Exponential Functions

Exponential Functions and Their Graphs

Irrational Exponents If b is a positive number and x is a real number, the expression bx always represents a positive number. It is also true that the familiar properties of exponents hold for irrational exponents.

Example 1: Use properties of exponents to simplify

Example 1: Use properties of exponents to simplify

Example 1: Use properties of exponents to simplify

Example 1: Use properties of exponents to simplify

Exponential Functions An exponential function with base b is defined by the equation x is a real number. The domain of any exponential function is the interval The range is the interval

Graphing Exponential Functions

Graphing Exponential Functions

Example 2: Let’s make a table and plot points to graph.

Example 2:

Example 2:

Properties: Exponential Functions

Example 3: Given a graph, find the value of b:

Example 3: Given a graph, find the value of b:

Increasing and Decreasing Functions

One-to-One Exponential Functions

Compound Interest

Example 4: The parents of a newborn child invest $8,000 in a plan that earns 9% interest, compounded quarterly. If the money is left untouched, how much will the child have in the account in 55 years?

Future value of account Example 4 Solution: Using the compound interest formula: Future value of account in 55 years

Base e Exponential Functions Sometimes called the natural base, often appears as the base of an exponential functions. It is the base of the continuous compound interest formula:

Example 5: If the parents of the newborn child in Example 4 had invested $8,000 at an annual rate of 9%, compounded continuously, how much would the child have in the account in 55 years?

Future value of account Example 5 Solution: Future value of account in 55 years

Graphing Make a table and plot points:

Exponential Functions Horizontal asymptote Function increases y-intercept (0,1) Domain all real numbers Range: y > 0

Translations Up k units Down k units Right k units Left k units For k>0 y = f(x) + k y = f(x) – k y = f(x - k) y = f(x + k) Up k units Down k units Right k units Left k units

Example 6: On one set of axes, graph

Example 6: On one set of axes, graph Up 3

Example 7: On one set of axes, graph Right 3

Non-Rigid Transformations Exponential Functions with the form f(x)=kbx and f(x)=bkx are vertical and horizontal stretchings of the graph f(x)=bx. Use a graphing calculator to graph these functions.