In this video, we are going to look at graphs of exponential functions. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.
Example of an Exponential Function Graph
The graph of
grows steeper in a positive direction as x values increase after plotting points as shown in the video.
The graph of
grows steeper in a negative direction as x values increase after plotting points, and is a reflection over the x-axis of the first graph.
The graph of
grows steeper at a slower rate as x values increase after plotting points, and is a reflection over the y-axis of
Example of Graphs of Exponential Functions
Example 1
When then ,
then ,
then ,
and then
The graph will be:
Example 2
When then ,
then ,
then ,
and then
The graph will be:
Video-Lesson Transcript
Let’s go over graphs of exponential functions.
First of all, what is an exponential function?
Exponential function has a form of
Something that should stand out is that is any exponent. And and are constant.
Let’s first look at functions that .
So, we’ll solve for and .
For demonstration purposes, let’s have
Let’s look at some specific ones.
When then ,
then ,
then ,
then ,
then ,
and then .
The thing I want to point out is when then .
When we graph the positive exponent, it will not be a straight line. It increases at a much higher rate. It even curves up drastically.
When we graph the negative exponents, it will curve but will not touch the -axis.
Because even if the exponent is negative, it just changes to a fraction and not a negative number.
So this is the basic graph of an exponential fraction.
This is .
Let’s look at the graph of .
Let us have the same points here.
Let’s start with then .
When then ,
then ,
then ,
then ,
then ,
and then .
This one is very steep on the positive exponents and very low on the negative exponents.
But still won’t touch the -axis.
Let’s graph .
Let me change the values a little bit.
Again let’s start with .
When then ,
when then ,
and when then .
If you look at it, it’s a mirror image of our first example.
When then ,
when then ,
and then .
It will be very close to the -axis.
The negative will be reflected in the -axis.
Let’s look at another one.
Let’s have
So let’s start with again, because any number raised to the power of is .
When then ,
when then ,
and when then .
Let’s go backward.
When then ,
when then ,
and when then .
Now let’s graph this.
Compared to the first example, this graph is reflected in the -axis.
Think about this.
We have a constant that’s greater than raised to an exponent.
It will constantly raise on one side if the exponent is greater than . Since you’re multiplying it by itself over and over again.
Whereas with the other one, we have a fraction. Which is less than one and raised to an exponent.
When multiplied by itself, it becomes smaller.
If you have a base that’s greater than , you will have a line that will grow rapidly.
But if you have a base that’s less than but still positive, it’s going to decrease or decay.