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Hello, Mr. Robson here.
Welcome to maths.
It's gonna be a glorious day today because we're looking at combinations of transformations of graphs.
This is a wonderful bit of maths.
I think you're gonna like it as much as I do.
Our learning outcome is that we'll be able to recognise the effect of applying a combination of transformations to a function.
Transformation's a keyword you're gonna hear many times throughout this lesson.
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 looking at combinations of transformations.
We can transform the graph of a function in a variety of ways.
f of x plus a is a translation of a in the y-direction.
f of x plus a is a translation of negative a in the x-direction.
Negative f of x is a reflection in the x-axis.
F of negative x is a reflection in the y-axis.
Af of x is a stretch by a scale factor of a in the y-direction, whereas f of axe is a stretch by a scale factor of one over a in the x-direction.
You're going to need all of those transformations throughout this lesson, so if you need to, pause now, and make some notes.
We can combine these transformations and change the graph of a function in multiple ways.
A simple example is a combination of these two transformations, negative f of x, which is a reflection in the x-axis and af of x, a stretch by a scale factor of a in the y-direction.
Combining those two transformations might look like this.
Negative 2f of x, the negative will reflect the graph f of x and the two will stretch f of x.
In this case, the order will not matter.
For the transformation negative 2f of x, we could reflect, then stretch, and we see the graph y equals negative 2f of x.
Alternatively, we could stretch, then reflect, and see the same graph y equals negative 2f of x.
Quick check you've got this.
Where will the two marked points be after the transformation negative 2f of x is performed on f of x in this case? Pause, work that out, now.
Welcome back.
That is the graph y equals negative 2f of x.
There's one point and another point.
In the case of the transformation negative 2f of x, the outputs of f of x are multiplied by two, then multiplied by negative one.
You might consider it just one multiplication by negative two.
So for the point 4,5 the input x did not change, but the output y was multiplied by negative two, hence the new coordinate pair 4,-10.
Something similar happened for that other coordinate pair on y equals f of x, we saw 6,1 that was transformed to 6,-2.
That output one was multiplied by negative two.
With other combinations of transformations, order will matter.
This is a transformation 2f of x plus three.
The times two will stretch the function.
The plus three will translate the function.
Here's some Oak pupils to discuss what they think's going to happen.
Sam says, "I think we stretch then translate." Aisha says, "I think we translate then stretch." Jacob says, "We have seen this problem before in maths." If we stretch, then translate, we end up with that graph for the transformation 2f of x plus three.
What if we translate, then stretch? I wanna show you that again.
It's important you understand what happens here.
We translate, and then from here, we stretch.
Watch that y-intercept change from 0,3 to 0,6.
We've got two different graphs.
We shouldn't have two different graphs.
2f of x plus three is just one combination of transformations.
We should just have one graph so the order clearly matters.
Jacob made a very valid point about this transformation.
Jacob says, "We have seen this problem before in maths." Did you spot what Jacob meant? What is the transformation 2f of x plus three doing to f of x? The output of f of x is being multiplied by two and then having three added to it.
If such a problem were numeric, what would you do first? If I said two lots of 10 plus three, what would you be doing first? Jacob's got it.
He says, "It's priority of operations." You do the multiplication first.
We do two lots of 10 and then we add the three.
We get 23.
Something similar happens with our transformation to f of x plus three.
Two lots of f of x plus three.
We stretch first, i.
e.
we do the multiplication, the times two, then we add next, i.
e.
the translation.
Quick check you've got this.
Where will the two marked points be after the transformation 2f of x minus one is performed on f of x.
Remember, the order here is going to matter.
Pause, see if you can spot where those two points are going to be.
Welcome back.
Hopefully, you appreciate that y equals 2f of x minus one is in that position.
Our point 0,4 became the point 0,7 and our point 2,0 became the 2,-1.
In the case of the transformation 2f of x minus one, it's the outputs of f of x that are multiplied by two and then have negative one added.
In the case of the point 0,4, the input x did not change but the output y was multiplied by two, then negative one added, hence the new coordinate pair 0,7.
We can also perform multiple transformations on trigonometric graphs such as f of x equals cos x as you see graphed here.
Let's look at the transformation 2f of x minus 60.
We could consider any one point on the graph and we can multiply first, i.
e.
stretch by a scale factor two in the y-direction.
That point will be transformed to there.
Let's subtract next, i.
e.
the x minus 60.
That means we're gonna translate by positive 60 in the x-direction like so.
So the point 0,1 moves to 60,2.
The graph of 2f of x minus 60 looks like so if you do that to every single point on the line.
We can do this order the other way around and get the exact same result.
Do you believe me? Let me show you.
Let's subtract first, i.
e.
perform that translation by positive 60 in the x-direction.
Let's multiply next.
That's our stretch by scale of factor two in the y-direction and we get to 60, 2 again.
The same point.
And if you apply that same combination of transformations in that order, you get that exact same graph for 2f of x minus 60.
It's important to understand why we get the exact same graph and why the order did not matter in this case.
We saw examples earlier where I stress to you the order matters, but the order doesn't matter in the case of 2f of x minus 60.
Why? It's because in this case, one transformation affects our inputs, our x values, whereas the other transformation affects our outputs, our y values.
Think about the graph of y equals f of x.
It's made up of infinitely many points, coordinate pairs, with an x value and a y value.
Our transformation is gonna take those x values and translate them by positive 60 in the x-direction.
It's going to take those y values and multiply them by a scale factor of two, as a transformation happening to the x value and a transformation happening to the y value and the happening independent of each other.
That's why in this case the order doesn't matter.
Order only matters when you're doing multiple things to the same coordinates.
Like the examples we saw earlier where we were doing multiple things to the output of f of x.
In the case of two lots of f of x minus one, the order did matter because both the translation and the stretch were happening to the outputs, the y values.
Okay, if you've got that, you can get this.
I'd like you to categorise these combinations of transformations into ones where the order in which we perform the transformations will matter and ones where it will not.
There's six combinations of transformations there.
Pause and categorise them now.
Welcome back.
I hope you enjoyed sorting those.
We should have categorised them like so.
In the case of our order matters examples, 3f of x plus 90 and 5f of x minus two, both transformations were affecting the output of f of x, so order matters.
In those two cases, you'd be performing that multiplication, that stretch, before you perform your addition or subtraction, your translation.
In the case of the order does not matter transformations, one transformation was affecting the x values, our inputs, the other was affecting the y values, our outputs.
Therefore, the order does not matter in the case of those four combinations of transformations.
Practise time now.
Question one.
On each graph, I'd like you to draw the combined transformation.
For A, I've given you y equals f of x, I'd like you to draw negative 4f of x.
For B, it's a parabola, a quadratic.
I'd like you to take f of x and draw negative 3f of x.
C, how nice is that, a cubic curve.
I'd like you to draw negative 2f of x in that case.
Pause and do it now.
Question two.
Again, on each graph I'd like you to draw the combined transformation, but slightly different transformations this time.
For A, I'd like to draw 3f of x minus two.
For B, I'd like to draw 2f of x plus four and for C, I'd like to draw negative 3f of x plus one.
Remember, sometimes order matters.
For question three, part A, I'd like you to draw 2f of x minus six and state the new turning point of the graph.
For B, I'd like you to draw 2f of x minus six and state the new turning point of the graph.
They sound like awfully similar combinations of transformations, so will we get the same answer? I'll leave you to figure that one out.
Pause and do that now.
Question four, f of x equals sine x.
For part A, I'd like you to draw the combination of transformations negative f of x minus one.
For part B, I'd like you to draw 3f of 3x.
Pause and try these two, now.
Feedback time, let's see how we did.
Part A you were drawing negative 4f of x, you should have drawn that line there.
Slightly tricky to check your work here so let me give you a coordinate pair from f of x and the combined transformation.
We've got an invariant point of -2,0, and on f of x your 0,2 coordinate pair should have become 0,-8 on negative 4f of x.
That might enable you to check your work.
For B, drawing negative 3f of x should look like so.
We've got an invariant point at 1,0, and the coordinate pair 3,4 on f of x should have become the coordinate pair 3,-12 on negative 3f of x.
For part C, drawing negative 2f of x, the curve should look like that and your point -2,3 on f of x should have become -2,-6, on negative 2f of x.
Question two.
Part A has you draw 3f of x minus two, it should look like so.
Those coordinate pairs will help you to mark your work.
For B, 2f of x plus four should look like so.
Those coordinate pairs might help you to check your work.
For Part C.
Negative 3f of x plus one will look like that.
In this case, those coordinate pairs will help you to check your work.
Question three, part A and part B looked awfully similar.
2f of x minus six and 2f of x minus six and I asked you to state the new turning point of the graph in each case.
For part A, your combined transformation should look like so with a turning point at -3,-2.
For part B, your combined transformation will look like so with a turning point at 3,4.
For question four, these two were beautiful.
Part A, your negative f of x minus one when f of x equals sine x, that would look like that.
Those coordinate pairs will help you to check your work.
For part B, 3f of 3x.
Well, that's exciting.
That's a stretch by a scale factor of three in the y-direction whilst stretching by a scale factor of one over three in the x-direction.
That should look like this.
Those coordinate pairs will help you to check your work.
Onto the second half of our learning now where we're going to look at identifying transformations.
Now that we know how to apply multiple transformations to graphs, we can start to spot the multiple transformations that have occurred when given a graph.
The first question we should ask is what sort of transformation has occurred? Has the shape of the graph been affected? In this example, we can see not.
The distance between key points has not changed.
The curves are congruent, so we know that this is not a stretch.
We have the exact same graph, but in a different position.
If that's the case, then it must be a reflection or a translation.
Our original graph, f of x is a positive cubic curve.
If we reflected that in either the x or y-axis, that would change.
A reflection in the x-axis would give us a negative cubic curve.
A reflection in the y-axis would do the same.
Therefore, we know this is not a reflection, this must be a translation.
Can you see the process of elimination that we're working through here? If it must be a translation, we can examine the graph point by point and see that all points are translated by positive three in the y-direction.
So this must be the graph of f of x plus three.
Multiple transformations have happened to this graph, but we go through the same process.
It's the same shape.
The two curves are congruent, so we know this is not a stretch.
It's the same nature for f of x and our combined transformation.
We see a positive cubic curve in both cases.
Because of that, we know this is not a reflection, so it must be a translation.
When we check, we see that every point has moved in the same way.
Translation by positive five in the x-direction and positive three in the y-direction.
So this must be the graph of f of x minus five plus three.
Quick check you've got that, f of x equals cos x.
What combination of transformations has happened to f of x to give us our new function here? Pause, see if you can spot it.
Welcome back, I wonder what you said.
It's a really interesting answer this one.
So let's go through it.
We know it's the same shape in each case, so we know this is not a stretch.
We can see that every point has been translated by negative 90 in the x-direction and positive two in the y-direction.
So this might be the graph of f of x plus 90 plus two.
Why do I say it might be the graph? Because we might also argue that every point has been translated by positive 270 in the x-direction and positive two in the y-direction.
So this might be the graph of f of x minus 270 plus two.
There were actually infinitely many answers to that one.
Here's another example with multiple possible answers.
It's the same shape, so we know it's not a stretch, but we went from a positive traveller to a negative one.
So a reflection has occurred.
F of x may first have been reflected in the x-axis, then translated by positive six in the x-direction.
That would make our graph the transformation negative f x minus six.
Alternatively, f of x may have first been reflected in the x-axis, then reflected in the y-axis.
That would make our graph the transformation negative f of negative x.
Quick check you've got that.
When presented with a graph, we can state the exact combination of transformations that has occurred.
Is that always the case? Sometimes the case or never the case? Pause, tell a person next to you or say it aloud to me on screen.
Welcome back, hopefully, you said B, sometimes.
In many cases, we can find the one and only possible combination of transformations that were applied.
However in some cases, there will be more than one possible answer and we'd need more information to declare the precise combination of transformations that were applied.
Looking for a change in shape enables you to spot a stretch.
With a curve like this one, look for a change in distance between turning points.
Look here, we can see there's no change in the x-direction between the two turning points, but look here.
In the y-direction, we can see a stretch by a scale factor of two.
F of x was stretched in the y-direction by a scale factor of two.
So is our new graph 2f of x? No, something else has happened, but by graphing 2f of x, that enables us to spot the next transformation.
Have you got it? Well done, it's a translation.
A translation of negative five in the x-direction.
Therefore, this is the transformation 2f of x plus five.
Quick check you've got that.
I'd like you to identify the combination of transformations in this case.
Pause and try this now.
Welcome back.
Well done for having a go at that tricky little problem this one.
When we look in the y-direction, there's no change in shape, but when we look in the x-direction, we see a change in shape, a stretch, by a scale factor of 1/2.
So this could be the graph y equals f of 2x, but it's not.
There's another transformation.
The graph of f of 2x enables us to spot.
Now the next transformation is a reflection in the x-axis.
So our combined transformation is negative f of 2x.
Practise time now.
Question one, I'd like to identify the transformation or combination of transformations on these two graphs.
Pause and do that now.
Question two, there's more than one answer to the possible combination of transformations in this case.
I'd like you to find two solutions.
Pause and do this now.
Question three.
You can see on the graph y equal f of x and in this case, f of x equals sine x.
So what is our combined transformation? There's more than one answer to the possible combination of transformations in this case.
I'd like you to find two solutions.
Pause and do this now.
Question four, f of x equals cos x, and you can see that on both graphs.
In each case, I'd like you to identify the combinations of transformations.
Pause and do this, now.
Welcome back, feedback time.
Let's see how you got on with question one.
Hopefully, you identified then both cases, there were congruent curves, so no stretches had occurred.
For part A, you should have said that it's a transformation f of x plus four.
For B, it was a combination of transformations, f of x plus six plus four.
Question two, I told you there's more than one answer to the possible combination of transformations in this case and asked you to find two solutions.
This could be the graph of negative f of x plus six, but it could also be the graph of negative f of negative x.
Question three, again, multiple possible combinations of transformations.
In this case, I asked you to find two solutions.
This could be the graph of f of x minus 180 plus two.
It could also be the graph of negative f of x plus two.
Many other answers were possible, in fact, infinitely many in this case.
If you've got a list that includes some that I haven't shown you here, then you can use tech to check them.
Go to Desmos.
com and do some graphing.
Question four, part A, the combination of transformations was 2f of x plus three.
And for part B, the combination of transformations was 4f of 2x.
We're at the end of the lesson now, sadly, but I hope you found that as super interesting as I did.
We learned that we can apply a combination of transformations to a function and also, recognise a combination of transformations that have been applied when given a graph.
We also learned that when graphing a combination whereby the output is affected more than once, we must apply priority of operations.
We know that if the input, the x value is affected once and the output, the y value is affected once, the order does not matter in that case.
Well, I thoroughly enjoyed that lesson and I hope to see you again soon for more mathematics.
Goodbye for now.
