1. exponential functions

2. Let’s examine exponential functions
Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.
Let’s look at the graph of this function by plotting some points.
x x 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
BASE
Recall what a negative exponent means: /2 /4 /8

3. Compare the graphs 2x, 3x , and 4x
Characteristics about the Graph of an Exponential Function where a > 1
1. Domain is all real numbers
2. Range is positive real numbers
3. There are no x intercepts because there is no x value that you can put in the function to make it = 0
Can you see the horizontal asymptote for these functions?
What is the x intercept of these exponential functions?
What is the range of an exponential function?
What is the domain of an exponential function?
What is the y intercept of these exponential functions?
Are these exponential functions increasing or decreasing?
4. The y intercept is always (0,1) because a 0 = 1
5. The graph is always increasing
6. The x-axis (where y = 0) is a horizontal asymptote for x  - 

4. All of the transformations that you learned apply to all functions, so what would the graph of look like?
up 3
right 2
down 1
Reflected over x axis
up 1

5. Reflected about y-axis
This equation could be rewritten in a different form:
So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.
There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.

6. Characteristics of Exponential Functions

7. A function that can be expressed in the form
and is positive, is called an Exponential Function.
Exponential Functions with positive values of x are increasing, one-to-one functions.
The parent form of the graph has a y-intercept at (0,1) and passes through (1,b).
The value of b determines the steepness of the curve.
There is no local extrema.

8. More Characteristics of
The domain is
The range is
End Behavior: As
The y-intercept is
The horizontal asymptote is
There is no x-intercept.
This is a continuous function.
It is concave up.

9. How would you graph How would you graph Domain: Range: Y-intercept:
Horizontal
Asymptote:
Inc/dec?
increasing
Concavity? up
How would you graph
Domain:
Range:
Y-intercept:
Horizontal
Asymptote:
Inc/dec?
increasing
Concavity? up

10. Recall that if then the graph of is a reflection of about the y-axis.
Thus, if then
Domain:
Range:
Y-intercept:
Horizontal
Asymptote:
Concavity? up

11. Notice that the reflection is decreasing, so the end behavior is:
How would you graph
Is this graph increasing
or decreasing?
Decreasing.
Notice that the reflection is decreasing, so the end behavior is:

12. Transformations
Exponential graphs, like other functions we have studied, can be dilated,
reflected and translated.
It is important to maintain the same base as you analyze the transformations.
x-axis
Vertical stretch 3
Vertical shift down 1
Vertical shift up 3

13. More Transformations Reflect about the x-axis. Vertical shrink ½ .
Horizontal shift left 2.
Horizontal shift right 1.
Vertical shift up 1.
Vertical shift down 3.
Domain:
Domain:
Range:
Range:
Horizontal
Asymptote:
Horizontal
Asymptote:
Y-intercept:
Y-intercept:
Inc/dec?
decreasing
Inc/dec?
increasing
Concavity?
down
Concavity? up