320 likes | 626 Views
Exponential Functions. Exponential Functions and Their Graphs. Irrational Exponents. If b is a positive number and x is a real number, the expression b x always represents a positive number. It is also true that the familiar properties of exponents hold for irrational exponents . Example 1:.
E N D
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
Example 2: Let’s make a table and plot points to graph.
Properties: Exponential Functions
Example 3: • Given a graph, find the value of b:
Example 3: • Given a graph, find the value of b:
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?
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?
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 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.