Lesson video

Hello, Mr. Robson here.

Welcome to maths.

Lovely to see you again.

Key features of exponential graphs today.

Well, exponential graphs are awesome.

You're awesome.

This lesson should be awesome.

Let's go.

Our learning outcome is that we'll be able to identify the key features of an exponential graph.

Exponential is one of the key words we'll hear throughout our lesson.

The general form for an exponential equation is y = ab^x where a is the coefficient, b is the base, and x is the exponent.

You'll also hear the word asymptote frequently.

An asymptote is a line that a curve approaches but never touches.

For example, for the graph y = 1/x, the x and y axes are both asymptotes.

Two parts to our learning today.

And the first part is all about asymptotes.

Graphs in the form y = b^x form an exponential curve with rapidly increasing y values.

This is true when the value of b is greater than 1.

So in these examples, y = 2^x, y = 3^x, and y = 4^x.

If we look at where our table of values for y = 4^x came from, you'll understand why the values are so rapidly increasing.

It's because the exponent is a variable.

When x = 3, y = 4^3.

So four has been repeatedly multiplied three times and the number grows really quickly.

Graphs in the form y = b^x can be identified by their key features.

If I show you the tables of values for y = 2^x, y = 3^x, and y = 4^x, you'll spot a key feature.

Right there.

Have you noticed something? Well done.

They have a common point, (0, 1).

That's because in the form y = b^x, when x = 0, b^0 is 1.

No matter how large we make that b value, b^0 will always be 1.

So graphs of the form y = b^x will intercept the y-axis at (0, 1).

I said key features, not just feature.

Let's see if we can spot another key feature in these tables of values.

Can you see it? When x equals 1, the y value is the base.

For y = 2^x, oh look, x = 1, y = 2, y = 3^x, x = 1, y = 3.

This is because when x = 1, b^x is b^1, which is of course, b.

Let's check you've got all that so far.

Select the statements that are true for graphs of the form y = b^x when b > 1.

Four statements to choose from.

Which ones are true? Welcome back.

Hopefully you said a is true.

The y values increase rapidly as the x values increase.

Hopefully you said b is not true.

The y values don't decrease rapidly as x values increase.

Hopefully you said they have a coordinate pair (0, 1) and a coordinate pair (1, b).

We can see those two key features if we graph this, y = b^x when b > 1, we'll have a graph that looks like that with a y intercept at (0, 1) and a coordinate (1, b).

When x is negative, graphs in the form y = b^x have y values below 1.

Again, this is when b has a value greater than 1.

So why do we get y values less than 1 when x is negative? Let's have a look at when x = -1.

We substitute it into the equation and y = 2^(-1).

You remember from your learning on number that 2^(-1) is the same as 1/2^1.

So it's a half.

Another example, when x = -2, a similar thing happens.

y = 2^(-2).

That's the same as 1/2^2.

So y has a value of a quarter when x = -2.

Will that y value ever reach zero? If I kept going with this table of values and made the negative x value really large indeed, will the y value ever reach zero? Let's have a look.

When x = -100, y = 2^x would be 2^(-100), which would be 1/2^100.

2^100 is 1.

267 and so on times 10^30.

It's an enormous number.

So 1/2^100 is a really small fraction.

But the fractions will get smaller and smaller and smaller, but they'll never reach zero.

There's no negative x value you can insert that would give you a y value of zero.

So they'd never reach zero.

Because those y values will never reach zero, they'll tend towards it, but they'll never reach it, we can say that the x-axis is an asymptote to graphs of this form.

Graphs in the form y = b^x have an asymptote at the x-axis.

That's when b is greater than 1.

Whilst there are infinitely many horizontal lines that this curve will never touch, we call the x-axis the asymptote because it is the line that the curve continually approaches, but never touches.

Graphs in the form y = b^x, where the base is between 0 and 1, also have an asymptote at the x-axis, but they have decreasing y values.

Look at this example.

y = (1/2)^x.

The y values are decreasing as the x values increase.

No matter how far we extend that x value, so if I went off this table of values and said, What about when x equals 7? Well, y would equal a half to the power of 7.

That's 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 which gives us 1/128, which is indeed a small fraction.

And those fractions will get smaller and smaller and smaller.

But you know what I'm going to say? They will never reach zero.

So once again, the x axis will be an asymptote.

Indices explain this for all cases in this form.

y = (c/d)^x is the same as (c^x)/(d^x) And because c is less than d, the fraction (c^x)/(d^x) will get smaller and smaller as x increases but it will never reach zero.

Graphs in the form y = b^x, where 0 < b < 1, also have a y-intercept at (0, 1).

There.

Well, of course they do.

If x = 0, (1/2)^0 is 1.

We could substitute any b value in there that's between 0 and 1.

What if it was y = (1/4)^x? (1/4)^0 would also be 1.

Graphs of this form will always have a y-intercept at (0, 1).

Quick check you've got all that.

Select the statements that are true for graphs of the form y = b^x when that b value is greater than 0 and less than 1.

Three statements to choose from.

Pick the ones that are true.

Pause and do that now.

Welcome back.

I hope you said a is true.

On this occasion, the y values do decrease as the x values increase.

They don't, however, have a coordinate pair (-1, b), but they do have a coordinate pair (0, 1).

That'll be a y-intercept.

Let's look at this graphically.

That's a graph of the form y = b^x when b is greater than 0, less than 1.

It'll intercept the y-axis at (0, 1) and it'll have the coordinate (-1, b^(-1)).

Those key features are important.

You might want to pause and sketch that graph down.

Graphs of the form y = -b^x also have an asymptote at the x-axis.

Look at the difference between these two graphs.

y = 2^x, y = -2^x.

Can you see that y = -2^x is a reflection of y = 2^x in the x-axis? Look at the table of values.

Same y values, just they're negative.

Well, if it's a reflection in the x-axis, and the x-axis is an asymptote of y = 2^x, then it must also be an asymptote of y = -2^x and indeed, it is.

You'll find that graphs of the form y = -b^x, whatever that value of b, will have an asymptote at the x-axis.

They'll have other key features too.

Graphs of the form y = -b^x also have a common point, (0, -1).

And they'll have a point at (1, -b).

But of course they do.

When x = 1, y = -b^1, which is just -b.

Quick check you've got that.

Select the statements that are true for graphs of the form y = -b^x.

Four statements there.

Some of them are true.

Pick them out.

Pause and do that now.

Welcome back.

Let's see how we did.

Hopefully you said a is not true.

They do not intercept the x-axis.

Indeed, the x-axis is an asymptote.

B was true.

They intercept the y-axis.

They have a coordinate pair (0, -1).

In fact, that's exactly where they intercept the y-axis.

They do not have a coordinate pair (1, b).

If we look at these key features on a graph, that's a graph of the form y = -b^x.

The x-axis is an asymptote.

They have a coordinate pair (0, -1).

That is indeed the y-intercept.

And they will have a coordinate at (1, -b), not (1, b), (1, -b).

Why is that? You could reason this out.

When x = -1, y must equal -b^1, which is just -b.

Does this mean that all exponential graphs have an asymptote at the x-axis? It looks like it.

All of the forms we've seen so far do.

Well, it is true for these three forms, but these are not the only forms that exponential graphs can take.

Consider this graph, y = 2^x + 5.

When I populate it as a table of values, you'll notice something's changed.

In the positive x direction, has anything really changed? As x increases, the y values will continue to increase infinitely.

We've seen lots of cases so far where that's true.

But as the x value decreases, what will happen to those y values? And this is where this case is different.

What if we extended that range of x values to include x = -10? Well, y would equal 2^(-10) + 5.

That's going to be 1/(2^10) + 5, which is 5 and 1/1024.

The fraction part of the mixed number tends towards but never reaches zero.

Therefore, the y value never reaches five as we extend in the negative x direction.

If we look at this graph after plotting you'll see this as an asymptote at y = 5.

The x-axis is no longer the asymptote.

Our line approaches but never reaches the horizontal line y = 5.

Quick check you've got this.

I'd like to complete this table of values, plot the graph and find the asymptote.

That's for the equation y = 2^x - 3.

Pause.

Give this a go.

Welcome back.

Hopefully, your table of values looked like so.

And then your graph looked just like that.

Can you see it? The asymptote is now at y = -3.

And what was different about this graph versus ones we've seen previously? Our exponential graph now intercepts the x-axis as well as the y-axis.

Practise time now.

For question one, I'd like to describe the key features of these exponential graphs.

When describing key features of a graph, I'd like you to be sure to mention any intercepts and any asymptotes.

Three graphs there and you'll need to write at least two sentences for each.

Pause and do that now.

Question two.

I'd like you to complete the table of values, plot the graph and find the asymptotes for these two exponential graphs.

Pause and do that now.

Feedback time now.

Let's see how we did.

Question one, I was asking you to describe the key features of some exponential graphs.

For a, we had the graph of y = 9^x.

You might have written the shape is an exponential curve with y values rising rapidly as x values increase.

It's good to describe the shape of a graph.

We'll have a y-intercept at (0, 1).

We'll have an asymptote at the x-axis.

For b, y = 1/(9^x), you might have written, the shape is an exponential curve with y values decreasing as x values increase.

You should have spotted our base there is between 0 and 1.

The y values will be decreasing as x increases.

Other key features, y-intercept at (0, 1).

And an asymptote at the x-axis.

For c, y = -9^x.

You might have written, the shape is an exponential curve with y values decreasing rapidly as x values increase.

y-intercept at (0, -1) this time.

An asymptote at the x-axis.

And this is what those three graphs would have looked like.

That'll be y = 9^x.

y = (1/9)^x.

Both intercepts in the y-axis is (0, 1).

And finally, our graph of y = -9^x.

And (0, -1) as the y-intercept for that one.

Look how your descriptions would have enabled you to sketch those graphs.

For question 2, I ask you to complete the tables of values, plot the graphs and find the asymptotes.

For equation a, that's y = 3^x - 7.

We should have that table of values.

And for b, y = -2^x + 12, that table of values.

The graphs would look like so.

An asymptote at y = -7 for the graph of y = 3^x - 7.

And an asymptote at y = 12 for y = -2^x + 12.

On to the second half of our learning now.

This is finding asymptotes from the equation.

y = 3^x has an asymptote at the x-axis.

That's when y = 0.

y = 3^x - 7 has an asymptote at y = -7.

A table of values help us to explain this.

There's a table of values for y = 3^x and a table of values for y = 3^x - 7.

Can you spot anything that links the y values when you compare the two tables? Have you spotted it? 27 - 7 is 20.

9 - 7 is 2.

3 - 7.

1 - 7.

1/3 - 7.

1/9 - 7.

And so on.

We had a constant of -7 added to this equation.

And that -7 constant changes our y values by -7.

So our asymptote changed by -7.

Hence the asymptote was at y = -7 rather than y = 0.

So we just saw that the graph of y = 3^x - 7 has an asymptote at y = -7.

Earlier we saw the graph of y = -2^x + 12.

It had an asymptote at y = 12.

Quick question for you.

Where do you think the asymptote of y = 4^x + 8 will be? Well done.

It'll be at y = 8.

Why will the asymptote be at y = 8? Well, as x increases in value, the value of 4^x also increases.

So y will keep increasing.

As x decreases in value, the value of 4^x will tend towards but never reach zero.

Therefore, 4^x + 8 will never reach eight.

So we've got these three examples.

Let's add one more.

What if we change that base to a value between zero and one? Where do you think the asymptote of y = (1/4)^x + 3 will be? Well done.

It'll be at y = 3.

Why? (1/4)^x will tend towards zero as x increases, but it'll never reach zero.

Therefore, y = (1/4)^x + 3 will never reach three.

So, let's generalise.

Graphs of the form y = b^x + c and y = -b^x + c have a horizontal asymptote at y = c.

That sentence is important.

You'll want to pause and write it down.

Graphs of the form y = b^x + c and y = -b^x + c have a horizontal asymptote at y = c.

We've got examples there where b is greater than one and b is between zero and one for y = b^x + c.

We've got an example of y = -b^x + c where b is greater than one.

And you'll notice in all cases there's an asymptote at y = c.

Question for you.

Will any of these graphs have a vertical asymptote? I can see the horizontal asymptote for each of them.

Will any of them have a vertical asymptote? What do you think? Pause, have a conversation with the person next to you or say an answer aloud to me back at the screen.

Welcome back.

I do hope you said no.

Why is it no? Well, there's no limit to the x values we can input into these expressions.

We can keep going infinitely in a positive direction, infinitely in a negative direction with those x values, and they will all generate a real y value.

Hence, there's no vertical asymptote.

Quick check you've got this.

I'd like you to match each equation to its asymptote.

Pause and give this a go now.

Welcome back.

Let's see how we did.

Hopefully you matched 8^x - 9 to an asymptote of y = -9.

<v ->9^x + 7 to an asymptote of y = 7.

</v> 9^x + 8 to an asymptote of y = 8.

(1/7)^x - 8 to an asymptote of y = -8.

And -7 + 9^x to an asymptote of y = -7.

You would have noticed that there were two on the right-hand side that were left out.

None of these graphs had a constant term of nine.

So the asymptote will not be nine, y = 9 for any of them.

And none of these graphs will have a vertical asymptote.

x = 8 is a vertical line.

None of these graphs will have a vertical asymptote.

We can write a generalisation for the y-intercept for graphs of the form y = b^x + c.

The y-intercept happens when x = 0.

So when x = 0, y = b^x + c becomes b^0 + c, b^0.

Oh yes, that's one.

So our y value becomes 1 + c.

So we'll always get a coordinate (0, 1 + c) but look, the y-intercepts.

The generalisation for the y-intercept for graphs of the form y = -b^x + c is slightly different.

But there's some familiarity.

Let's take a look.

When x = 0, y = -b^x + c becomes -b^0 + c.

Oh look, our y value is -1 + c.

We'll get the coordinate pair (0, c - 1), and that will be our y-intercept.

Let's check you've got this.

I'd like you to match each equation to its y-intercept.

Pause.

Get matching now.

Welcome back.

Hopefully, you matched 8^x - 9 to the y-intercept (0, -8).

<v ->9^x + 7, you should have matched it to (0, 6).

</v> 9^x + 8, you should have matched to (0, 9).

(1/7)^x - 8, you should have matched to (0, -7).

And -7 + 9^x, you should have matched to a y-intercept of (0, -6).

Practise time now.

I'd like you to match these graphs to their equations.

Four graphs, four equations, match them up.

Pause and do that now.

For question two, I'd like you to sketch the below graphs.

I showed you some sketches earlier.

I'd like you to sketch the below graphs of y = 2^x - 8 and y = -(1/3)^x - 9.

A sketch does not include a scale on the axes.

I'm not asking you to plot these with a table of values.

But I do want you to include key features like asymptotes and intercepts.

So make sure they're marked on your sketches.

Pause.

Give this a go now.

Question three, part a, an exponential graph has a y-intercept (0, 1), y values that increase as x values increase, and (1, 5) is a point on the graph.

What is the equation of the curve? For part b, a second exponential graph has the asymptote y = 2, y values that increase as x values increase, a y-intercept (0, 3), and (1, 5) is a point on the graph.

What is the equation of the curve? Two tricky problems, those, but I'm sure with enough thought, you'll get through it.

Give it a go.

Welcome back.

Feedback time.

Question one was matching graphs to their equations.

We should have matched them like so.

The second graph was y = 5^x + 5.

The fourth graph was y = 5^x - 5.

Look at the difference between the y-intercepts.

The first graph was y = 5^(-x) and d, y = 5^(-x) - 5.

Oh look, it had a negative or y-coordinate was negative at the y-intercept, hence we matched that pair together.

For question 2, I asked you to sketch these graphs.

y = 2^x - 8 should have looked like that.

Your sketch should include the fact that there's an asymptote at y = -8.

You should also label the y-intercept.

In this case, that was (0, -7).

The x-intercept should be labelled 2, 3, 0.

Your sketch should look like that.

We should clearly see those three key features.

We should be able to see the shape of your graph also.

Part b, -(1/3)^x - 9.

The curve should look like that.

We should have labelled our asymptote at y = -9 and our y-intercept at (0, -10).

When practising this skill, it's a good idea to use tech to check.

Graphing technology like Desmos quickly reassures us that these key features are accurate.

I went to desmos.

com, used their graphing calculator, type in the graph of y = 2^x - 8.

And can you see how that technology confirmed that my sketch was correct? Question 3.

An exponential graph had a y-intercept of (0, 1).

y values increase as x values increase.

(1, 5) is a point on the graph.

What's the equation of a curve? So you know it's of the form y = b^x.

When x = 1, the y value is the base.

Therefore, it's five.

It must be y = 5^x.

Question 3, part b.

There is a lot going on here.

That's a lot of information.

For me, I think a sketch will really help.

It helps me to digest all this information that we're given.

I can now see an asymptote at y = 2.

I can see my y-intercept at (0, 3).

And I can see that known coordinate at (1, 5).

Aha, I see something else now.

That must be the base.

It must be 3^x + 2 because we've got an asymptote at y = 2.

Now, at this point, I think that is the equation of this graph.

But we're given a known coordinate, (1, 5).

If I substitute x = 1, y = 5 into my equation, I can check that I'm right.

5 = 3^1 + 2.

It does indeed.

The values of that coordinate pair satisfy my equation.

I believe I'm right.

Sadly, we're at the end of the lesson now.

But we've learned how to identify the key features of an exponential graph.

We can find the asymptotes and intercepts when presented with a graph or the equation.

And we can sketch exponential graphs if all the key features are known.

I hope you've enjoyed this lesson as much as I have.

I look forward to seeing you again soon for more mathematics.

Goodbye for now.