Lesson video

Hello.

Mr. Robson here.

Excellent choice to join me for maths today, especially because we're doing transformations of graphs in the form y equals negative f of x.

Now you would've seen transformations in mathematics before, but not ones as nice as these.

Let's see what it's all about.

Our learning outcome is I'll be able to recognise the effect of applying the transformation y equals negative f of x to a graph.

Keyword you'll hear today, transformation.

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.

We're going to begin by exploring the transformation y equals negative f of x.

We know that we can transform a graph by translation, for example, f of x plus a, that's a transformation, and it'll be a translation of f of x in the y-direction by a.

What do we mean by that? Well, if on the left-hand side of the screen, that's y equals f of x.

On the right-hand side of the screen, you can see y equals f of x plus 2.

f of x plus 2 is a translation of f of x in the y-direction by positive two.

You can see that translation there.

All of the points have moved positive 2 in the y-direction.

Aisha, Sam, Jacob, and Laura are speculating about this transformation, negative f of x.

Aisha starts the conversation by saying, "If f of x plus a is a translation of f of x, what kind of transformation do we think negative f of x will be?" It's good to hypothesise and speculate in maths.

This is how discovery happens.

Jacob says, "Negative f of x is multiplication by a negative.

Multiplication makes things bigger.

I think it's an enlargement." Sam says, "I think negative f of x will make positives negative and vice-versa.

Therefore, it will be a reflection of f of x." Laura speculates, "I think negative f of x would turn the point five, five into negative five, negative five.

That will make it a rotation around the origin." What do you think? With which pupil do you agree? Pause and have a think.

What do you think negative f of x is going to do? Welcome back.

I wonder what you thought.

I wonder what you said.

We can find out if negative f of x is an enlargement, a reflection, or a rotation by exploring some graphs.

Let's start with a really simple graph, f of x equals x.

That's the graph of f of x.

That's the graph of negative f of x.

Here's our Oak pupils again.

Jacob says, "Well, negative f of x could be a reflection in the y-axis." Sam says, "Negative f of x could be a reflection in the x-axis." And Laura says, "Negative f of x could be a 90-degree rotation around the origin." And Aisha's right to say, "We need further exploration.

Let's try another function." That's a great idea.

So let's look at f of x equals 2x minus four.

It's another linear function but it's got some important differences to f of x.

There is negative f of x in this case.

This gives our Oak pupils some more ammunition to add to their thinking.

Jacob says, "Negative f of x is not a reflection in the y-axis." Sam says, "Negative f of x could be a reflection in the x-axis." And Laura says, "Negative f of x is not a rotation around the origin." We've ruled out some of the possibilities for what negative f of x is as a transformation.

Aisha says, "We are getting closer!" We are indeed.

Should we look at one more example? The impact of the transformation of negative f of x is more easily seen in quadratics.

There's a graph of f of x equals x squared, and the graph of negative f of x.

Let's have a look at another quadratic.

f of x equals x plus two squared.

That is f of x, and that's negative f of x.

Have you spotted what kind of transformation this is? Our Oak pupils have.

They say, "Negative f of x transforms a graph by reflection in the x-axis." Well done, you people.

That's an important sentence, so you'll want to pause this video and write it down.

Negative f of x transforms a graph by reflection in the x-axis.

Quick check you've got that.

Negative f of x transforms a graph by.

Which one of those four options is it? Pause, tell the person next to you, or say it aloud to me on screen.

I hope you said C, reflection.

Negative f of x transforms a graph by reflection.

But that's not the full story.

To finish the story, we need to answer this one.

Negative f of x transforms a graph by reflection in the.

Which one of those four options completes that sentence? Pause, tell the person next to you, or say it aloud to me on screen.

Welcome back.

I do hope you said B, x-axis.

That sentence should be negative f of x transforms a graph by reflection in the x-axis.

You can see that in that diagram on screen.

Knowing that negative f of x is a transformation by reflection in the x-axis enables us to sketch negative f of x.

Any known point can be reflected in the x-axis to show us where negative f of x is located.

I'll take this known point.

It's two in the positive direction from the x-axis.

So I'm going to plot it two in the negative direction from the x-axis.

I'm drawing a reflection in the x-axis.

I can take these two points and do something similar.

They're three in a positive direction from the x-axis, so I'm going to plot them three in the negative direction from the x-axis.

Can you see how I'm building my reflection? And complete the sketch like so.

There's f of x and its transformation, negative f of x.

You can see that negative f of x is a reflection of f of x in the x-axis, and importantly, we can perform this reflection point by point.

Quick check you've got this.

This is the graph f of x equals negative x squared plus 4x minus five.

Which coordinate pairs will be on the graph of negative f of x? Pause.

See if you can spot which pairs will be on that graph.

Welcome back.

Hopefully you said A, one, two will be a coordinate pair on the graph of negative f of x.

And that will be a reflection at that point there.

B, the coordinate pair two, one will also be there.

You can see it reflected in the image now.

C and D will not be.

C and D would be there.

They would've been reflections of the same two points but in the y-axis, which is not the transformation negative f of x.

Transformation negative f of x is a reflection in the x-axis.

There it is, negative f of x.

We don't actually need to know the exact function of f of x in order to sketch negative f of x.

This is a cubic graph and we've been given no other information.

We don't what this function is.

But we don't need to.

Any point can be reflected in the x-axis to show us where negative f of x is located.

I can take that point on f of x and reflect it through the x-axis.

I'll do the same for that point.

And that point.

And that point.

And that point.

Then we can sketch negative f of x.

Quick check you can do this now.

This is f of x, another unknown cubic function.

I'd like you to sketch negative f of x.

You're going to need some squared paper.

You're going to need to copy down f of x as it looks on this screen and then you're going to sketch negative f of x on the same grid.

Pause and do that now.

Welcome back.

Hopefully you identified certain points on f of x by their distance from the x and y-axes and then you reflected them in the x-axis.

We can do this point by point by point by point, and that enables us to sketch negative f of x.

Sometimes, we will see negative f of x intersect f of x.

In this case, f of x equals sin x.

We can identify some points on the function.

Like so.

When we reflect them, we can see that our graphs will intersect one another.

There's the reflection of those points.

And there's the graph of negative f of x.

Notice that the intersections are on the x-axis.

These three points.

When we reflect a point in the x-axis, if its distance from the x-axis is zero, it remains there.

We call these invariant points.

Invariant means they have not varied.

They haven't changed.

That was applicable to those three points in this case.

A couple of little checks for you now, the first of which I'm going to ask you to sketch negative f of x for this function.

That's f of x equals x squared minus 6x plus five.

You want to copy that function and then sketch negative f of x on the same grid.

Pause and do this now.

Welcome back.

You should have identified certain points on f of x, reflected those in the x-axis, and drawn negative f of x like so.

Next little check with that same transformation.

During the transformation, which coordinate pairs remained invariant? Pause.

Have a think about that now.

Welcome back.

Hopefully you said B and C.

That was those two points.

Why are they invariant? Well, these two points were not varied by the transformation.

If they're not varied, then they are invariant.

They remained as one, zero and five, zero.

Practise time now.

Question one, part A and B, I've given you f of x.

I'd like you to, on both graphs, draw negative f of x.

Pause and do this now.

For question two.

This is the graph of an unknown cubic function f of x.

I'd like you to list three coordinate pairs that will be on the graph of negative f of x.

And then for part B, I'd like you to draw negative f of x.

Pause and do this now.

Question three.

Here is the graph f of x equals cos of x plus one in the domain x is greater than or equal to negative 360 and less than or equal to positive 360.

For part A, I'd like you to draw negative f of x.

For part B, I'd like you to tell me which coordinate pairs were invariant? Pause and do this now.

Feedback time.

Question one, you had f of x, and I asked you to draw negative f of x in both cases.

Take any coordinate pairs, reflect them in the x-axis, and you can draw negative f of x.

There's your answer for part A.

For part B, any coordinate pairs reflected in the x-axis enables us to draw negative f of x.

Question two.

I asked you to list three coordinate pairs that will be on the graph of negative f of x first.

There are the coordinate pairs on f of x.

When reflected, they'll be there.

And so you could have said any three coordinate pairs from that list.

For part B, you were drawing negative f of x, which should look like so.

Question three, I gave you the graph of f of x equals cos x plus one and asked you to draw negative f of x.

We can take those coordinate pairs, reflect them in the x-axis, and sketch negative f of x like so.

For part B, which coordinate pairs were invariant? The invariant points were negative 180, zero and positive 180, zero.

On to the second half of our learning now where we're going to be sketching negative f of x.

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

If you can understand and explain why things do what they do in mathematics, you are a very powerful mathematician indeed.

I'm looking forward to exploring this concept with you.

Let's begin with this question.

Mathematically, what do each of these things mean? Pause.

See if you can write a sentence to explain what each of those four things means.

Welcome back.

Let's have a look.

What do these things mean? f of x.

Well, that's a function f being performed on an input x.

f of x plus two.

Well, that's the output of f of x having two added to it.

Two f of x.

Well, that's the output of f of x being multiplied by two.

Now, last one.

Negative f of x.

Well, that's the output of f of x being multiplied by negative one.

So the transformation negative f of x is multiplying our outputs by negative one.

It's affecting our outputs.

Let's consider that for a function.

It's a simple linear function.

f of x equals x plus three, and you can see from my table of values how I graphed that function.

Let's have a look at the table of values for negative f of x.

The most efficient way to populate our table of values for negative f of x is to multiply our f of x outputs by negative one.

That's those outputs from f of x multiplied by negative one.

We can see the impact of the outputs being multiplied by negative one.

Zero, three becomes zero, negative three.

We can reflect that on the graph here.

One, four becomes one, negative four, as is shown by those points.

Two, five becomes two, negative five.

There we go.

Three, six becomes three, negative six.

See that one on the graph.

This process continues.

Our inputs have not changed.

All of the outputs, our y-values, have changed.

They've changed from positive to negative.

Hence, we see a reflection in the x-axis.

It's because the transformation negative f of x affects our outputs.

It makes all of those positive outputs negative.

Quick check you've got that.

This is a table of values and the graph of f of x equals x squared.

I'd like you to populate the table of negative f of x and plot the transformation.

Pause and do that now.

Welcome back.

Multiplying the outputs of f of x by negative one will give us our outputs for negative f of x.

When we plot those, we can see negative f of x there.

This example shows us what happens with invariant points.

Let's look at a table of values for f of x equals x squared minus 2x minus three.

Let's substitute in x equals negative two.

That gives us an output for f of x of five.

When x equals negative one, the output is zero.

When x equals zero, the output is negative three.

We can keep going with that process and populate the table like so.

Our table of values contains positive values, negative values, and zero values.

That's why this is a useful example to see what happens with invariant points.

There's f of x.

Now let's think about our table of values for negative f of x.

We know we can populate our table of values for negative f of x by multiplying all the outputs for f of x by negative one.

When we do that, we get these values in the table for negative f of x.

Positive values on f of x have become negative values on negative f of x.

Negative values become positive.

But zero multiplied by negative one remains unchanged.

There is the graph y equals negative f of x.

And because zero multiplied by negative one remains unchanged, when we graph negative f of x, these two points do not change.

They do not vary.

Therefore, they are invariant.

Whenever the output, our y-value, is zero, that point will remain invariant by the transformation negative f of x.

Quick check you've got that.

I'd like to calculate the values for f of x.

In this case, f of x equals x cubed minus 2x squared.

Then I'd like you to populate the table of values for negative f of x.

I'd like you to plot both the function and the transformation and highlight any invariant points.

Quite a lot to do here, so pause and I'll see you when you're finished.

Welcome back.

Hopefully you populated your table of values for f of x like so and graphed f of x like this.

We get this beautiful cubic curve.

Next, the easiest way to populate our table of values for negative f of x is to take those outputs of f of x and multiply them by negative one.

From here, we can graph y equals negative f of x.

And the last thing I asked you for was invariant points.

You should have identified these two points.

The invariant points are zero, zero and two, zero.

They were not varied by the transformation.

Hence, we call them invariant points.

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

Consider this example.

f of x equals x squared plus 8x plus 12.

We know the shape of this graph.

It's going to be a positive parabola.

When x equals zero, f of x equals 12.

So at zero, 12, we'll find the y-intercept.

We can factorise the quadratics to identify the roots at negative two and negative six.

We can rewrite it in completing-the-square form to identify there's a turning point at negative four, negative four.

From here, we can sketch it.

Don't forget that a good sketch will label all of those key features, turning points and intercepts.

We can now plot the key features of negative f of x.

The roots are going to be invariant points.

The turning point is going to transform from negative four, negative four to negative four, positive four.

The y-intercept will transform from zero, 12 to zero, negative 12.

And it's a reflection in the x-axis, so we're going to see a negative parabola.

y equals negative f of x will look like that in this case.

Quick check you've got that.

I'd like you to sketch f of x equals x squared minus 6x.

Remember, a sketch, you'll label any turning points.

You'll label any intercepts of the axes.

Once you've sketched f of x, I'd like you to then sketch negative f of x.

Pause and have a go at this now.

Welcome back.

Starting with sketching f of x.

Well, we know it's a positive parabola.

We know when x is zero, f of x is zero.

That'll be the y-intercept.

We can factorise to find the roots at zero and six.

And we can write in completing-the-square form to identify a turning point at three, negative nine.

There is our sketch of y equals f of x with those key features labelled.

We can now start to graph negative f of x.

We know it's going to be a negative parabola because it's a reflection in the x-axis.

The roots are going to be invariant.

And the turning point's going to transform to three, positive nine.

So our sketch of y equals negative f of x will look like so.

Practise time now.

Question one.

Part A, you're going to sketch f of x equals 3x plus nine.

Just a sketch.

For part B, you'll then sketch negative f of x.

And for part C, you're going to comment on how the transformation negative f of x impacted both intercepts of f of x.

Write at least one sentence there.

Pause and do this now.

Question two.

You're going to sketch a parabola this time.

I'd like you to sketch f of x equals x squared plus 10x plus 21.

Once you have that sketch, for part B, you will then be able to sketch negative f of x.

Pause and do this now.

Question three.

f of x equals cos x minus 0.

5 in the domain x is greater than or equal to zero and less than or equal to 360.

You've got a table of values for f of x there.

I'd like you to complete the table for negative f of x and then graph that transformation.

Pause and do that now.

Question four.

Izzy graphs a function perfectly but has made an error when transforming it.

Izzy says, "Negative f of x is a vertical reflection so it looks like this." What I'd like you to do is write at least one sentence of advice that will help Izzy understand her error and improve her work.

So pause and get some advice written for Izzy.

Right, feedback time.

Question one, part A, our sketch of f of x.

Well, we know it's a linear graph.

We know when x equals zero, f of x equals nine.

That'll be our y-intercept.

And we know when f of x equals zero, x equals negative three.

That's our x-intercept.

Our sketch for y equals f of x looks like so.

From there, we can sketch negative f of x like so with those key features labelled.

For part C, I asked you to comment on how the transformation negative f of x impacted both intercepts of f of x.

You might have said, "The y-intercept transformed from zero, positive nine to zero, negative nine.

The output nine was multiplied by negative one.

The x-intercept remained invariant." I hope you used the word invariant there.

"Because the output zero was multiplied by negative one, so remains unchanged." Question two, part A, I asked you to sketch f of x equals x squared plus 10x plus 21.

We know that's a positive parabola.

When x equals zero, f of x equals 21.

That's our y-intercept.

We can factorise to find roots at negative three and negative seven.

We can write in completing-the-square form to find the turning point at negative five, negative four, so there's our sketch of y equals f of x.

We can use that sketch to sketch y equals negative f of x, and that should look like so with all those key features labelled.

Question three, part A asked you to complete the table for negative f of x.

Well, that's going to be taking the outputs of f of x and transforming them by multiplying by negative one.

You should have those values.

From there, we can graph the transformation negative f of x and that will look like so.

Isn't that beautiful? Finally, for question four, you were writing at least one sentence of advice to help Izzy understand her error and improve her work.

There's a lot you could have said here, but you might have said, "The transformation negative f of x is a reflection in the x-axis, not around its minimum point." That was Izzy's error.

She reflected at the turning point, not through the x-axis.

You might have gone on to say, "Consider any coordinate, such as the minimum point three, four.

The input, three, remains the same, but the output is multiplied by negative one.

The transformed coordinate pair is three, negative four.

Once you've plotted that point, this helps you to see the transformation." You may have shown Izzy that y equals negative f of x would actually be there on the graph.

We're at the end of the lesson now, sadly, but wasn't it interesting? We did a lot of learning.

We learned that we can recognise the effect of applying the transformation negative f of x to a graph.

The effect of negative f of x is to multiply the outputs of f of x by negative one.

Hence, the graph of negative f of x is a reflection of f of x in the x-axis.

Hope you enjoyed today's lesson as much as I did, and I'll look forward to seeing you again soon for more mathematics.

Goodbye for now.