Take a look at the transformation of curves with us. We look at translation and reflection to see how these can be used to change shapes. This is an AQA GCSE Maths topic, but the concepts are applicable to all levels of maths.
Functions of graphs can be transformed to show shifts and reflections. Graphic designers and 3D modellers use transformations of graphs to design objects and images.
Okay, so we're looking at transformations of curves and we're going to focus on the two main transformations that are translations, which is just moving a curve up or down or left or right, and then reflections, which is, as you probably guess, it's reflecting through an axis, either X or Y. So let's do translations first. So if we start off with a fairly nice example, let's just start off with our standard X squared. Our function FX is going to be X squared. Let's look now what happens if we do something to this function.
So let's say we want to do, we want to take our original function and at the very end we want to just add three to the output.
Well, what we're doing here is we're taking our original function of X squared and at the very end we're just adding three. So what effect is that going to have if I call this my green function, if I were to plot this on my axis, all we're doing is instead of plotting zero, if this is zero zero, all we do is we plot it at the very end, we just add three. So instead of plotting zero, I plot three. And if this is, say X equals one and that's equals two. So X equals one.
Instead of plotting one, squared equals one, what we do is we plot one squared equals one, and at the very end we add three onto it. So we do one plus three and we get four. And so that point there is going to be our new point. And finally, if we look at when X equals two, well, again, we do two squares to get four. And then at the very end we add three onto that and our four becomes seven.
And so we get that point there. And the same thing happens on the other side. If we were to do the same thing to those points, what we end up seeing is we get a new curve that looks like this. And what's actually happened is all the points on our original curve have just shifted up three spaces. And so what we see is when we make a change outside the brackets, outside the brackets, what we get is a change in the Y axis.
We're only changing the Y component of each of those coordinates. We're just moving it up in this case. So we move it along the Y axis and is what we'd expect, right? If we add three, we'd expect it to shift up three spaces. So it's what we'd expect.
Let's look now at a different example. Instead of making a change outside the brackets, instead of doing plus three outside these brackets, we'll do plus three inside the brackets and we'll see what happens there. So again, I'll copy out my original X squared function as we had before. This is my X squared function, slightly wonky, but there we go. And I'm actually going to now do F of X plus three inside these brackets.
What this means is, instead of the input to my function being just X, the input is now going to be x plus three. And so before, when the input was X, our output was X squared, right? So what the function is doing is it's saying we want to square whatever the input is. In this case, the input is x plus three. And so we take that x plus three and we square it.
Well, it might be tempting to expand on these brackets and plot that function and see what we get. But there's actually a simpler way of doing this to understand what's going on. So in this case, all we're really doing here, I'll use a different colour, I'm going to call this my blue function. All we're doing here really, is we're peering into the future. And I'll tell you what I mean by that.
So in this case, if we've got, say, minus one, minus two and minus three, well, what we do is, at the very start, we have to add three onto X before we do anything else. If you remember, Bibas, the first thing we do is we take care of inside the brackets, and so we add three to X and then we square it. And so in this case, if I add three to minus three, I get zero, right? I'm kind of looking ahead almost to zero and I know therefore, the zero square is just zero, right? I've already plotted that there.
And so I just plot that at that point there. And the maths side of things, all I'm doing is minus three plus three to get zero, and I'm squaring that. The effect that this has is we're peering into the future. We'll do one more example. And so if, say X is minus T, what I do minus two, plus three to get minus one, I square minus one to get plus one.
And so that's my output, right? And I plot that output on my graph there and let's do one more. Let's do minus one as well. So if x is minus one, we do minus one plus three to get two, and we then do two squared and we get four. So I'm going to plot four at that point now.
So probably what you can see happening is zero. We do three squared to get nine. What you can see happening is we're recreating this X squared shape, right? We end up with slightly wonky there, we end up with this new X squared shape. But what we can see is that every single point on our curve is actually been shifted over, everything has been shifted back by three spaces.
So it's kind of continuity, right? You'd expect that adding three inside the brackets will cause our graph to shift up three spaces, but it doesn't. It causes it to shift down three spaces. And so our rule is if it's inside the brackets, which means the x axis, and it's the opposite to what we expect.
What we expect. And that's really the only two rules that we need. And we can actually apply these rules as well to reflections. So we'll go ahead and apply those rules to reflections now. So let's look at another nice example.
I'll do just a random x squared kind of graph. Let's go for this.
In this case, if I want to reflect it well, let's look at what happens if this is my normal function, this is my original FX, my original function of x. If I do FX, and then I take that and I put a minus in front of them, so I make everything negative. What's going to happen? Well, in this case, I'll use a different colour. I'm going to call this my blue function.
In this case, what we're doing is at the very end, before we actually plot our values, before we plot the y component of our values, we actually put a minus in front of it. And so, for example, that point is going to become that point, that point is going to become that point, that point is going to say where it is, it's just zero, that point becomes that point. And what we can see happening is basically we reflect our entire graph about the x axis, so it kind of flips vertically. Let's go one more version of this. So let's look at what happens if we put them minus inside the brackets, just to go back to this for a second.
Outside the brackets, we know we're changing the y component of each coordinator, so only the y component of those coordinates is changing. Inside the brackets, we know we're going to be changing the x component of the coordinates, so it's going to be changing the horizontal position of each point on this curve. So inside the brackets, I know I'm changing the horizontal component. And I know that essentially what I'm doing is before I actually do anything, I put a minus in front of x. And so rather than plotting for say, x equals one and getting zero, what I actually do is I plot for x equals minus one instead.
Well, when x equals minus one, I get this point up here. And so I'm actually plotting this original point. Instead of being in line with minus one, I actually plot it being in line with one and vice versa. My x equals minus one, I'm actually going to plot the value of the function when x equals positive one. And so I plot for that point there.
And so what we see happening is everything, it gets kind of flipped around. This actually becomes my knee function, it looks like that. And what we see happening is every single point is basically flipped. I could pick that point there and that point there that is going to be completely flipped in the x axis, that's it in the y axis. So that is translation and reflections.
Let's go ahead and look at exam style questions to put this into practise. So we've got a graph of y equals FX and we want to do four different translations and want to understand what's going on here. So first of all, let's say this is going to be my blue function. So FX plus three, well, we're inside the brackets, so we know we're going to be changing the x component of each point on the graph. It's going to be a horizontal change and it's the opposite to what we'd expect.
And so in this case, I know that my curve is going to shift down three spaces. And so I'm going to end up with something that looks like this. And that point there, rather than being x equals one, is going to be x equals minus t. So that's my first solution for the second one. For part B.
I'm outside the brackets. My green function, I'm outside the brackets. So I'm now changing the why component. It's going to be a vertical change. And all I'm doing is at the very end I'm adding a minus in front of it, right, so rather than y equals four, I now have y equals minus 40 is going to stay the same at that point there.
When we end up seeing as we get this new function that looks a bit like that, right, everything is just flipped in the x axis. So rather than plus four, I get minus four. Right, looking at, I just put it in the wrong one. Looking at part C, we've got FX minus three. And again, we're outside the brackets.
We know therefore we're changing the y component and it's what we'd expect. So all that's going to happen is our original graph is going to shift down by three spaces. So we're going to end up with something that looks like this, incredibly wonky drawn. It's going to look something like that. Not much better.
There we go. So this original y component of four is now going to shift down three spaces to become one. And so we end up with the coordinate one, which looks about right, roughly one, one. Finally, our part D, let's call this our yellow function, y equals f of minus x. Well, we're inside the brackets, we know we're going to be changing the x component, it's going to be a horizontal change and it's the opposite to what we'd expect.
In this case, it's just a reflection really. So we don't really have to worry too much about the opposite effect here because there's only one reflection we can do. What we're doing is we're affecting in the kind of horizontal direction about the y axis. Does that make sense? So this is a kind of mirror line.
And so all that happens is that point becomes that point. That point there is going to become that point there. Everything is just flipped around that point on the outside. It's going to become that point there. And so we end up with a knee function that goes through those knee points.
And it looks like that. And so, again, all that's happened is our positive one X coordinate is going to become a negative one X coordinate. So end up with minus one four. And that is the end of transition. Hopefully that makes sense.
Yeah, that's pretty much all that you could get us in the exam.
Book a lesson with this tutor
A young, gifted tutor who uses his engineering experience to ignite your curiosity in Maths. A straight 'A' student, he knows exactly how you need to learn to help you ace your exams.