Learn About Graphs Of Exponential Functions

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

$y=2^x$

grows steeper in a positive direction as x values increase after plotting points as shown in the video.

The graph of

$y=-(2)^x$

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

$y=(\frac{1}{2})^x$

grows steeper at a slower rate as x values increase after plotting points, and is a reflection over the y-axis of $y=2^x$

Example of Graphs of Exponential Functions

Example 1

$y = 2 \times 5^{x}$

When $x = -2$ then $y = \dfrac{2}{25}$ ,

$x = -1$ then $y = \dfrac{2}{5}$ ,

$x = 0$ then $y = 2$ ,

and $x = 1$ then $y = 10$

The graph will be:

Example 2

$y = -3^{x+1}$

When $x = -2$ then $y = \dfrac{-1}{3}$ ,

$x = -1$ then $y = -1$ ,

$x = 0$ then $y = -3$ ,

and $x = 1$ then $y = -9$

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

$y = a \times b^x$

Something that should stand out is that $x$ is any exponent. And $a$ and $b$ are constant.

Let’s first look at functions that $a = 1$ .

So, we’ll solve for $b$ and $x$ .

For demonstration purposes, let’s have

$y = 2^x$

Let’s look at some specific ones.

When $x = -2$ then $y = \dfrac{1}{4}$ ,
$x = -1$ then $y = \dfrac{1}{2}$ ,
$x = 0$ then $y = 1$ ,
$x = 1$ then $y = 2$ ,
$x = 2$ then $y = 4$ ,
and $x = 3$ then $y = 8$ .

The thing I want to point out is when $x = 0$ then $y = 1$ .

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 $x$ -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 $y = 2^x$ .

Let’s look at the graph of $y = 3^x$ .

Let us have the same points here.

Let’s start with $x = 0$ then $y = 1$ .

When $x = -2$ then $y = \dfrac{1}{9}$ ,
$x = -1$ then $y = \dfrac{1}{3}$ ,
$x = 0$ then $y = 1$ ,
$x = 1$ then $y = 3$ ,
$x = 2$ then $y = 9$ ,
and $x = 3$ then $y = 27$ .

This one is very steep on the positive exponents and very low on the negative exponents.

But still won’t touch the $x$ -axis.

Let’s graph $y = -(2)^x$ .

Let me change the $x$ values a little bit.

Again let’s start with $x = 0$ . $y = -1$

When $x = 1$ then $y = -2$ ,
when $x = 2$ then $y = -4$ ,
and when $x = 3$ then $y = -8$ .

If you look at it, it’s a mirror image of our first example.

When $x = -1$ then $y = - \dfrac{1}{2}$ ,
when $x = -2$ then $y = - \dfrac{1}{4}$ ,
and $x = -3$ then $y = - \dfrac{1}{8}$ .

It will be very close to the $x$ -axis.

The negative will be reflected in the $x$ -axis.

Let’s look at another one.

Let’s have $y = (\dfrac{1}{2})^x$

So let’s start with $x = 0$ again, $y = 1$ because any number raised to the power of $0$ is $1$ .

When $x = 1$ then $y = \dfrac{1}{2}$ ,
when $x = 2$ then $y = \dfrac{1}{4}$ ,
and when $x = 3$ then $y = \dfrac{1}{8}$ .

Let’s go backward.

When $x = -1$ then $y = 2$ ,
when $x = -2$ then $y = 4$ ,
and when $x = -3$ then $y = 8$ .

Now let’s graph this.

Compared to the first example, this graph is reflected in the $y$ -axis.

Think about this.

We have a constant that’s greater than $1$ raised to an exponent.

It will constantly raise on one side if the exponent is greater than $0$ . 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 $1$ , you will have a line that will grow rapidly.

But if you have a base that’s less than $1$ but still positive, it’s going to decrease or decay.