ 1) and x is any real number.

5. Example: Evaluating an Exponential Function
We substitute 3 for x and evaluate the function. Solution:
After 3 hours at a shopping mall, the average amount spent is $160.

6. Example: Graphing an Exponential Function
Solution:
We set up a table of coordinates, then plot these points, connecting them with a smooth, continuous curve.

7. Example: Transformations Involving Exponential Functions (1 of 2)
We’ve identified three points and the asymptote.

8. Example: Transformations Involving Exponential Functions (2 of 2)
The graph will shift 1 unit to the right. Add 1 to each x-coordinate.

9. Characteristics of Exponential Functions of the Form f(x) = b to the power x

10. The Natural Base e
approaches as n gets larger and larger. As n → ∞, the approximate value of e to nine decimal places is

11. Example: Evaluating Functions with Base e
Solution:
Because 2017 is 39 years after 1978, we substitute 39 for x in the given function.
This indicates that the gray wolf population in the Western Great Lakes in the year 2017 is projected to be approximately wolves.

12. Formulas for Compound Interest
After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:
For n compounding periods per year:
For continuous compounding:

13. Example 1: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to quarterly compounding.
Solution:
We will use the formula for n compounding periods per year, with n = 4.
The balance in the account after 5 years subject to quarterly compounding will be $14,

14. Example 2: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to continuous compounding.
Solution:
We will use the formula for continuous compounding.
The balance in the account after 5 years subject to continuous compounding will be $14,