Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Lovely to see you again.

Today, we're drawing exponential graphs.

They sound cool.

Let's find out just how cool.

A learning outcome is I'll be able to generate coordinate pairs for an exponential graph from its equation and then draw the graph.

Keywords we'll hear today.

The general form for an exponential equation is y equals ab to the power of x, where a is the coefficient, b is the base, and x is the exponent.

Look out for those words throughout our lesson.

Two parts to our learning today.

We're gonna begin by plotting exponential graphs.

Sam and Aisha are plotting the graphs of equations.

Sam says, "I plotted the graph of y equals x multiplied by two." Aisha says, "I've plotted the graph of y equals 2x." What do you notice? Well done.

They're the same graph.

This is because x multiplied by two is equal to 2x.

So what about these two? y equals x squared, or x to the power of two, and y equals two to the power of x.

Sam says, "Does that mean the graphs of x squared and two to the power of x will be the same?" What do you think? What's your mathematical intuition telling you? Same graph, different? Aisha knows.

Aisha says, "I only need one x value to show you why they won't be the same.

Take x equals eight.

When x equals eight, y equals x squared.

We'll have y value of eight squared, which is eight eights, which is 64.

But for the graph of y equals two to the power of x, when x equals eight, y equals two to the power of eight.

That's two times two times two times two times two times two times two times two, which is 256.

So when x equals eight, we've got different y values for the two equations, so we won't get the same graph.

Sam says, "I'd forgotten how powerful powers are.

Two to the power of x could be quite an exciting graph." I couldn't agree more, Sam.

Let's compare the tables of values.

Aisha says, "Let's plot the two graphs and compare them." y equals x squared has a table of values like so, and we can find the y values for y equals two to the power of x by substituting in x values as we would for any other equation.

When x equals zero, y equals one.

When x equals one, y equals two.

When x equals two, two squared is four, y equals four.

When x equals three, two cubed is eight, so y equals eight.

Pop those into our table of values.

We could repeat that process, substituting in x equals four x, equals five x, equals six, and we'll get those y values.

Sam says, "Look at how quickly the y values have increased.

I told you two to the power of x was going to be exciting." "Did you notice something about the y values?" asks Aisha.

Sam says, "Yes, they form a geometric sequence." Oh look, the y values are in a geometric sequence with a common ratio of two.

That's because the base in the equation is two.

How nice.

When we go to plot these, we can start with y equals x squared.

Then, y equals two to the power of x looks like that.

Sam says, "Look how quickly the graph of two to the power of x increases." And Aisha says, "This is known as exponential growth." You might have heard that phrase before.

Exponential growth.

Really rapid growth.

For the equation, y equals two to the power of x, in this case, the base is constant and the exponent is the independent variable.

This is why this is called an exponential graph because it's the exponent that varies.

Quick check you've got this.

I'd like you to complete the table of values and find coordinate pairs for the exponential graph of y equals three to the power of x.

Pause and do that now.

Welcome back.

Let's see how we did.

Hopefully, you found that when x equals one, y equals three.

When x equals two, y equals nine.

When x equals three, y equals 27.

Pop those into our table of values and oh look, we don't need to work out three to the power of five because there's a pattern in those y values, a geometric sequence, common ratio of three 'cause our equation is a base of three.

81 times three was 243, so when x equals five, y equals 243.

We can put those coordinate pairs, draw a nice smooth curve through and it looks like so.

The y values have increased so quickly, we actually need a new scale to plot that last coordinate pair.

I couldn't fit the coordinate pair five, 243 on that grid.

Even when we zoom out and we change the scale, we still see the same shape.

We still see an exponential curve.

Next up, let's compare what happens when we start to plot negative x values for such graphs.

"For y equals x squared," Sam says, "I know exactly what happens when x has negative values, a parabolic curve, with y values increasing in both x directions.

It looks like so." Aisha explains, "Yes, this is because we know from our number lessons that when you square a negative value, the result is positive." That's a lovely link to make, Aisha, well done.

So what happens when y equals two to the power of x has negative values? What do you think? I'd like you to pause this video and consider where do you think this graph is going for x equals negative one, x equals negative two? Pause and have a think.

Welcome back.

I wonder what you thought.

Aisha knows.

Aisha says, "I remember this one from our Number lessons too, Sam." When x equals negative one, y equals two to the power of x becomes two to the power of negative one, which is equal to one over two to the power of positive one.

That's 1/2.

When x equals negative two, y equals two to the power of negative two, which is the same as one over two to the power of positive two, which is 1/4.

Pop those in our table of values.

Sam says, "Wait, I've got the last one." Oh look, it's that geometric sequence again.

So Sam writes an 1/8 as the y value for when x equals negative three.

Aisha says, "Of course.

The previous y value has to be half.

Well spotted, Sam." I agree.

If you are allowed a calculator, you can use that to calculate the y values or indeed, check them if you calculated them without the calculator.

So, two to the power of negative three absolutely is 1/8.

Two to the power of negative two, 1/4.

Two to the power of negative one, 1/2.

Two to the power of zero, one.

Two to the power of one, two.

Two to the power of two is four, two to the power of three is eight.

If you are allowed to, it's a good idea to use tech to check.

Quick check you've got this so far.

Laura was rushing when she completed this table of values.

Using tech, or otherwise, spot and correct her errors.

This is the equation y equals four to the power of x.

Some of those y values are right, some of them are not.

I allow you to pick out the wrong ones.

Pause, see you in a moment.

Welcome back.

Hopefully, you spotted that 1/64 was correct for when x equals negative three.

1/16 was correct for when x equals negative two, but negative four was not a correct y value.

Why not? Well, we could do this without a calculator.

When x equals negative one, y equals four to the power of negative one, which is the same as one over four to the power of positive one, which is 1/4.

Also, 0, 0 wasn't correct.

So common error this one, so do watch out for it.

When x equals zero, people are falling into thinking four to the power of zero, four times zero, zero.

So four to the power of zero must be zero.

It's not.

It's one.

And your calculator will reassure you that four to the power of zero is wrong.

1, 4 was a correct coordinate pair.

2, 16 was a correct coordinate pair.

3, 12 was not.

When x equals three, y equals four to the power of three.

Four by four by four is 64.

Laura says, "I forgot that if allowed, I should use tech to check." It's a good idea too, Laura.

If Laura had plotted these values, how would she have known immediately that these three coordinate pairs are incorrect? Pause, look at the graph.

How do you just know they're wrong? Welcome back.

They're the three incorrect coordinate pairs.

How did you know they're wrong? You might have said those three points don't fit a smooth exponential curve.

When you draw an exponential graph, it's going to have a shape like that so you'll be able to spot if you've got any incorrect coordinate pairs.

Next check.

Laura took her time, well done, and used a calculator, well done, to help her complete this table of values, but she has made an error.

Write a sentence or two to explain Laura's error to her.

This is for the graph y equals three to the power of x.

Pause now, spot the error.

Welcome back.

Did you get it? You might have written you varied the base in an exponential graph.

The base is constant, the exponent varies.

Oh look, there's the base being varied.

Laura's inadvertently given us a table of values for y equals x to the power of three, not three to the power of x.

Practise time now.

For Question 1, I'd like you to complete the table of values and plot the exponential graph of y equals five to the power of x.

Quick note for you, you can only approximately plot the fractional coordinate pairs that you'll have in your table of values.

Pause, draw this graph now.

Question 2.

I'd like you to use a calculator to complete this table of values for y equals 12 to the power of x.

No need to plot this graph.

This is me checking that you can accurately use your calculator.

Pause, work out these values now.

Welcome back.

Let's see how we did.

For Question 1, we were completing the table of values for y equals five to the power of x.

Your values should look like that.

When you plot those coordinate pairs on the graph and then join them with a nice smooth curve, it looks like so.

You might wanna pause now.

Just check that your table of values is the same as mine, your coordinate pairs in the same position, and you have a lovely smooth exponential curve.

Question 2, I asked you to use a calculator to complete a table of values for y equals 12 to the power of x.

When x equals negative three, your calculator display should have looked like so, giving you one over 1,728, absolutely leave that as a fraction.

Don't try to convert it to a decimal.

Goes in our table of values, when x equals negative two, when x equals negative one, when x equals zero.

Oh look, one again.

When x equals one, when x equals two, and when x equals three.

So, I didn't ask you to plot this one, but just to reassure you, if we had plotted it, we'd have got the same smooth exponential curve.

Onto the second half of the lesson now where we're going to do some further exploration of exponential graphs.

We've seen the graphs of y equals two to the power of x, y equals three to the power of x, and y equals four to the power of x.

They're here on this graph like so.

What do you think the graph of y equals one to the power of x looks like? You can see four to the power of x, three to the power of x, two to the power of x.

Hmm.

What's one to the power of x gonna look like? Pause, have a think, conversation with the person next to you perhaps, see you in a moment.

Welcome back.

I wonder what you thought.

I hope you spotted that y equals one to the power of x is something of a unique case.

One to the power of anything is one.

So we get those coordinate pairs and that line, that's y equals one to the power of x, or is it? This is not an exponential graph.

It is the straight line graph of y equals one.

Look, when the base is greater than one, when the base is one.

So what do you think these graphs will look like when the base is less than one? y equals 1/2 to the power of x, 1/3 to the power of x, 1/4 to the power of x.

Again, pause, have a think.

What do you think happens for those exponential graphs? Welcome back.

It's useful to consider y equals two to the power of x when thinking about y equals 1/2 to the power of x.

I've put a table of values there for y equals two to the power of x and we're gonna look at populating the table of values for y equals 1/2 to the power of x.

I'm gonna start with the easiest x values first.

When x equals one, 1/2 to the power of x, that's 1/2 to the power of one.

That's just 1/2.

When x equals two, y equals 1/2 to the power of two.

That's 1/2 multiplied by 1/2, which is 1/4.

When x equals three, 1/2 to the power of three, that's 1/2 cubed, 1/8.

Have you spotted a pattern? Look at the two tables of values.

Is anything jumping out at you? Pause, have a think or a conversation with a person next to you.

See you in a moment.

Welcome back.

Did you spot it? They are the same y values but in reverse.

Why does that happen? Well, if you think of 1/2 as two to the power of negative one, it explains this pattern.

y equals 1/2 to the power of x is the same as two to the power of negative one to the power of 1/2.

Now, we know from our laws of indices that we can simplify that to two to the power of negative x.

So we're now comparing y equals two to the power of positive x to y equals two to the power of negative x.

What happens now is the y value we got when x equals three is now the y value that we have when x equals negative three and vice versa.

That is why we have the same y values but in reverse in our tables of values.

How lovely.

We can use our knowledge of indices to test the coordinate pair negative three, eight.

y equals 1/2 to the power of x when x equals negative three.

Well, 1/2 to the power of negative three will become one over 1/2 to the power of positive three, which is one over 1/8.

One over 1/8, that's eight.

So we know it's right.

We can also use tech to check.

If we input it into our calculator, the exact same results when x equals negative three, y equals positive eight.

When we're working with more complex exponential graphs, we have so many checking mechanisms at our disposal, we shouldn't be afraid to use them.

When we plot these two graphs together, we see a beautiful symmetry in the y-axis.

That's y equals two to the power of x, which you've already seen.

Let's plot the coordinate pairs for y equals 1/2 to the power of x, which I'll remind you, is y equals two to the power of negative x.

That is the coordinate negative three, eight, negative two, four, negative one, two, and so on.

Are you spotting a lovely pattern? We join a nice smooth curve and that's the graph of y equals 1/2 to the power of x.

Somewhat beautifully, there's a line of symmetry at the y-axis between y equals two to the power of x and y equals 1/2 to the power of x.

How nice.

Exponential graphs of the general form y equals ab to the power of x.

b is the base.

If the base is between zero and one, then as the x values increase, the y values decrease.

We can see this in our graph of y equals 1/2 to the power of x.

You'd see a similar pattern for y equals 1/3 to the power of x, 1/4 to the power of x, a 1/10 to the power of x, 3/4 to the power of x.

If that base is between zero and one, we'll get a curve that looks like this one.

Quick check you've got that.

I'd like you to fill in the table of values and plot the graph of y equals 1/3 to the power of x.

I've been very kind.

The graph of y equals three to the power of x is done for you, so you might spot something that's very helpful.

Pause, fill in the table of values, plot the graph.

Welcome back.

I hope you populated your table of values for y equals 1/3 to the power of x, like so.

The same y values as three to the power of x, but in reverse.

When you plot the coordinates, you see the same symmetry that you saw in your table of values in your coordinate grid.

Nice smooth curve through there and we have the graph y equals 1/3 to the power of x.

Exponential graphs of the general form y equals ab to the power of x.

If the base is between zero and one and as x values increase, the y values decrease, the graphs are symmetrical because y equals 1/3 to the power of x is the same as three to the power of minus one to the power of x, which is the same as three to the power of negative x.

This can be generalised to work for any value between zero and one.

We know what the graph of y equals two to the power of x looks like and we now know what the graph of y equals 1/2 to the power of x looks like.

So what do you think the graph of y equals negative two to the power of x will look like? Pause, have a think, speculate.

What's that graph gonna look like? Welcome back.

Again, a comparison helps us to understand this graph.

It's useful to consider y equals two to the power of x when thinking about y equals negative two to the power of x.

There's a table of values for two to the power of x.

Let's populate that table of values for negative two to the power of x.

When x equals negative three, y equals negative 1/8.

When x equals negative two, y equals negative 1/8.

When x equals negative one, y equals negative 1/2.

x equals zero, y equals negative one and so on.

Have you spotted something? The same values as y equals two to the power of x but negative.

So when we come to plot y equals two to the power of x, it's all positive, it's all up there.

y equals negative two to the power of x, it's all negative, it's all down there.

Still, we draw a smooth curve, those coordinate pairs, and there's the graph, y equals negative two to the power of x.

This time, we've got a line of symmetry in the x-axis.

We've still got a smooth exponential curve, but we've got negative, decreasing y values.

Practise time now.

Question 1.

I'd like you to complete the tables of values and plot these exponential graphs.

That's the graph of y equals 1/5 to the power of x and y equals negative five to the power of x.

Pause, give these two graphs a go now.

For Question 2, a matching exercise.

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

Four graphs, four equations, pair them up.

Pause and give this a go now.

Feedback time.

Let's see how we did.

Hopefully, for y equals 1/5 to the power of x, your table of values looks like so.

When you pull up those coordinate pairs, draw a nice smooth curve, you'll get the graph y equals 1/5 to the power of x and it should look just like that.

y equals negative five to the power of x should give you those values in your table and the graph should appear there.

That's y equals negative five to the power of x.

You might wanna pause, just check that your tables of values match mine and your graphs match mine.

Question 2, we were matching graphs to their equations.

We should have paired a, y equals 10 to the power of x, to the second graph because it's positive, increasing y values.

We should have matched b, y equals negative 10 to the power of x to the first graph because it's negative, decreasing y values.

Oh look, those two graphs are symmetrical in the x-axis.

You should have matched c, y equals 1/10 to the power of x to the fourth graph because it's positive but decreasing y values, and that leaves d, y equals negative 1/10 to the power of x to be a third graph because it's negative values, but they're increasing.

Sadly, that's end of our lesson now.

We've learned now we can generate coordinate pairs for an exponential graph from its equation and then draw the graph.

Graphs of the form y equals ab to the power of x form a curve, with y values exponentially increasing when b is greater than one and a equals one.

Graphs of the form y equals ab to the power of x form a curve with y values decreasing when b is greater than zero but less than one and a equals one.

The graph of y equals ab to the power of x when a equals one is symmetrical to the graph of y equals ab to the power of x when a equals negative one with the x-axis being the line of symmetry.

I hope you've enjoyed this lesson as much as I have and I look forward to seeing you again soon for more maths.

Goodbye for now.