exponential functions

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 2x 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 3 8 2 4 BASE 1 2 0 1 Recall what a negative exponent means: -1 1/2 -2 1/4 -3 1/8

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  - 

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

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.

Characteristics of Exponential Functions

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.

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.

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

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

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:

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. Reflect @ x-axis Vertical stretch 3 Vertical shift down 1 Vertical shift up 3

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