Lesson video

Hello, Mr. Robson here.

Smart choice to join me for maths today, especially exciting today because we're learning about transformations of graphs of the form y equals af(x).

This is a pretty gorgeous bit of mathematics, so let's not hang around here.

Let's get stuck in.

Our learning outcome is I'll be able to recognise the effect of applying the transformation.

Y equals af(x) to a graph.

Transformation's a keyword you're gonna hear a lot throughout today's learning.

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

There's two parts to our learning today, and we're gonna begin by exploring Y equals af(x).

We know graphs can be transformed in a number of ways.

Let me show you some examples.

When we transform the graph y equals f(x) by y equals negative f(x), we see a reflection in the x axis.

That's what the transformation negative f(x) does.

Let's have a look at another example.

F of negative x transforms f(x) by reflection in the y axis.

Have a look at another example.

f(x) plus A transforms f(x) by translation of A in the y direction.

There are lots of ways that we can transform that graph of f(x).

So what about this one? What about the transformation af(x)? Our Oak pupils are speculating as to what it might do.

Aisha starts the conversation with, "We've seen transformation of a graph by reflection and by translation.

What kind of transformation do we think af(x) will be?" Jacob says, "af(x) equals A multiplied by f(x).

It's multiplication that makes things change size, af(x) must be an enlargement." That's an interesting idea, but Sam comments, "The graph of f(x) is already infinitely long.

How can it possibly be enlarged?" Aisha chips in with, "Good point, Sam, but you are right Jacob.

af(x) equals a multiplied by f(x), so something must be multiplied." What do you think's going on here? What do you think will happen? We take the graph f(x) and apply transformation af(x).

How do you think it will affect it? Pause, have a conversation with the person next to you.

Good think to yourself, see you in a moment.

Welcome back.

Wonder what you thought, what you said, what your hypothesis is for what af(x) is going to do to a graph.

We can find out how af(x) affects f(x) by exploring some graphs.

Let's start by looking at this simple linear graph.

f(x) equals X.

That's y equals 2f(x).

Here is y calls 3f(x).

What sort of transformation do you see? Again, pause, have a conversation with a person next to you or good think to yourself, what transformation are you seeing there? Welcome back.

Let's explore some of those ideas.

Here's Sam and Jacob again.

Sam says, "For 2f(x), I saw a stretch in the y direction." Sam is seeing this graph stretch like that.

Jacobs sees something different.

Jacob says, "I saw a stretch towards the y axis." Jacob sees it stretching like that.

These are good conjectures, but they're in conflict with each other.

They can't both be right.

We can test both conjectures by exploring another example.

How does the transformation af(x) look if f(x) equals x plus two.

There's something crucially different about this linear function.

When we apply the transformation 2f(x), we get that graph.

When we apply the transformation 3f(x), we get that graph.

Jacob says, "Af(x) cannot be a stretch towards the y axis.

We've ruled out that option." Jacob adds "The x intercepts have not moved in that direction." Can you see how this transformation is not a stretch towards the y axis? It's not a stretch in the x direction.

We can see clearly what's going on by looking at this example, do you see in which direction the graph is stretching in each case? Jacob says, "You were right, Sam.

It is a stretch in the y direction." The transformation af(x) is a stretch by a scale factor of A in the y direction.

It's a really important sentence that one.

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

Welcome back.

Any coordinate pair on the graph of f(x), has an input of x and an output of f(x).

Any coordinate pair on the graph 2f(x) has an input of C and an output of 2f(x).

Can you see how 2f(x) has affected our graph? It is the output, f(x), the y coordinate which is multiplied or enlarged you might say, by a scale factor of A, the x coordinate is unchanged by this transformation.

For the transformation 2f(x), the y coordinate is multiplied by a scale factor of two.

So the coordinate pair zero, two in this case became zero, four.

The pair one, three became one, six, negative four, negative two became negative four, negative four.

In each case, the x coordinates are unchanged by this transformation, whereas the y coordinates were multiplied by two.

Let's see if that's the case for another example.

For the transformation 3f(x), the y coordinate is multiplied by a scale factor of three.

The coordinate pair zero two became zero, six.

Negative one, one became negative one, three and negative three negative one became negative three, negative three.

The x coordinates are unchanged.

The y coordinates are multiplied by that scale factor of three.

Let's check you've got that.

What I'd like you to do is fill in the blanks.

There are some statements about the transformation for f(x) and I'd like you to complete them.

Pause and do so now.

Welcome back.

Let's see if we completed those statements correctly.

It should read, for the transformation 4f(x), the y coordinate is multiplied by scale factor of four.

The coordinate pair zero, two became zero, eight.

The coordinate pair one, three became one, 12.

The coordinate pair negative four, negative two became negative four, negative eight.

The x coordinates are unchanged.

The y coordinates are multiplied by four.

Well done, another check.

The coordinate pair three, five is on the line f(x).

What I'd like you to do is match the transformation with the new coordinate pair.

Pause, get matching.

Welcome back, hopefully you paired them up like so.

The transformation af(x) multiplies the outputs by a scale factor of A.

The inputs x are unchanged.

As we explore af(x), some graphs will look like they've been stretched towards the y axis.

It'll look like a stretch in the x direction.

Let's look at this example.

That's y it equals f(x).

That's y equals 2f(x), and when you see this, you might be convinced that's stretching in the x direction, it's stretching towards the y axis.

So does the transformation af(x) have a different effect on quadratic graphs? Absolutely not, but we're gonna need to test other examples to answer that question for us.

Transforming the graph f(x) equals x squared plus 4x reveals the truth.

Can you see a crucial difference between this parabola and the one we saw in the last example? The turning point this time is not the origin.

The turning point here is at negative two, negative four.

So when we apply a transformation to f(x), we see the truth.

There's no stretch towards the y axis.

There's not a stretch in the x direction.

The stretching is happening in the y direction.

We can see that the graph is again stretched in the y direction.

This will be true for all graphs, linear graphs, quadratic graphs, cubic graphs, all graphs.

For the transformation 2f(x), The coordinate pair one, five became one, 10.

The turning point negative two, negative four became negative two, negative eight.

Quick check you've got this.

The turning point of f(x) equals negative x squared minus four x minus 15 is negative two, negative 11.

What will the turning point of the transformation 5f(x) be? Pause, take a pick from those three options.

Welcome back, hopefully you said option C.

The transformation af(x) affects linear and non-linear graphs in the same way, it multiplies the outputs by a scale factor of A.

The input, negative two, is unchanged.

The output, negative 11, is multiplied by that scale factor of five to become negative 55.

Well done, another quick check.

A local minimum on f(x) equals x cubed minus 36 squared plus 225x is 15 zero.

What will the local minimum be on the graph of 4f(x)? Three options to choose from.

Pause, see if you can reason which one it is.

Welcome back.

Well done if you spotted it's A, 15, zero.

Hold on, you told me the local minimum was 15, zero and you're now telling me after transformation it's 15, zero.

Where's the multiplication in that example? The transformation af(x) affects linear and non-linear graphs in the same way.

It multiplies the outputs by a scale factor of A.

In this case, the output is zero.

When we multiply that by four, it remains zero.

Something very interesting about the transformation af(x).

Every non-zero y value was transformed.

The two points with a zero y value were invariant.

So in the transformation of that graph, zero, zero and 15, zero were invariant points.

The transformation y equals af(x) is particularly beautiful when applied to trigonometric graphs.

Notice in this example how f(x) equals sine X is affected by this type of transformation.

There's y equals 2f(x), there's y equals 4f(x).

How beautiful is this graph looking now? the same rules have applied.

It's a stretch in the y direction by a given scale factor.

So the point 91 on f(x) became 92 on 2f(x), it became 94 on 4f(x), whereas the point 180, zero didn't transform anywhere because the output of zero, well, you can multiply that by anything, you're staying at zero.

The same rules apply with this transformation when we use them on trigonometric graphs.

Let's check you've got that.

This is f(x) equals cos X for x values greater than or equal to zero, less than or equal to 360.

What will the graph of 5f(x) look like? Pause and draw that one for me.

Welcome back, hopefully for the transformation 5f(x), you drew that.

It almost looks like that pair of graphs is smiling at you and I'm certainly smiling and enjoying the beauty of this transformation.

Useful starting point when you're transforming trigonometric graphs in the form af(x), the invariant points anywhere where y equals zero.

When you multiply the output by the scale factor, it remains zero.

You have invariant points, you plot the rest of the graph around those, all other points where Y was not equal to zero, they were stretched by scale factor five in the y direction.

Zero, one became zero, five, 180, negative one became 180, negative five and so on.

Practise time now, question one, part A, I've given you the linear graph of f(x).

I'd like you to draw 2f(x).

For B, I've drawn the parabola f(x).

I'd like you to draw 3f(x), pause and do these two now.

Question two.

This is the graph of f(x) equals x cubed minus five x squared plus seven x minus three.

On that same grid part A, I'd like you to draw 4f(x).

For part B, I'd like you to comment on the impact the transformation had on the intercepts.

The x intercepts the y intercept.

For this you'll need to write at least one sentence.

pause and do this now.

Question three, this function, Y equals f(x) is unknown but many coordinate pairs unknown.

What I'd like you to do is list five coordinate pairs that are on the transformation 12f(x), pause and do this now.

Question four.

This is the graph of f(x) equals cos x in that domain of x values.

For part A on the same grid, I'd like you to draw 2f(x), for part B, I'd like you to write a comment on the impact the transformation had on the range of the function.

Pause and do that now.

Feedback time, question one, part A, I asked you to draw 2f(x).

It should have looked like so.

Your x intercept remained at two, zero.

Your y intercept should be at zero, eight.

If you've got a straight line going through those two points, you're right.

Part B joined three F of x would've looked like so.

That turning 0.

3 negative one should now be at three, negative three and the y intercept has accelerated so far in the y direction, we can no longer fit it on this graph.

Question two part A, I asked you to draw 4f(x).

Your graph should look like so.

And for B, I asked to comment on the impact the transformation had on the intercepts.

You might have written, the x intercepts were invariant points because they had an output y equals zero.

Be sure to include the word invariant in your sentence there.

The transformation af(x) affects the outputs by multiplication, zero multiplied by anything remains zero.

Hence they remained invariant points.

On the y intercept, you might have written, the y intercept was transformed from zero negative three to zero negative 12.

The input x equals zero remained the same, but the output the y value was multiplied by the scale factor of four.

Question three, I asked you to list five coordinate pairs on the transformation 12f(x).

You should have listed these five.

Question four part A, I asked you to draw 2f(x), you should have drawn that, and then for B, I asked you to comment on the impact that the transformation had on the range of the function.

There is the range of f(x).

Let's compare it to the range of 2f(x) and you might have written, the range of the graph was also multiplied by the scale factor of two.

Well done.

Onto the second half of our learning now where we're gonna look at sketching y equals af(x).

A useful thing to remember about transformations of graphs is mathematically what they mean.

When we look at these things algebraically, what's going on here? I'd like you to pause and see if you can answer this question.

Mathematically, what do each of these things mean? Welcome back, f(x) minus one, well that means the input has one subtracted from it before the function F is performed on it.

What about F of negative X? Well, that's the input being multiplied by negative one before the function F is performed on it.

How about negative f(x)? How does that one differ? Well, that's the output of f(x) being multiplied by negative one.

So how about 2f(x)? Well, that's the output of f(x) being multiplied by two.

So this transformation af(x) is multiplying our outputs by A.

The transformation af(x) is affecting our outputs.

Here are a table of values and graph for f(x) equals x cubed minus 3x minus one.

Here's a table of values for 3f(x), but I've left those 3f(x) output values outta the table.

To populate a table of values to graph 3f(x), we just have to multiply the outputs of f(x) by three.

'Cause remember this transformation affects the outputs, so we take the outputs of f(x) and multiply them by three and we get the outputs of 3f(x).

We can plot those values on the graph and then graph y equal 3f(x).

Quick check you've got this.

Here's a table of values and graph for f(x) equals negative x cubed plus 3X plus two.

What I'd like you to do is populate the table of values for the transformation 5f(x).

Pause and do that now.

Welcome back, hopefully you took those outputs of f(x), multiplied them all by five and found the outputs of 5f(x).

Now that we have a table of values for 5f(x), we can graph it, we'll plot those coordinate pairs and we can draw the function y equals 5f(x).

The outputs have been multiplied by five.

Hence we've stretched our graph by a scale factor of five in the y direction.

Think about it.

Those outputs are the y values we're plotting when plotting the coordinates.

When we multiply those by five, we're obviously stretching in a y direction.

To enable us to quickly sketch graphs of transformations of the form af(x), it's important we notice what's happened to the key features of the graph.

Let's compare y equals f(x) to y equals 5f(x).

Well, the shape remained a negative cubic curve.

Lots has changed after transformation, but the essence of this graph hasn't.

It's still a negative cubic curve.

The x intercepts were invariant points because the output is zero at those points.

You can multiply that by whatever you like.

It's still zero.

The y intercept was transformed by a scale factor A in the y direction and the same thing happens with a turning point.

Unless that turning point is on the x axis, the turning point with a non-zero y value is transformed by a scale factor of A in the y direction.

If we spot the key features and understand how the transformation affects them, we can sketch the graph of f(x) equals negative three x squared plus six x plus nine, and the transformation for f(x).

The shape of f(x) is in negative parabola in this case, it will have a y intercept at zero, nine.

We can factorise and find its roots at negative one and three and we can rearrange to the completeness square form to find a turning point at one, 12.

So there is our sketch of y equals f(x) and don't forget a good sketch highlights the key features, intercepts, turning points.

How will these key features be affected by the transformation for f(x)? Well, the shapes going to remain a negative parabola.

The y intercept will transform from zero, nine to zero, 36.

The roots will be in variant points, negative one, zero, three, zero.

They'll be in variant points and the turning point will transform from one, 12 to one, 48.

That's where those coordinate pairs will be, and that is y equals 4f(x) or a sketch of y equals 4f(x).

Quick check you've got this.

I'd like you to start by defining the key features of f(x) equals 2x squared minus 12x plus 10.

Once you've found those key features, I'd like you to then describe how they will be affected by the transformation 6f(x).

Pause and do this now.

Welcome back, let's see how we got on with this one.

We should have started by defining the key features of f(x).

Well, the shape is that of a positive parabola, there's a y intercept at zero, 10, there's roots at one and five.

You can see those by factorization.

We can rearrange to find the turning point at three, negative eight.

You put those points on a sketch, there is y it equals f(x).

Next, let's see how you got on with, how will those key features be affected by the transformation 6f(x).

Hopefully you said the shape's unaffected, still gonna be positive parabola.

The y will be transformed from zero, 10 to zero, 60.

That output being multiplied by that scale factor six.

The roots will be invariant points at one, zero and five, zero.

The output is zero.

Multiply that by six, it's still zero.

That turning point will also be transformed.

Three, negative eight, transforming to three, negative 48.

So our sketch of y equals 6f(x) would've been there.

Another interesting thing about this transformation is that the value of A in af(x) can be fractional.

Let's take a look at an example.

We can graph half of f(x) by halving the outputs of f(x).

You see a table of values and a graph here for F of x equals cos X.

Let's populate a table of value for half f(x).

We're just gonna take the outputs of f(x) and multiply them by half.

When we graph that, can you see what this is gonna look like? How nice is that? That's y equals a half f(x).

At first, the transformation looks like a compression towards the x axis, but we don't call it that.

It's a stretch, we still call it a stretch.

When A is greater than zero and less than one, the transformation is a stretch by a fractional scale factor.

The outputs, the y coordinates, therefore move closer to the x axis.

That's why it has the appearance of a compression, but it's still a stretch and it's important that you call it so.

Quick check you've got this.

This is the graph of the function f(x) equals cos x.

What will the graph of 1/4 f(x) look like? Pause and draw that one now.

Welcome back.

You should have drawn something like that.

Four invariant points, every other point stretched in the y direction by our fractional scale factor, in this case a quarter.

So zero, one, for example, was transformed to zero, a quarter or zero, 0.

25.

Practise time now, question one, part A, I'd like you to sketch f(x) equals 3x plus nine.

Be sure to label any key features such as intercepts.

For part B, you're then gonna sketch 6f(x).

For part C, you're gonna comment on how the transformation 6f(x), impacted both intercepts of f(x).

You'll write at least one sentence for part C.

Pause and do this now.

Question two, I'd like to sketch f(x) equals x squared minus 4x plus three.

Again, be sure to label any key features, intercepts, turning points.

For part B, I'd like to sketch the transformation 3f(x), pause and do this now.

For question three.

This is the graph of the function f(x) equals sine X for x is greater than or equal to zero, less than or equal to 360.

For part A, I'd like you to graph the transformation 3f(x).

For part B, answer the question, what is the range of 3f(x)? For part C, I'd like to graph half of f(x) and for D, what is the range of half f(x)? Pause, do those things now.

Feedback time.

Let's see how we got on with question one.

Our sketch of f(x) equals 3x plus nine should have looked like so.

Our sketch for part B of 6f(x) should have looked like so.

Commenting on the transformation of the intercepts you might have written, the x intercept is an invariant point because the output is zero, negative three, zero remained invariant.

The y intercept however is transformed to zero, 54.

The output nine was multiplied by a scale factor of six.

Question two part A, if we're gonna sketch f(x) equals x squared minus 4x plus three, we need to know a few things about the key features, so it's a positive parabola with a y intercept zero, three, roots at one and three and a turning point at two, negative one.

There's our sketch of y equals f(x).

We can use that sketch to sketch 3f(x), that will have those coordinate points.

That is the sketch y equals 3f(x).

Question three part A, I asked you to graph 3f(x).

That should have looked like so.

For part B, what's the range of the transformation 3f(x)? Well, the range will be 3f(x) is greater than or equal to negative three and less than or equal to positive three.

Graphing half f(x) should have looked like so, and the range for that graph, a half f(x) is greater than equal to negative a half, less than or equal to positive a half.

Well done.

We're at the end of the lesson now sadly, but we have done some learning today.

We have learned to recognise the effect of applying the transformation, y equals af(x) to a graph.

The transformation y equals af(x) affects the outputs, the y values.

It therefore stretches the graph by a scale factor of A in the y direction.

If A is greater than zero and less than one, we still see the graph stretched, but the scale factor is fractional.

The outputs therefore get closer to the x axis.

I hope you've enjoyed this bit of mathematics as much as I enjoy it and I look forward to seeing you again soon for more maths.

Goodbye for now.