Exponential Functions Exponential Functions and Their Graphs Irrational
- Slides: 31
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: n Given a graph, find the value of b:
Example 3: n Given a graph, find the value of b:
Increasing and Decreasing Functions
One-to-One Exponential Functions
Compound Interest
Example 4: n 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: n 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 n Make a table and plot points:
Exponential Functions Horizontal asymptote n Function increases n y-intercept (0, 1) n Domain all real numbers n Range: y > 0 n
Translations For k>0 n y = f(x) + k n y = f(x) – k n y = f(x - k) n y = f(x + k) Up k units Down k units Right k units Left k units
Example 6: n On one set of axes, graph
Example 6: n On one set of axes, graph Up 3
Example 7: n On one set of axes, graph Right 3
Non-Rigid Transformations n 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.
- Exponential functions and their graphs
- Polynomial function form
- Unit 3 lesson 3 rational functions and their graphs
- Quadratic functions and their graphs
- Removable and nonremovable discontinuity
- Chapter 1 functions and their graphs
- Sketch the graph of the following rational function
- Common functions and their graphs
- Number of terms example
- Polynomial functions and their graphs
- Polynomial functions and their graphs
- Chapter 2 functions and their graphs answers
- Rational functions and their graphs
- Lesson 3 rational functions and their graphs
- Investigating graphs of polynomial functions
- Investigating graphs of functions for their properties
- What is state graph in software testing
- Graphs that enlighten and graphs that deceive
- Irrational exponential function
- Graphs that compare distance and time are called
- Integration of rational and irrational functions
- Examples of linear quadratic and exponential functions
- Linear quadratic and exponential graphs
- Exponential function formula
- 9-1 quadratic graphs and their properties
- Expander graphs and their applications
- 3-1 inequalities and their graphs
- Inequalities and their graphs 3-1
- 3-1 inequalities and their graphs
- 3-1 inequalities and their graphs
- Rational sum
- Rational numbers