1. Graphing Exponential Functions
Section 10.3
Graphing Exponential Functions

2. Graphing Exponential Functions with b > 1
Example
Graph by hand.
Solution
List input–output pairs (see table)
Input increases by 1 and output multiplies by 2
Plot these points (see next slide)

3. Graphing Exponential Functions with b > 1
Solution Continued
Use graphing calculator to verify

4. Graphing Exponential Functions with 0< b < 1
Example
Graph by hand.
Solution
List input–output pairs (see table)
For example
(–1, 8) is a solution
x increases by 1, y is multiplied by ½

5. Graphing Exponential Functions with 0 < b < 1
Solution Continued

6. Property Illustration
Base Multiplier Property; Increase or Decreasing Property
Base Multiplier Property
Property
For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b.
For the function , as the value of x increases by 1, the value of y is multiplied by 3
For the function , as the value of x
increases by 1, the value of y is multiplied by 3/4
Illustration

7. Increase or Decrease Property
Base Multiplier Property
Property
Let , where a > 0. Then
If b > 1, then the function f is increasing. We say that the function grows exponentially (left).
If 0 < b < 1, then the function f is decreasing. We say that the function decays exponentially (right).

8. y-intercept of an Exponential Function
Intercepts
Property
For an exponential function of the form
the y-intercept is (0, a).
The function , the y-intercept is (0, 5)
The function , the y-intercept is (0, 4)
Illustration

9. Intercepts and Graph of an Exponential Function
Warning
Exponential function of the form , the y- intercept is not (0, b). By writing , we see that the y-intercept is (0, 1).
For example, for , the y-intercept is (0, 1).
Let
1. Find the y-intercept of f.
Example

10. Intercepts and Graph of an Exponential Function
Solution
is of the form ,
We know that the y-intercept is (0, a), or (0, 6).
2. Find the x-intercept of f.
By base multiplier property, x increases by 1, y value multiplies by ½
Example
Solution

11. Intercepts and Graph of an Exponential Function
Solution Continued
No number of halvings will result in zero
As x grows large, y gets closer to the x-axis
Called horizontal asymptote
3. Graph f by hand.
Example

12. Plot solutions from the table
Intercepts and Graph of an Exponential Function
Intercepts
Solution
Plot solutions from the table
Verify on graphing calculator

13. Sketch and compare the graphs of .
Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
Sketch and compare the graphs of
Solution
Input–output pairs are listed in the table
g is a reflection of f across the x-axis

14. Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
Find the domain and range of f.
5(3)x is defined for any real number x
Domain: All real numbers
Range: All positive real numbers
Solution

15. Graphs of Functions of the Form y=abx and y= –abx
Reflection Property
Example
Find the domain and range of g.
– 5(3)x is defined for any real number x
Domain: All real numbers
Range: All negative real numbers
Solution

16. Reflection Property Property Illustration
The graphs of are reflections of each other across the x-axis.
For all exponential functions the x-axis is a horizontal asymptote
The range of an exponential function f(x) = abx is the set of all positive real numbers if a > 0, and the range is the set of all negative real number if a< 0.
Illustration

17. Reflection Property Continued
b > 1 (left) and 0 < b < 1 (right)

18. Finding Values of a Function from Its Graph
Reflection Property
Example
The graph of an exponential function f is shown.
Find f(2).
Blue arrow shows input of x = 2 leads to an output y = 8
f(2) = 8
Solution

19. Finding Values of a Function from Its Graph
Reflection Property
Example
2. Find x when f(x) = 2.
Red arrow shows output of y = 2 leads to an input x = -2
x = –2 when f(x) = 2
Solution

20. Finding Values of a Function from Its Graph
Reflection Property
Example
3. Find x when f(x) = 0.
Graphs of exponential functions get close to zero, but never reaches x-axis
No value of x where f(x) = 0
Solution