College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.2 Exponential Functions

Concepts 1. Graph Exponential Functions 2. Evaluate the Exponential Function Base e 3. Use Exponential Functions to Compute Compound Interest 4. Use Exponential Functions in Applications

Definition of an Exponential Function Let b be a constant real number such that b > 0 and b ≠ 1. Then for any real number x, a function of the form is called an exponential function. Previously: Now:

Example 1: Graph the function.

Example 2: Graph the function.

Example 3: Graph the function.

Example 3 continued: Use transformations to graph the function. Parent function: If a < 0 reflect across the x-axis. Shrink vertically if 0 < |a| < 1. Stretch vertically if |a| > 1. If k > 0, shift upward. If k < 0, shift downward. If h > 0, shift to the right. If h < 0, shift to the left.

Concepts 1. Graph Exponential Functions 2. Evaluate the Exponential Function Base e 3. Use Exponential Functions to Compute Compound Interest 4. Use Exponential Functions in Applications

Evaluate the Exponential Function Base e e is a universal constant (like the number  ) and an irrational number. e ≈

Examples 4 – 7: Evaluate. Round to 4 decimal places

Example 8: Graph the function.

Concepts 1. Graph Exponential Functions 2. Evaluate the Exponential Function Base e 3. Use Exponential Functions to Compute Compound Interest 4. Use Exponential Functions in Applications

Use Exponential Functions to Compute Compound Interest Suppose that P dollars in principle is invested (or borrowed) at an annual interest rate r for t years. Then: Amount of simple interest I (in $) Amount A (in $) in the account after t years and n compounding periods per year. Amount A (in $) in the account after t years under continuous compounding.

Example 9: Suppose that $15,000 is invested with 2.5% interest under the following compounding options. Determine the amount in the account at the end of 7 years for each option. a) Compounded annually

Example 9 continued: b) Compounded quarterly c) Compounded continuously

Concepts 1. Graph Exponential Functions 2. Evaluate the Exponential Function Base e 3. Use Exponential Functions to Compute Compound Interest 4. Use Exponential Functions in Applications

Example 10: Weapon-grade plutonium is composed of approximately 93% plutonium-239 (Pu-239). The half-life of Pu-239 is 24,000 years. In a sample originally containing 0.5 kilograms, the amount left after t years is given by Evaluate the function for the given values of t and interpret the meaning in context.

Example 10 continued: a) b)

Example 11: A 2010 estimate of the population of Mexico is 111 million people with a projected growth rate of 0.994% per year. At this rate, the population P(t) (in millions) can be approximated by where t is the time in years since 2010.

Example 11 continued: