Lesson video

Hello, Mr. Robson here, welcome to maths.

Great choice to join me today, especially because we're looking at transformations of graphs of the form y equals f of axe.

You might have seen transformations of graphs before in mathematics, but I doubt you've seen one as beautiful as the one we're going to be looking at today.

Let's take a look.

Our learning outcome is that we'll be able to recognise the effect of applying the transformation y equals f of axe to a graph.

Transformation's a keyword that we'll hear a lot throughout the learning today.

A transformation is a process that may change the size, orientation, or position of a shape or graph.

Two parts to our learning.

Let's begin by exploring y equals f of axe.

We know that the transformation af of x stretches the graph of fx in the y-direction by a scale factor of a.

In this example, we can see y equals f of x and its transformation, y equals 2f of x.

It's a stretch in the y-direction.

This stretch happened because 2f of x equals two multiplied by f of x, or f of x multiplied by two.

The outputs of f of x, that's our y values, will multiply by a scale factor of two.

If you take a coordinate pair and multiply the y value, you're stretching in the y-direction.

What about the transformation f of axe? Can you see the difference? The a is inside the bracket.

Aisha says, "We know that the transformation af of x "is a stretch by scale factor a in the y-direction.

"What effect do we think f of axe will have?" Jacob speculates, "I think we'll see a stretch "by scale factor a in the x-direction." Whereas Sam speculates, "axe equals a multiplied by x, "so it's another multiplication transformation.

"I think we'll see a transformation "by scale factor a in the y-direction again." What do you think the effect of f of axe will be? Pause, have a conversation with a person next to you or good think to yourself, see you in a moment.

Welcome back, I wonder what you think.

What's your prediction? What's f of axe going to do to a graph? The transformation f of axe is a stretch.

Here's y equals f of x.

That is y equals f of 2x.

Sam says, "It's definitely not a stretch in the y-direction.

"The range has not changed." Oh, this is good, we ruled out that option.

Jacob says, "But it's not stretching away from the y-axis.

"In fact, f of 2x has stretched "f of x towards the y-axis." Can you see what Jacob means on that graph there? The transformation f of axe is a stretch by a scale factor of one over a in the x-direction.

That's a really important sentence.

So I'm gonna ask you to pause now and copy that down.

Welcome back, the transformation f of axe is a stretch by a scale factor one over a in the x-direction.

The coordinate pair (90, 1) became (45, 1) when we transformed f of x by f of 2x.

Funnily enough, it's a scale factor of one over two.

The coordinate pair (270, -1) became (135, -1), again, a scale factor of one over two.

But it's just those x coordinates that are being multiplied by that scale factor, hence by stretching in the x-direction.

The transformation f of 3x is a stretch by a scale factor of one over three in the x-direction.

So in this case, (12, 0) becomes (4, 0).

That's where that point will transform two.

The y value does not change.

The x value is multiplied by a third.

Therefore, (6, -36) becomes (2, -36).

The y value doesn't change.

The x value is multiplied by a third.

(-3, 45) will become (-1, 45).

The y value doesn't change.

The x value is multiplied by a third.

So now we can sketch y equals 3f of x.

Sam says, "It looks like we are squishing the graph, "not stretching it." Jacob, "Correct, Sam.

"Squish is not good technical language, Sam.

"We still call this a stretch, "just by a fractional scale factor." In this case, it was by the fractional scale factor of a third.

Notice we had one invariant point.

This is because x equals zero in this coordinate pair.

We multiply zero by any scale factor, it will not change.

So that remained an invariant point.

Quick check you've got this.

In this case, f of x equals negative x squared plus 10x.

What will happen to these three coordinate pairs when the transformation f of 5x is applied? Pause and think about this now.

Welcome back, the y values do not change.

The x values are multiplied by a scale factor of one over five.

So (5, 25) becomes (1, 25).

(10, 0) will transform to (2, 0) and the point (0, 0) is an invariant point because it's x value is zero.

Zero multiplied by one over five is still zero.

Once we've got those points, we can now draw y equals f of 5x.

f of fx is a stretch by one over five in the x-direction.

In some cases, we'll see values of a such that a is greater than zero and less than one.

So far, we've only seen positive integer values for a in this transformation.

But what if a is greater than zero and less than one, i.

e.

a fraction? Let's look at this example.

You can see on the graph y equals f of x.

We can graph the transformation f of a half x.

The stretch is still by a scale factor one over a in the x-direction, nothing's changed there.

But in this case, one over a is one over a half, which is two.

The coordinate pair (90, 1) will become the coordinate pair (180, 1) or x coordinate value being multiplied by a scale factor of one over a, one over half, a is multiplied by two.

(180, 0) will become (360, 0) and the graph will look like so.

That's y equals f a half x.

Quick check you've got that.

Here's f of x equals sin x in this domain of x values.

What I'd like you to do is graph the transformation f of half x.

Pause and do this now.

Welcome back, hopefully you identified there'll be a scale factor one over a, one over a half, it'll be two.

A scale factor of two.

Therefore, (90, 1) will transform to (180, 1).

(180, 0) will transform to (360, 0).

(270, -1) will transform to (540, -1) and (360, 0) will transform to (720, 0), giving you that graph, y equals f of a half x.

Practise time now.

Question one, f of x equals cos x.

Part A, which of these three graphs shows the correct transformation f of 2x? For part B, and this is the crucial part 'cause this'll tell me you really understand this, part B, I'd like you to write a sentence to justify your answer.

Pause and do this now.

Question two, f of x equals negative x squared minus 12x plus 45.

For part A, I'd like to write a sentence to explain what will happen to each of these coordinate pairs when the transformation f of 3x is applied.

Comment on both the x and y values.

For B, I'd like you to list four coordinate pairs on the graph of f of 3x.

And for C, I'd like you to identify which point is invariant in the transformation f of 3x.

Pause and do this now.

Question three, f of x equals 2x minus five, a linear function this time.

For part A, I'd like you to draw the transformation f of 5x.

Draw that on the same grid as you see y equals f of x.

For part B, again on the same grid, draw the transformation f of a quarter x.

Pause and do those now.

Welcome back, feedback time.

Let's see how we did.

Question one, part A, we were identifying which graph showed the correct transformation for f of 2x.

We should have said it was C.

A sentence to justify your answer you might have written: "The transformation f of 2x is a stretch by a scale factor "of one over two in the x-direction." You can see that stretch on the graph like so.

Question two, I ask you to write a sentence to explain what will happen to each of these coordinate pairs when the transformation f of 3x is applied.

You might have written, "The y values do not change.

"The x values are multiplied by a third." For part B, I asked you to list four coordinate pairs on the graph of f of 3x.

You might have listed those four coordinate pairs.

For part C, which point is invariant in the transformation f of 3x? That would be (0, 45).

It's that point which is invariant.

Question three, part A, I ask you to draw the transformation of f of 5x.

We can identify some coordinate pairs, stretching by scale factor of one over five in the x-direction, and then draw y equals f of 5x.

For part B, transformation f of a quarter x.

Those x values are gonna change by scale factor one over a quarter, that's four.

So (5, 5) becomes (20, 5) and (-5, -15) becomes (-20, -15).

We can then draw the line y equals f of a quarter x.

Onto the second half of our learning now where we're gonna be sketching y equals f of axe.

We know that f of axe transforms a graph by a stretch of scale factor one over a in the x-direction, but we don't yet understand why this happens.

If you can explain why something happens in mathematics, you are a very powerful mathematician indeed.

So let's look at deepening our understanding of this transformation.

To deepen our understanding, it's worth thinking about this question.

Mathematically, what do each of these things mean? I'd like to pause this video, copy those down, then write a sentence for each.

What do they each mean? Pause, I'll be back in a moment to run through the answers.

Welcome back, the negative f of x, should have identified that that means the output of f of x is multiplied by negative one.

For af of x, we should have identified that the output of f of x is multiplied by a.

For f of x plus a, something's changed.

The input is affected.

The input of a added before the function f is performed on it.

Therefore, the f of axe, it's the input which is multiplied by a before the function f is performed on it.

So we conclude the transformation f of axe affects our inputs.

We can understand the effect f of axe has by looking at a table of values.

It's a table of values for f of x equals sin x.

We know that (0, 0) will be an invariant point, so let's consider some other points.

(90, 1), well, that means that an input of 90 into our function gives us an output of one, i.

e.

sin 90 equals one.

(180, 0), well, that means an input of 180 gives us an output of zero.

(270, -1), an input of 270 gives us an output of negative one.

We can make a similar statement about any of these coordinate pairs.

Let's think about what inputs we now need to generate those same outputs if we were to graph f of 2x.

Notice in my table of values for f of 2x, I have maintained the same outputs as our table of values for f of x.

What input is now going to give us an output of one? Well done, sin 90 equals one.

We're now graphing f of 2x.

Well, that's sin two lots of 45, making one.

So for f of 2x, we need an input of 45 to give us an output of one.

We can drop that in the table.

Next, what input now gives us this output of zero for f of 2x? Well, we know from f of x that 180 gives us an output of zero because sin 180 equals zero.

So we could write that as sin two lots of 90 equals zero.

For f of 2x, we need an input of 90 to give us this output of zero.

Next, what input now gives us an output of negative one? We know that sin 270 equals negative one, so sin two lots of 135 equals negative one.

For f of 2x, we need an input of 135 to give us an output of negative one.

Finally, what input now gives us this output of zero? Sin 360 equals zero, oh, that's sin two lots of 180 equaling zero.

So for f of 2x, we need an input of 180 to give us this output of zero.

The effect that the transformation f of 2x had on our inputs is now clear.

Our inputs were multiplied by a scale factor of a half.

Remember, f of 2x is going to stretch our graph by scale factor one over two in the x-direction.

Can you see from these comparable table of values why this happens? When we graph these two sets of values, we see the physical effect.

There's our coordinate pairs on the graph of f of x and there is their transformation to become f of 2x.

A stretch by scale factor one over two in the x-direction because we were affecting those inputs, those x values.

One little point to note, if you were asked to draw f of 2x on this graph, you would continue that line f of 2x.

You would fill this space.

Quick check you've got that.

I'd like you to populate the table of values for the transformation f of 3x in this case.

Pause and do that now.

Welcome back, hopefully you populate the table of values with 30, 60, 90, 120.

The inputs were multiplied by a scale factor of one over three to generate the same sets of outputs.

Now that we've got a populated table of values for f of 3x, I'd like you to draw the transformation, f of 3x.

Pause and do this now.

Welcome back, hopefully you plotted the coordinate pairs for f of 3x there and didn't just draw the graph of f of 3x in that domain.

You filled the whole space, filled the whole graph, and your graph of f of 3x looked like so, quite delightful.

A stretch by scale factor one over three in the x-direction.

It's not just in trigonometric graphs that we see this effect in the table of values.

Look at this example, this is a table of values for f of x equals x squared minus 4x minus 96.

We could look at the table of values for f of 4x.

Notice I've included the same outputs.

What inputs into f of 4x will give us these same outputs? Well, there's an invariant point, (0, -96).

What about this point, (12, 0) on f of x? If f of 12 equals zero, how do we get the output of zero in f of 4x? Well, if f of 12 equals zero, then f of four lots of three equals zero.

So three must be the input into f of 4x to give us that output of zero.

Let's take another example.

How about this coordinate pair, (-4, -64)? Well, that means in f of x, f of negative four equals negative 64.

So that must mean f of four lots of negative one equals negative 64.

So the input into f of 4x to get the output of negative 64 is negative one.

We can populate the whole table of values using this idea.

Our inputs were multiplied by a scale factor of one over four.

Selecting key features will enable us to sketch both graphs.

For example, we can see the y intercept.

We can spot a root, and another root.

Then we can see the turning point.

f of x equals x squared minus 4x minus 96, that's a parabola, it's going to have a turning point.

We can see that turning point by symmetry in the outputs.

Those are the key features we picked out of the table of values of f of x, hence we can sketch y equals f of x.

Those are the same key features on the graph of f of 4x, hence we can draw y equals f of 4x.

Quick check you've got that.

I'd like to find the key features of and sketch g of x equals x squared plus 6x minus 27.

Remember, a sketch will include key features such as intercepts and turning points.

Once you've sketched g of x, I'd like you to, on the same grid, sketch g of 3x.

Pause and do this now.

Welcome back, all sorts of things we can identify about g of x.

We can see by factorising how it's got roots at x equals negative nine and x equals three.

We can rearrange that quadratic to find a turning point at (-3, -36).

We know there'll be a y intercept at (0, -27).

Hence we can sketch y equals g of x like so.

So on the same grid, we're gonna sketch g of 3x.

Let's look at those key features.

For g of 3x, we're gonna multiply all the inputs, the x values, by one over three.

So (-9, 0) becomes (-3, 0).

(3, 0) becomes (1, 0) and that turning point, (-3, -36) becomes (-1, -36).

With this information, we're able to sketch y equals g of 3x.

Practise time now, question one.

We know one coordinate pair on the linear function f of x.

For each case, I'd like you to select the correct transformation from the transformations at the bottom of the screen.

In each case, I'd like you to write a sentence to justify your decision.

Pause and do this now.

Question two, I'd like to find the key features of and sketch g of x equals negative x squared minus 25 x minus 150.

Then, using those key features of g of x, on the same grid, allow you to sketch g of 5x.

Pause and do this now.

Question three, the table in the graph tells that f of x equals cos x has two points whereby f of x equals one in the domain x is greater than or equal to zero and x is less than or equal to 360.

For part A, how many points are there on the graph of f of 2x where f of 2x equals one in the domain x is greater than or equal to zero and less than or equal to 360? For part B, how many on the graph of f of 3x, i.

e.

in the same domain, how many times does f of 3x have an output of one? For part C, you're gonna consider the same question for f of 4x, and then for D, you're gonna generalise.

Pause and have a think about this problem now.

Welcome back, feedback time.

Question one, we were matching each graph to the correct transformation.

For A, we should have said that's f of 2x, for B, f of 8x, and for C, that's f of a half x.

Next, I ask you to write a sentence to justify your decision in each case.

For A, f of 2x, you might have written, "This is the transformation f of 2x "because the y value, the output, remained the same "but the x value, the input, was multiplied "by scale factor of one over two." For B, f of 8x, you might have written, "This is the transformation f of 8x "because the y value, the output, remained the same, "but the x value, the input, was multiplied "by scale factor of one over eight." For C, f of a half x, you might have written, "This is the transformation f of a half x "because the y value, output, remained the same, "but the x value, the input, was multiplied "by scale factor one over one half, which is two." For question two, we're finding the key features of g of x to begin with.

That would be those routes, that y intercept and that turning point.

So you can sketch y equals g of x like so.

A negative parabola this time, because we've got a negative x squared coefficient.

Using those key features, we can multiply all of those inputs, the x values by one over five, to find these key features on g of 5x and therefore draw y equals g of 5x.

Question three, part A, how many points are on the graph of f of 2x where f of 2x equals one in this domain? There's three points on the graph of f of 2x.

For part B, how many points for the graph of f of 3x? There is the graph of f of 3x and you see there's four points.

Part C, how about the graph of f of 4x? That's five points.

For D, I ask you to generalise and you might have written something along the lines of, "f of axe has a plus one points "at f of axe equals one in the domain x is greater than "or equal to zero and less than or equal to 360." Well done, we're at the end of the lesson now, sadly.

But we've learned, we've learned to recognise that the effect of applying the transformation y equals f of axe to a graph is a stretch by scale factor one over a in the x-direction.

Some magnificent mathematics today.

I hope you enjoyed it as much as I enjoyed it and I look forward to seeing you again soon for more learning, goodbye for now.

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