Lesson video

Hello, Mr. Robson here.

Super choice to join me for maths today, especially because we're exploring the transformation, y = f(-x).

This one is a cracker.

Enjoy.

Our learning outcome is I'll be able to recognise the effect of applying the transformation y = f(-x) to a graph.

Transformation will be a key word today.

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

We've got two parts to our learning.

We're gonna begin by exploring y = f(-x).

We know that the transformation -f(x) will be a reflection of f(x) in the x-axis.

Here's an example of that.

There's y = f(x) and y = -f(x).

The coordinate pair (1, 3) becomes the coordinate pair (1, -3).

All other coordinate pairs reflect like this giving us a reflection in the x-axis.

The transformation -f(x) is a reflection in the x-axis.

Our Oak pupils are speculating about today's transformation, f(-x).

Aisha says, "If -f(x) is a reflection of f(x), what kind of transformation do we think f(-x) will be?" Jacob says, "I think the coordinate pair (1, 3) will become (-1, 3), I think f(-x) will be a reflection in the y-axis".

Sam says, "I think coordinate pair (1, 3) will become (1, -3) again, I think f(-x) will also be a reflection in the x-axis".

Laura says, "I think f(-x) would turn the point (1, 3) into (-1, -3).

That will make it a rotation around the origin".

With whom do you agree? What do you think the transformation f(-x) is going to do? Pause, have a conversation with a person next to you or a good think to yourself.

Welcome back.

I wonder what you think.

I wonder with whom you agreed.

Let's explore this transformation.

We can find out if f(-x) is a reflection in the y-axis, the x-axis, or it's a rotation by exploring some graphs.

There's y = f(x) and y = f(-x) for this linear function.

The coordinate pair (1, 3) became (-1, 3).

This looks like a reflection in the y-axis.

When you have a conjecture in maths, it's sensible to test it with another example, so we're not just gonna take this evidence that f(-x) is a reflection in the y-axis, we'll test another function.

If f(-x) is a reflection in the y-axis, we should see the same thing for this function, there's y = f(x), there's y = f(-x).

Yes, it is.

f(-x) Transforms the graph of f(x) by reflection in the y-axis.

That's important that sentence.

You're gonna lean on it throughout today's learning, so pause and write it down.

We can see every coordinate pair of f(x) is reflected in the y-axis to generate the coordinate pairs of f(-x).

For that pair we can see the reflection through the y-axis.

Same for this pair, and this pair, and this pair, this pair, and this pair.

All of the points on y = f(x) are reflected in the y-axis to generate y = f(-x).

Quick check you've got this, f(-x) transforms a graph by which one of those four words completes that sentence? Pause, tell the person next to you or say it's aloud to me on screen.

Welcome back.

Well done.

I can hear you saying it's option C, reflection.

That sentence should read f(-x) transforms a graph by reflection.

Let's add a bit more to that sentence, f(-x) transforms a graph by reflection in the.

Again, four options, which one completes the sentence? Pause, tell the person next to you or say it's aloud to me on screen.

Welcome back.

Well done.

I could really hear you that time, saying option C, y-axis.

The sentence should read f(-x) transforms a graph by reflection in the y-axis.

Here are our Oak pupils again, and one of them is delighted.

Jacob says, "I was right.

The transformation f(-x) is a reflection in the y-axis".

The other pupils say, "You were right, Jacob".

This means we can draw f(-x) by reflecting the coordinate pairs of f(x) in the y-axis.

Let's see that in action for this function.

We can take that coordinate pair, and note that it's a distance of one away from the y-axis, a positive one distance in the x direction.

So we're going to draw a coordinate that's a distance of one away from the y-axis but in a -x direction.

We'll do the same for this coordinate pair, that's two away from the y-axis in a positive x direction.

Let's reflect it so it's two away from the y-axis in a -x direction.

We keep doing that for every single point.

And eventually we have enough evidence to enable us to draw y = f(-x).

f(-x) Transforms the graph of f(x) by reflection in the y-axis.

We don't even need to know the function f(x) in order to draw f(-x).

We just reflect the points in the y-axis.

Let's check that you can do that.

I'd like you to draw the transformation of f(-x) in this case.

Pause, and do that now.

Welcome back.

Hopefully you took every point on y = f(x) and reflected them through the y-axis.

Once you've done that for a few points, you're able to draw y = f(-x).

Well done.

In this example, the graphs of f(x) and f(-x) will eventually intersect, just not in the region we can see here.

Let's zoom out a little.

There we go.

We can see the two graphs intersecting.

This point on f(x) is two from the y-axis.

The point on f(-x) is therefore the same.

The position of this point changed.

It varied.

This point on f(x) has a distance of zero from the y-axis, so a reflection in the y-axis does not see it move.

The position of this point is unchanged, it did not vary by the transformation.

Therefore we call this an invariant point.

Quick check you've got that.

I'd like you to draw the transformation f(-x) for this beautiful cubic graph.

And then I'd like you to tell me which coordinate pair is invariant.

Pause, and do those things now.

Welcome back.

We should be reflecting each point in the y-axis to be able to draw y = f(-x).

Once we've reflected some points, we can graph y = f(-x), and it was that point we were interested in with regard invariant points.

The coordinate pair (0, 1) is invariant.

It remains unchanged by the transformation.

Practise time now.

Question one, I'll give you f(x) and I'll ask you to draw f(-x).

Do that for both of these cases.

Pause, and do it now.

Question two.

Here's a graph of an unknown cubic function f(x).

List three coordinate pairs that will be on the graph of f(-x).

For part B of this question, I'd like you to draw f(-x).

Pause, and do this now.

Question three.

Here's the graph f(x) = sin(x) in the domain x is greater than or equal to -180 and less than or equal to 180.

For part A, I'd like you to draw f(-x), and for part B, I'd like you to answer the question which coordinate pair was invariant by the transformation? Pause, and do this now.

Feedback time.

Let's see how we did.

Question one part A, I gave you f(x).

This was a linear function, 3x + 2.

Your f(-x) should have looked like so.

For part B, f(x) was a quadratic function in this case, your graph of f(-x) should have looked like so.

Question two part A, I asked you to list three coordinate pairs that'll be on the graph of f(-x).

Let's look at some coordinate pairs that are on f(x), and then we can reflect those in the y-axis, and come up with our list of coordinate pairs that'll be on f(-x), you could have said any three from those four.

We can use those coordinate pairs to draw f(-x) and it looks like so.

Question three part A, I asked you to draw f(-x) if f(x) = sin(x).

Your graph should have looked like that.

And then for part B, I asked you which coordinate pair was invariant.

The invariant point was the origin (0, 0).

Onto the second half of our learning now, where we'll be sketching y = f(-x).

We know that f(-x) transforms a graph by reflection in the y-axis, but we don't yet understand why this happens.

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

So let's get powerful.

To start this exploration, I'd like you to answer this question.

Mathematically, what do each of these things mean? Pause this video, and see if you can write a sentence for each of those four.

What do they mean? See you in a moment.

Welcome back.

I wonder what you said.

In the case of f(x) - 1, the output of f(x) has one subtracted from it.

So for -f(x), that's the output of f(x), being multiplied by -1.

How about f(x - 1)? That's the input having one subtracted from it, before the function f, is performed upon it.

So in the last example, and this is the one that we're really interested in today, the input is multiplied by -1 before the function, f is performed on it.

So the transformation f(-x) affects our inputs.

Let's consider that for a function.

f(x) - (x -3)².

You can see that graph on the axes there.

What input gives us an output of zero? Well, the answer to that one's quite simple.

It's x = 3.

When we substitute three into the function, f(3) gives us an output of zero.

For f(x) an input of three, gives us an output of zero.

Let's consider that for f(-x) now.

What would the function f(-x) be? Well, it wouldn't be (x - 3)², it would be (-(x) - 3)².

After the transformation f(-x), what input now gives us an output of zero? Well done.

It's when x = -3.

f(-(-3)) Gives us an output of zero.

So for f(-x), we need an input of -3 to get an output of zero.

This transformation f(-x) is all about inputs and outputs.

For f(-x) the input, our x value, had to change from 3 to -3 to give us the same output of zero for this function.

Our inputs change from positive to negative, whilst the output remains the same.

It's a really important sentence, so I'm gonna say it again.

Our inputs change from positive to negative, whilst the output remains the same.

Only the x coordinate changes.

The y coordinate doesn't.

Hence all our points are reflected in the y-axis.

That is how we generate the transformation y = f(-x).

Quick check you've got this.

I'd like to fill in the blanks.

There's four spaces there to complete these two sentences.

Pause, what's going in those gaps? Welcome back.

Well done.

That first sentence should indeed read, the transformation f(-x) means the input is multiplied by -1 before the function, f is performed on it.

For our second sentence, the transformation f(-x) affects our input, hence it is the x coordinate which is affected.

It is because of this that the transformation creates a reflection in the y-axis.

Well done.

Let's consider the transformation f(-x) in a table of values.

f(x) = 2x - 3, I've populated that table of values.

For an x input of -3, we get an f(x) output of -9.

Let's consider the table of values for f(-x).

What input is now gonna give us an output of -9 for f(-x)? Well done.

It's x = 3.

For f(x), we need an input of -3 to generate an output of -9.

For f(-x), we'll need an input of 3 to generate an output of -9.

Let's drop that in the table.

What input now gives us an output of -7? Well done.

It's x = 2.

Drop that in the table.

We can keep going with this question.

What input now gives us an output of -5? And you know the answer.

It's x = 1.

What input now gives us an output of -3? x = 0, We'll drop that in the table.

And do you notice something interesting about that moment? With the whole table populated, let's have a look at that x = 0 moment.

It's the only moment where this happens.

This being the exact same input generates the exact same output after the transformation f(-x).

Let's draw the line y = f(-x).

And you notice when we transform a graph by f(-x), the coordinate pair with an x value of zero is invariant.

That's the one point in this function that has not varied by that transformation.

Quick check you've got that.

Which points are invariant in the transformation f(-x)? Pause, think about these five options.

Welcome back.

Which points are invariant in the transformation f(-x)? Hopefully you said B, all points on the y-axis, and D, all coordinate pairs with a value x = 0.

Well, think about it.

A coordinate pair with a value x = 0 will be on the y-axis, and when we're reflecting through the y-axis, it'll stay there, it'll be invariant.

The contentious one here was the origin (0, 0).

If (0, 0) is a point on f(x), then this will also be an invariant point.

But we can't say that (0, 0) is always invariant, because it might not be a point on the function f(x).

Izzy is exploring the transformation f(-x) just like we are.

Izzy says, "I found a graph which when transformed by f(-x), all of the points are invariant".

Well, that's awesome, Izzy.

Let's have a look.

f(x) = cos(x).

"When I graph f(-x), not a single point has moved".

There's f(-x).

Do you agree with Izzy? Pause, have a conversation with the person next to you, or a good think to yourself.

I'll see you in a moment.

Welcome back.

I wonder what you thought.

Did you agree with Izzy? Izzy's been tricked unfortunately.

It might look true, but, the point (360, 1) on f(x) transforms to (-360, 1) on f(-x).

That's that point there, and it transforms to that point there.

It's not invariant.

It has been transformed.

The same happens in lots of positions on the graph, we can see it happening in lots of positions.

The point (180, -1) on f(x) transforms to (-180, -1) on f(-x).

Can you see again, our inputs are changing from positive to negative, our outputs are not varying? There are infinitely many points on this function that vary.

Their position is transformed.

There's only one invariant point, and that is (0, 1) where x = 0, or that coordinate pair is on the y-axis.

The trick occurs because for this function, f(x) = cos(x), the y-axis is already a line of symmetry.

Therefore, all points of the transformation f(-x) are plotted upon a position that already exists on f(x).

Do watch out for this problem.

Quick check you've got this.

True or false? If a function has the y-axis as a line of symmetry, then all points will be invariant when we graph the transformation f(-x).

Is that true? Is it false? Once you've decided, I'd like you to select one of the two statements at the bottom of the screen to justify your answer.

Pause, and do this now.

Welcome back.

I hope you said that's false, and justified the answer with the only invariant point or points, will be the y-intercept or y-intercepts.

All other points on the graph will be transformed.

If you said that, well done.

We can quickly sketch f(x) and f(-x) by considering just the key features of a graph.

Let's look at this example, f(x) = x² + 8x + 12.

What do we know about it? Well, we know it's a positive parabola.

We know that when x = 0, f(x) = 12, so (0, 12) will be our y-intercept.

We can factorise that quadratic expression.

That'll tell us there's roots at -2 and -6.

We could rewrite it, and complete in the square form to find the turning point at (-4, -4).

This enables us to sketch the graph.

Don't forget that a sketch will always include these key features like intercepts and turning points.

We can now plot the key features of f(-x).

If this is f(x) that we can see here, what do we know about f(-x)? Well, we know that the y-intercept is going to be an invariant point.

The turning point is going to transform the output, <v ->4 will remain the same,</v> but the input -4 will become 4.

So that point transforms. The same thing will happen to the roots.

The roots won't be (-6, 0) (-2, 0), they'll be (6, 0) and (2, 0).

The reflection in the y-axis will mean it remains a positive parabola.

So, we can sketch y = f(-x) like that.

Quick check you've got this.

I'd like you to sketch f(x) = x² - 6x, and thus sketch f(-x) on the same grid.

Pause, and do this now.

Welcome back.

Let's start with the key features of f(x), it's a positive parabola.

It has roots at 0 and 6.

It has a turning point at (3, -9).

It'll look like that.

Okay, if that's y = f(x), what's y = f(-x) going to look like? It'll be a positive parabola.

Also, if we're reflecting that one in the y-axis, it'll have a root at (0, 0), which is an invariant point.

The other root will transform from (6, 0) to (-6, 0), and the turning point will transform from (3, -9) to (-3, -9).

So our sketch of y = f(-x) will be there.

Practise time now.

Question one, I'd like to sketch f(x) = 3x + 9.

For part B, I'd then like you to sketch f(-x).

For both those sketches, don't forget to include the coordinate pairs for any intercepts.

Question one part C, I'd like you to comment on how the transformation f(-x) impacted both the x- and y-intercepts of f(x).

I'd like you to write two sentences for that part please.

Pause, and do this now.

Question two, part A I'd like you to sketch f(x).

f(x) = x² + 10x + 21.

For part B, you're then going to use those key features from part A to sketch f(-x).

Don't forget to label all of the key features on f(-x).

Pause, and do this now.

Question three.

Here's an unknown function f(x), and isn't that a delightful looking function? We don't need to know what that function is in order to do part A, sketch f(-x).

You can do that.

For part B, on your sketch of f(-x), you're going to label the positions, write the coordinates, of the three given points.

Pause, try this problem now.

Feedback time.

Question one part A, sketching f(x) = 3x + 9.

It's a linear graph.

When x = 0, f(x) = 9, that's our y-intercept.

And when f(x) = 0, x = -3, that'll be our x-intercept.

Our sketch of f(x) will look like so.

We can use those key features to sketch f(-x).

And then for part C, I asked you to comment on how the transformation f(-x) impacted both the x- and y-intercepts of f(x).

You might have said, the x-intercept transformed from (-3, 0) to (3, 0), a reflection in the y-axis.

The y-intercept remained invariant.

I do hope you use the word 'invariant'.

The y-intercept remained invariant, because the input x = 0 is unaffected by this transformation.

Well done.

Question two part A, we're sketching f(x) = x² + 10x + 21.

What do we know about it? It's positive parabola.

When x = 0, f(x) = 21, that'll be the y-intercept.

There are roots at -3 and -7, and there's a turning point at (-5, -4).

f(x) Looks like so.

Sketching f(-x) for part B therefore, we'll see as transform some points but not the invariant point.

And we can sketch y = f(-x) there.

I hope you've labelled all of the key features of both of your graphs.

Question three, we were looking at this lovely unknown function, and I challenged you to sketch y = f(-x).

Hopefully you drew that, a reflection of f(x) in the y-axis.

For part B, I asked you to on your sketch of y = f(-x), label the positions of the three given points.

If you get this, you truly get this topic.

(-a, b) Transformed to (a, b).

(b) Was the output, it was unaffected.

(-a) The input was transformed to (a).

For the point (c, -d) that was transformed to (-c, -d).

The input (c) became (-c).

The output (-d) remained unaffected.

For that last point (0, e) well, (0, e) is an invariant point.

That's where x = 0, it's on the y-axis.

It's an invariant point, so that label remained unchanged.

We're at the end of the lesson now, sadly.

I hope you found it as interesting as I have.

We have learned all sorts.

We've learned that we can recognise the effect of applying the transformation f(-x) to a graph.

The effect of f(-x) is to multiply the input by -1 before performing the function, f.

Hence, the graph of f(-x) is a reflection of f(x) in the y-axis.

I love this maths, and I hope that you loved it too, and I look forward to seeing you soon, for more wonderful mathematics.

Goodbye for now.