PEGEF - Characteristics of Exponential Functions (Lesson)

Characteristics Exponential Functions

After graphing exponential functions, it is important we are able to recognize the key features. Below is a table outlining the key features you should be able to identify and state.

Feature	Definition	Picture and Example
Domain	- The set of all possible values to input for x. - For all exponential functions the domain is all real numbers because you are allowed to put any value in for x. - To represent all real numbers, we say: $LaTeX: x\in\left(-\infty,\infty\right)$ $x\in\left(-\infty,\infty\right)$
Asymptote	- The line the curve approaches but never touches. - For exponential functions, the asymptote is always y = k.
Range	- The set of all possible output values. - The range for exponential functions will either be all values above the asymptote or below the asymptote. The range will never include the asymptote because the curve never touches that value!
y-intercept	- The point where the curve crosses the y-axis. - Set x=0 and solve for y.
Intervals of increase or decrease	- The set of all x-values for which the function is increasing or decreasing.
End Behavior	- A statement that tells us what the "ends" of the curve are doing

Watch this video to practice finding these characteristics.

Average Rate of Change

Average rate of change graph In earlier lessons, we have learned that the rate of change is the slope between two points.

Given two points, (2, 10) and (1, 4), calculate the slope between them: $LaTeX: m=\frac{10-4}{2-1}=\frac{6}{1}=6$ $m=\frac{10-4}{2-1}=\frac{6}{1}=6$

If you were to draw a line between the points, the slope of the line would be 6.

If you are not given both the x- and y-parts of the points, you may need to plug in the x-part to find the y-part.

Calculate the rate of change for the function f(x) = 2^x + 3 over the interval $LaTeX: 1\le x\le3$ $1\le x\le3$ .

You are given the x-parts of each point.

So first you should find the y-parts.

f(1) = 2¹ +3 = 2 + 3 = 5 so f(1) = 5 is on the curve. f(3) = 2³ + 3 = 8 + 3 = 11 so f(3) = 11 so (3, 11) is on the curve.

Now calculate the slope between those points: $LaTeX: m=\frac{11-5}{3-1}=\frac{6}{2}=3$ $m=\frac{11-5}{3-1}=\frac{6}{2}=3$ .

So the rate of change over that interval is 3.

When given a table of values, you can recognize an exponential function by the rate of change. The output values in an exponential function increase or decrease by a factor.

f(x) = 3^x
x	f(x)
0	1
1	3
2	9
3	27
4	81

Notice that each of the y-values is tripled, or increased by a factor of 3!

Characteristics of Exponential Functions Practice

What do you know? List the equation, asymptote, y-intercept, domain, range, interval of increase, and interval of decrease of each graph.

Calculate the average rate of change.

1. Calculate the rate of change for the function f(x) = 3^x + 1 over the interval $LaTeX: 2\le x\le4$ $2\le x\le4$ .

2. Calculate the rate of change for the function f(x) = -2(1/2)^x over the interval $LaTeX: 0\le x\le1$ $0\le x\le1$ .

3. Calculate the rate of change for the function f(x) = 4(2)^x + 3over the interval $LaTeX: 1\le x\le3$ $1\le x\le3$ .

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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