# 9.4 – Graphs of Exponential Functions

## Key Terms

• Horizontal Asymptote – A horizontal line that the graph of a function approaches but never intersects.
• A function has a horizontal asymptote at every y-value where it is undefined and near which the function’s values become very large positive or negative numbers.
• Think of an asymptote like an electric fence.  You can’t cross it!  You can’t even touch it!
• Range – The set of a function’s output values (y-values).
• y-intercept – A point where the graph of a function crosses the y-axis.
• A function has at most one y-intercept.
• The y-intercept of the line with the equation y = mx + b is the point (0, b).

## Review

Graph Behavior
• Increasing graphs slant or curve up from left to right
• Decreasing graphs slant or curve down from left to right
• All exponential graphs are curved, not straight.

## Notes

The Exponential Function’s Base
• The value of the base determines whether the graph increases or decreases from left to right.
• The base for an exponential function can never be a negative number.
• A negative value of b would make the function undefined for many values of x.

• If b > 1, then the graph increases

• If 0 < b < 1, then the graph decreases

Intercepts, Points, and Asymptotes
• The graph of $F(x)=b^{x}$ will always have the y-intercept of (0, 1).
• The graph of $F(x)=b^{x}$ will always contain the point (1, b), where b is the base of the expression.
• The x-axis is a horizontal asymptote: y = 0
• Horizontal Asymptote: y = 0

• Point (1, b) and y-intercept (0, 1)

• Example

Domain and Range
• The domain of a function is all the input values it will accept.
• Domain ends with -in for input.

• The range of a function is all the output values it will return.
• Range starts with r for return.

• For $a+b^{x}$, the ranges is all positive real numbers greater than a.

(1, b) and (1, ab)
• You can write $F(x)=b^{x}$ in the form $F(x)=a\bullet b^{x}$ by making a = 1.
• The rule that the y-intercept = (0, a) and the other known point = (1, ab) is true for both forms of exponential functions.
• The difference between the graphs below is that the y-intercept of the function $F(x)=a\bullet b^{x}$ is the point (0, a), and its other known point is (1, ab).

Vertical Translations (Shifting Up and Down)
• Shifting affects the range, but not the domain.
• The domain is always “All real numbers.”
• Range will always be greater than the shift.
• The range of the top graph is y > 5
• The range of the bottom graph is y > -3

Horizontal Translations (Shifting Left and Right)
• Horizontal shifts do not affect domain or range.
• Horizontal shifts appear when a number is added to or subtracted from the input variable.
• Ex. $G(x)=2^{x-1}$ is shifted 1 unit to the right.
• Ex. $H(x)=2^{x+6}$ is shifted 6 units to the left.

Vertical Stretching and Compressing
• Multiplying a function by a negative number flips the graph across the x-axis.
• When the function is flipped across the x-axis, the inequality sign of the range is reversed.

• To horizontally stretch an exponential function, you multiply the input variable by a number between zero and 1 (a positive fraction).
• To horizontally compress it, you multiply the input variable by a number greater than 1.

Translations (All Together)

Exponential Functions and Graphs: Chart

## Examples

 Ex 1. The domain of the function given below is the set of all real numbers. $f(x)=(\frac{1}{7})^{x}$ True: Any number can be an input. Ex 2. What are the properties of $F(x)=8\bullet 3^{x}$: The domain is all positive real numbers The y-intercept is (0, 8) The graph is increasing Ex 3. The range of the function given below is the set of all positive real numbers greater than 9. Answer: $F(x)=9+3^{x}$ Ex 4. The following facts are true for the graph of the function below $F(x)=(\frac{2}{7})^{x}$ The range of F(x) is y > 0 The y-intercept is (0, 1) It is decreasing Ex 5. The following facts are true for the graph of the function below. $F(x)=7\bullet 5^{x}$ The domain of F(x) is all real numbers The y-intercept is (0, 7) It is increasing Ex 6. The graph below could be the graph of the following exponential function: Answer: $F(x)=3\bullet (1.2)^{x}$ Ex 7. The graph below could be the graph of the following exponential function: Answer: $F(x)=2\bullet (0.5)^{x}$