Lesson video

Hi, I'm Mr. Bond.

And in this lesson, we're going to learn how to recognise, sketch and interpret graphs of exponential functions.

The first thing that we need to understand about exponential functions is that they're written in the form Y is equal to K to the power of X.

And that these graphs differ, depending on the value of K.

Let's start with our first example.

We'll sketch the graph of Y is equal to three to the power X.

As we normally do when we sketch graphs, we could start by thinking about a table of values.

Here, we've got a table of values with some values of X from negative two to two, and we need to find the values of Y.

You might be able to work these out in your head, or you might want to use a calculator to find these values for Y.

If we then sketch these on coordinate axes, we'll get something that looks like this.

Now let's consider changing the value of K.

What about if K were equal to four? Okay, if we choose the same values for X, we could find the values of Y either in our heads or using a calculator, and we'd get this.

If we plotted these points on coordinate axes, we'd get this.

So the first thing I want us to think about is, what's the same and what's different? It's probably easier to compare them if we draw one graph on top of the other.

Here we have that image a little bit larger, and I still want us to think, what's the same and what's different? Hopefully, you've spotted that both graphs have the same Y intercept, zero one, and you may even have spotted this from the table of values.

Again, hopefully, you've spotted that as X increases, Y increases for both graphs.

And, Y is greater than zero for all values of X.

At no point do these values go below the X axis.

Let's look at some ways in which the graphs differ.

The rate of increase differs.

And also, four to the power X is greater than three to the power X when X is greater than zero, but four to the power X is less than three to the power X when X is less than zero, so on the left-hand side of the Y axis.

Now, we're going to consider exponential functions where the value of K is a decimal.

We'll start by considering the sketch of Y is equal to 0.

3 to the power X.

We'll use a table of values again.

This time, you will almost certainly want to use a calculator to help you find the values of Y.

You should get this.

And if we plot these points on coordinate axes, we'll get something that looks like this.

Now consider the sketch of Y is equal to 0.

4 to the power of X.

Again, we'll use a table of values.

Using a calculator will give us the following values for Y.

And then if we plot these points onto a coordinate axes, we'll get something that looks like this.

Again we're going to think, what's the same and what's different? Once again, it's useful to draw these graphs one on top of the other so that we can compare them.

Here are those graphs a little bit larger.

What did you notice that was the same, and what did you notice that was different? Just like our previous pair of graphs, they have the same Y intercept, zero one.

Are you starting to realise why this is the case? This time, as X increases, Y decreases.

However, Y is greater than zero for all values of X for both graphs.

Again, they don't go below the X axis.

Let's look at what's different.

The rate of decrease differs.

And 0.

4 to the power X is greater than 0.

3 to the power X when X is greater than zero.

But, 0.

4 to the X is less than 0.

3 to the X when X is less than 0.

What about if K were equal to zero? What about if K were less than zero? You can investigate this using online graphical software.

Here are some questions for you to try.

Pause the video to have a go, and resume the video when you've finished.

Here are the answers.

In this question, we had to form generalisations about exponential functions of the form Y is equal to K to the power X.

The first statement was that the graph intercepts the Y axis at the point zero one.

Hopefully, you noticed in all of the examples that we've looked at so far that this is always true.

And it's always true because anything to the power zero is always equal to one.

The next statement was the graph passes through the third quadrant.

Well you might remember me saying that for each of the graphs we've looked at, that we don't go below the X axis.

So this never will go through the third quadrant.

And lastly, as X increases, Y increases exponentially.

Well this is only sometimes true.

It's true for values of K greater than one, but not true for values of K between zero and one.

Here's another question for you to try.

Again, pause the video to complete the task and resume the video when you've finished.

Here are the answers.

So in this question, you simply had to sketch the graph, starting by completing the table of values and then plotting these values onto coordinate axes.

Here's a third question for you to try.

Sketch the graph of Y is equal to 0.

5 to the power of X.

Pause the video to complete the task, and resume the video when you've finished.

Here are the answers.

In this question, you had to sketch the graph of Y is equal to 0.

5 to power X.

You might have used a table of values for this.

Also, you needed to make sure that as X increased, Y decreases.

And here's today's final question.

Again, pause the video to complete your task and resume the video when you've finished.

Here are the answers.

This question was a great opportunity to compare several exponential functions on the same graph.

It makes it really clear that when the exponential is in the form Y is equal to K to the power X, these will always go through the Y axis at zero one.

That's all for this lesson.

Thanks for watching.