1. Graphing Exponential Functions
Section 4.3
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)
Section 4.3
Slide 2

3. Graphing Exponential Functions with b > 1
Solution Continued
Use graphing calculator to verify
Section 4.3
Slide 3

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
Section 4.3
Slide 4

5. Graphing Exponential Functions with 0< b < 1
Solution Continued
Section 4.3
Slide 5

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
Section 4.3
Slide 6

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).
Section 4.3
Slide 7

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
Section 4.3
Slide 8

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
Section 4.3
Slide 9

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
Section 4.3
Slide 10

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
Section 4.3
Slide 11

12. Plot solutions from the table
Intercepts and Graph of an Exponential Function
Intercepts
Solution
Plot solutions from the table
Verify on graphing calculator
Section 4.3
Slide 12

13. 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
Section 4.3
Slide 13

14. 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
Section 4.3
Slide 14

15. 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
Section 4.3
Slide 15