In this video, we are going to look at graphs of exponential functions.

For example:

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

## 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.

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