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've plotted the graph of y equals x multiplied by 2." 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 2 is equal to 2x.

So what about these 2? Y equals x squared or x to the power of 2 and y equals 2 to the power of x.

Sam says, does that mean the graphs of x squared, and 2 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 8.

When x equals 8, y equals x squared, we'll have y value, we have 8 squared, which is 8/8ths, which is 64.

But for the graph of y equals 2 to the power of x, when x equals 8, y equals 2 to the power of 8.

That's 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2, which is 256.

So when x equals 8, 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 2 to the power of x by substituting in x values, as we would for any other equation.

When x equals 0, y equals 1.

When x equals 1, y equals 2.

When x equals 2, 2 squared is 4, y equals 4.

When x equals 3, 2 cubed is 8, so y equals 8.

Pop those into our table of values.

We could repeat that process.

Substituting in x equals 4x equals 5x equals 6, and we'll get those y values.

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

I told you 2 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." And look, the y values are in a geometric sequence with a common ratio of 2.

That's because the base in the equation is 2.

How nice.

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

Then y equals 2 to the power of x looks like that.

Sam says, "Look how quickly the graph of 2 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 2 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 3 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 1, y equals 3; when X equals 2, y equals 9; when X equals 3, y equals 27.

Pop those into our table of values, and oh look, we don't need to work out 3 to the power of 5 because as a pattern, in those y values, a geometric sequence, common ratio 3, 'cause our equation is a base of 3.

81 times 3 was 243, so an x equals 5, y equals 243.

We can put those coordinate pairs, draw a nice smooth curve through, and it looks 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 5 243 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 2 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 1, x equals negative 2? 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 1, y equals 2 to the power of x becomes 2 to the power of negative 1, which is equal to 1 over 2 to the power of positive 1.

That's a half.

When x equals negative 2, y equals 2 to the power of negative 2, which is the same as 1, over 2 to the power of positive 2, which is a quarter." 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 8th as the y value for when x equals negative 3.

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.

Same, 2 is the power of negative 3 absolutely is an 8th.

2 to the power of negative 2, a quarter; 2 to the power of negative one, a half; 2 to the power of 0, 1; 2 to the power of 1, 2; 2 to the power of 2 is 4; 2 to the power of 3 is 8.

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 4 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.

Maybe you spotted that 1 over 64 was correct for when x equals negative 3.

1/16th was correct for when x equals negative 2, but negative 4 was not a correct y value.

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

X equals negative 1, y equals 4 to the power of negative 1, which is the same as 1 over 4 to the power of positive 1, which is a quarter.

Also, 0,0 wasn't correct.

So common error, this one, so do watch out for when x equals 0, people are fooled into thinking 4 to the power of 0, 4 times 0,0, so 4 to the power of 0 must be 0.

It's not, it's 1.

And your calculator will reassure you that 4 to the power of 0 is wrong.

1/4 was a correct coordinate pair.

2/16 was a correct coordinate pair.

3/12 was not.

When x equals 3, y equals 4 to the power of 3, 4 by 4 by 4 is 64.

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

If Laura 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 3 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 3, not 3 to the power of x.

Practise time now.

For question one, I'd like you to complete the table of values and plot the exponential graph of y equals 5 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 two.

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 one, we were completing the table of values for y equals 5 to the pair of x.

Your value should look like that when you plop 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 lovely smooth exponential curve.

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

When x equals negative 3, your calculator display should have looked like so, giving you 1 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 2, when x equals negative 1, when x equals 0, oh look, 1 again, where x equals 1, when x equals 2, and when x equals 3.

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 2 the power of x, y equals 3 to the power of x, and y equals 4 to the power of x.

They're here on this graph, like so.

What do you think the graph of y equals 1 to the power of x looks like? You can see 4 to the power of x, 3 to the power of x, 2 to the power of x.

Hmm, what's 1 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 1 equals 1 to the power of x is something unique case.

One to the power of anything is one.

So we get those coordinate pairs and that line.

That's y equals 1 to the power of x, or is it? This is not an exponential graph.

It is the straight line graph of y equals 1.

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 1? Y equals a half to the power of x, a third to the power of x, a quarter 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 2 to the power of x when thinking about y equals half to the power of x.

I've put a table of values there for y equals 2 to the power of x.

And we're gonna look at populating the table of values for y equals a half to the power of x.

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

When x equals 1, a half to the power of x, that's a half to the power of 1.

That's just a half.

When X equals 2, y equals a half to the power of 2, that's a half multiplied by a half, which is a quarter.

When x equals 3, a half to the power of 3, that's a half 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, conversation with the 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 a half as 2 to the power of negative one, it explains this pattern.

Y equals a half to the power of x is the same as 2 to the power of negative 1 to the power of a half.

Now, we know from our laws of indices that we can simplify that to 2 to the power of negative x.

So we're now comparing y equals 2 to the power of positive x to y equals 2 to the power of negative x.

What happens now is the y value we got when x equals 3 is now the y value that we have when x equals negative 3 and vice versa.

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

How lovely.

We can use our knowledge of indices to test the coordinate pair - negative 3, 8.

Y equals a half to the power of x when x equals negative 3.

Well, half to the power of negative 3 will become 1, over half to the power of positive 3, which is one over 1/8.

One over 1/8, that's 8.

So we know it's right.

We can also use tech to check.

if we input it into our calculator, we exempt same results.

And x equals negative 3, y equals positive 8.

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 2 to the power of x, which you've already seen.

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

That is the coordinate - negative 3 8, negative 2,4, negative 1,2, and so on.

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

Somewhat beautifully, there's a line of symmetry at the y axis between y equals 2 to the power of x and y equals a half 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 0 and 1, then as the x values increase, the y values decrease.

We can see this in our graph of y equals a half to the power of x.

You'd see a similar pattern for y equals a third to the power of x, a course to the power of x, a 10th to the power of x, 3 quarters to the power of x.

If that base is between 0 and 1, 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 3 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 a third to the power of x like so.

The same y values is 3 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 a third to the power of x.

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

If the base is between 0 and 1, and as x values increase, the y values decrease, the graphs are symmetrical because y equals a third of the power of X is the same as 3 to the power of minus 1 to the power of x, which is the same as 3 to the power of negative x.

This can be generalised to work for any value between 0 and 1.

We know what the graph of y equals 2 to the power of x looks like, and we now know what the graph of y equals a half to the power of x looks like.

So what do you think the graph of y equals negative 2 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 2 to the power of x when thinking about Y equals negative 2 to the power of x, as a table of values for 2 to the power of X.

Let's populate that table of values for negative 2 to the power of x.

When x equals negative 3, y equals negative 1/8.

When x equals negative 2, y equals negative of a quarter.

When x equals negative 1, Y equals negative or half.

X equals 0, y equals negative 1, and so on.

Have you spotted something? The same values as y equals 2 to the power of x, but negative.

So when we come to plot, y equals 2 to the power of x, it's all positive, it's all up there.

Y equals negative 2 to the power of x, it's all negative.

It's all down there.

Still, we draw a smooth curve as coordinate pairs, and there's the graph, y equals negative 2 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.

Quick check, you've got this.

I'd like to complete the table of values and plot y equals negative 4 to the power of X.

Once again, I've been very kind.

There's a table of values there for y equals positive 4 to the power of x.

I've even drawn that graph for you, so hopefully, this won't take long.

Pause, see you in a moment.

Welcome back.

Hopefully, you populated your table values like so.

The same y values is for the power of x but negative.

When you plot this coordinate pairs and your nice smooth curve, we have the line y equals negative 4 to the power of x.

The exponential graph of y equals negative b to the power of x will be symmetrical to the graph of y equals b to the power of x in the x axis.

That feels important.

You might wanna pause and write that down.

Practise time now.

For question one, 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 5 to the power of x.

Pause, give these two graphs a go.

Now, for question two, a matching exercise.

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

4 graphs, 4 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 5 to the power of x should give you those values in your table, and the graph should appear there.

That's why it calls negative 5 to the power of X.

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

Question two, we are 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 1/10th to the power of x to the 4th graph because it's positive but decreasing y values.

And that leaves (d), y equals negative 10th to the power of x to be the third graph because it's negative values, but they're increasing.

Sadly, let's send a lesson now.

We've learned that 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 1 and a equals 1.

Graphs of the form y equals ab to the power of x form a curve with Y values decreasing when B is greater than 0 but less than 1 and A equals 1.

The graph of y equals ab to the power of x when equals one is symmetrical to the graph of y equals ab to the power of x when a equals negative 1, 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.