Chapter 4: Exponential and Logarithmic Functions

4.1 Exponential Functions

Exponential functions are a fundamental class of functions in algebra, characterized by a constant base raised to a variable exponent. They are widely used in modeling growth and decay in natural and financial contexts.

Definition of the Exponential Function

• Exponential Function: A function of the form , where is a positive real number and .

• The base determines the rate and direction of growth or decay.

• The domain of is all real numbers, and the range is all positive real numbers.

Evaluating Exponential Functions

• To evaluate an exponential function, substitute the given value of into the formula and compute the result.

• Example: Evaluate at .

Solution:

Graphing Exponential Functions

To graph an exponential function, create a table of values for selected -values, plot the corresponding points, and connect them with a smooth curve. The graph will have a horizontal asymptote at .

• For , the function shows exponential growth.

• For , the function shows exponential decay.

• All exponential functions of this form pass through the point .

Transformations Involving Exponential Functions

Exponential function graphs can be shifted, reflected, or stretched/compressed using transformations:

• Horizontal Shifts: shifts the graph units to the right.

• Vertical Shifts: shifts the graph units up.

• Reflections: reflects the graph across the -axis.

• Horizontal Asymptote: The line is a horizontal asymptote for all basic exponential functions.

Example: The graph of is the graph of shifted 1 unit to the right.

The Natural Base

• The number is an important mathematical constant known as the natural base.

• Exponential functions with base are called natural exponential functions: .

• These functions are widely used in calculus, science, and finance.

Applications: Compound Interest

Exponential functions are used to model compound interest, where interest is earned on both the initial principal and accumulated interest.

• Compound Interest Formula (n times per year):

• = final amount

• = principal (initial amount)

• = annual interest rate (decimal)

• = number of compounding periods per year

• = number of years

• Continuous Compound Interest Formula:

• Interest is compounded continuously at rate for years.

Example: If , , , , then

Calculate to find the final amount.

Applications: Population Growth

• Exponential functions can model population growth, radioactive decay, and other real-world phenomena.

• Example: The population of a species increases according to , where is the initial population and is the growth rate.