Video: Transforming Functions: Vertical Stretch

Video Transcript

In this video, we’re going to take a couple of simple functions and carry out transformations that will stretch or squash their graphs in the same direction as the 𝑦-axis. We’ll also take a look at the effect of these transformations on the equations of the functions. And come up with a general rule about stretch and squash transformations parallel to the 𝑦-axis.

First then, here’s a graph of the function 𝑓 of 𝑥 is equal to 𝑥 squared. Let’s think about what would happen to that graph if we plotted 𝑦 equals two times 𝑓 of 𝑥. Well, we’d be doubling all of the 𝑦-coordinates from the graph 𝑦 equals 𝑓 of 𝑥. So two times all of the 𝑦-coordinates from 𝑓 of 𝑥. So in the original 𝑓 of 𝑥 function, if the 𝑦-coordinate was zero, two times zero is zero. So that point would stay where it was. And if the 𝑦-coordinate had been one, we would double that to make it two. So this point here doubles its 𝑦-coordinate and ends up here. And again, at this point, the 𝑦-coordinate is going to double from one to two and end up here.

Now, for points on the graph of 𝑦 equals 𝑓 of 𝑥 that had a 𝑦-coordinate of two, we’re gonna be doubling those to make them four. So this point is gonna move up to here, and this point is gonna move up to here. And again, if the 𝑦-coordinate in the 𝑓 of 𝑥 function would have been full, we’ll double that to make it eight. And we can join up all those points to see what our curve 𝑦 equals two 𝑓 of 𝑥 would look like. And it’s something like this. Now, it’s important to remember what we’ve done. We’ve doubled the 𝑦-coordinates. And instead of our 𝑦-coordinate going up to here, we’ve doubled that distance to bring it up to here. Instead of this 𝑦-coordinate going up to here, we’ve doubled it to come up to here.

Now, a quick visual inspection of that might make it look like we’ve just squashed the curve towards the 𝑦-axis. But that really isn’t what’s happened, remember. We have doubled all of our 𝑦-coordinates. The transformation two times 𝑓 of 𝑥 has moved the 𝑦 equals 𝑓 of 𝑥 curve twice as far away from the 𝑥-axis. It’s been stretched times two in the 𝑦-direction.

Let’s look at another function now, 𝑔 of 𝑥. In fact, 𝑔 of 𝑥 is equal to a function called sign of 𝑥. But don’t worry about that for now. The point is it’s got positive and negative 𝑦-coordinates above and below the 𝑥-axis. Now, we’re gonna plot 𝑦 equals three times 𝑓 of 𝑥, which means we’re going to multiply all the 𝑦-coordinates by three. Now, that means that whenever the 𝑦-coordinate on the original 𝑓 of 𝑥 curve was zero — so here, here, here, and here — then we’re gonna multiply that by three. And it will still be zero. So those points on our transformed function will stay exactly where they were on the original function.

Now, the points on the original function that had a 𝑦-coordinate of one, so here and here, will have that multiplied by three. And three times one is three. So they will become three. So this point is gonna move up to here. And this point is gonna move up to here. And every point in between will also have its 𝑦-coordinate tripled. So our function is going to look — or the graph of our function is gonna look something like that. And in this section, it will come down like that. And tripling all our 𝑦-coordinates in that section will make it look something like that.

So now, let’s consider when 𝑦 would be equal to negative one. Well, three times negative one is equal to negative three. So this point is gonna move down to here. And this point is gonna move down to here. And again, tripling all of those negative 𝑦-coordinates now is gonna look something like that and that. So carrying out the transformation 𝑦 equals three times 𝑓 of 𝑥 takes all of the 𝑦-coordinates and multiplies them by three. So if they were positive, they’re gonna get three times higher up the graph. And if they were negative, they’re gonna get three times lower down the graph, further away from the 𝑥-axis. And it’s important to remember that any points that were on the 𝑥-axis are gonna stay exactly where they were before the transformation happened. So it’s like we’ve locked down the 𝑥-axis and we’re stretching that curve away from the 𝑥-axis above and below.

Now, starting again, we’ve got 𝑦 equals 𝑔 of 𝑥. Now I’ve zoomed in on the 𝑦-axis. So the 𝑦-axis and 𝑥-axis have got a slightly different scale on them now. And if we want to plot 𝑦 equals a half times 𝑔 of 𝑥, we’re gonna be multiplying all of our 𝑦-coordinates by a half. Now, any points that, on- for 𝑔 of 𝑥, were on the 𝑥-axis that had a 𝑦-coordinate of zero, when we multiply that by a half, it’s still gonna be zero. So these points are gonna be transformed to themselves. Again, we’re locking down the 𝑥-axis. Any points that were on the 𝑥-axis are gonna stay there. And if our 𝑦-coordinate was one, so here and here, on the 𝑔 of 𝑥 function, when we multiply that by a half, those 𝑦-coordinates are gonna be only a half. They’re gonna be squashed towards the 𝑥-axis.

And this same effect is gonna happen to all of the points on the curve in between those points that we just looked at. So halving all the 𝑦-coordinates in that section will give us a curve that looks like this. And halving all of those 𝑦-coordinates in that section from 𝑔 of 𝑥 will give us a curve that looks like this. 𝑦 equals a half 𝑔 of 𝑥. And again, let’s consider the points in 𝑦 equals 𝑔 of 𝑥 which had a 𝑦-coordinate of negative one. We’re gonna halve that to be negative a half. So this point is gonna move up to here. And this point is gonna move up to here. And now, if we halve all of the 𝑦-coordinates in this section of the graph, that part of the 𝑦 equals a half 𝑔 of 𝑥 curve is gonna look like that. And that part of the 𝑦 equals a half 𝑔 of 𝑥 curve is gonna look like that. So we’ve locked off the 𝑥-axis again. All points on the 𝑥-axis stay where they are. But this time, we’re squashing our curve towards the 𝑥-axis. We’re multiplying all of our 𝑦-coordinates by a half.

So we’ve been looking at transforming a function by multiplying it by a constant. For example, mapping 𝑦 equals 𝑓 of 𝑥 onto 𝑦 equals 𝑎 times 𝑓 of 𝑥. And as we’ve seen, that ends up with just multiplying all of the 𝑦-coordinates by the constant 𝑎. And we saw that when 𝑎 is greater than one, then the function gets stretched away from the 𝑥-axis. And when 𝑎 is between zero and one, then 𝑓 of 𝑥 gets squashed towards the 𝑥-axis. So what do you think would happen if 𝑎 was equal to one? Well, then, we’d be multiplying all of our 𝑦-coordinates by one. So 𝑓 of 𝑥 would stay exactly where it is. Okay then, what about if 𝑎 was negative? Well, let’s try one with 𝑎 is equal to negative one.

Here, we’ve got a function 𝑦 equals 𝑓 of 𝑥. So now, let’s define a function 𝑔 of 𝑥 which is equal to negative one times 𝑓 of 𝑥. So our 𝑎-value in the previous slide would’ve been negative one. So now, let’s plot 𝑦 equals 𝑔 of 𝑥. Well, in the 𝑓 of 𝑥 function, if our 𝑦-coordinate was zero, if we multiply that by negative one, negative one times zero is still zero, so this point here maps onto itself. And when on the 𝑓 of 𝑥 function our 𝑦-coordinate was one, so here and here, that’s gonna be multiplied by negative one. So that’s gonna become negative one. And when the 𝑦-coordinate would’ve been two in our 𝑓 of 𝑥 function, those 𝑦-coordinates get multiplied by negative one to make negative two. So those points are gonna map down here, and the same for three and four and so on. So our 𝑦 equals 𝑔 of 𝑥 curve is gonna look like this. So this is a reflection in the 𝑥-axis. The 𝑦-coordinates are the same distance from the 𝑥-axis but they’re on the other side. So when our 𝑎 transformation here, our 𝑎-value, is negative one, we get a reflection in the 𝑥-axis.

Well, now, let’s think about when 𝑎 is less than negative one. So, for example, when 𝑎 is negative two. Let’s plot 𝑦 equals 𝑎 times 𝑓 of 𝑥 when 𝑎 is negative two. Here’s our 𝑓 of 𝑥 curve. Now, when we multiply all of our 𝑦-coordinates by negative two, we’re getting that reflection in the 𝑥-axis. But all of the distances away from the 𝑥-axis are doubling. So we’re getting a reflection and a stretch. So look, that distance here is doubled to that distance there. It moves to the other side of the 𝑥-axis and doubles. But again, any points that were on the 𝑥-axis, it doesn’t matter what you multiply it by. If it started off at zero, you multiply it by a number, it’s still gonna stay at zero. So the points that started on the 𝑥-axis stay on the 𝑥-axis. So this transformation of multiplying the 𝑓 of 𝑥 function by negative two is creating a reflection in the 𝑥-axis followed by a stretch times two parallel to the 𝑦-axis.

So finally, let’s think about when negative one is less than 𝑎 is less than zero. So 𝑎 is between zero and negative one. So specifically, let’s try when 𝑎 is equal to negative a half. So we’ve got to plot 𝑦 equals 𝑎 times 𝑓 of 𝑥, where 𝑎 is negative a half. So we’re plotting 𝑦 equals negative a half times 𝑓 of 𝑥. Now, we’re given the graph of the 𝑓 of 𝑥 function there. So we’re just gonna go through and work out what happens to all our 𝑦-coordinates. Again, if your 𝑦-coordinate started off as being zero, it’s gonna be multiplied by negative a half. It’s still gonna be zero. So that’s gonna stay where it is. And now, multiplying all of our 𝑦-coordinates by negative a half, they’re going to the opposite side of the 𝑥-axis. But they’re being halved. Their distance away from the 𝑥-axis is half of what it was for the 𝑓 of 𝑥 function. So the curve is being squashed towards the 𝑥-axis and any points that were on the 𝑥-axis stay there. But we’ve also got this reflection in the 𝑥-axis going on. So that’s reflected in the 𝑥-axis and then squashed times a half parallel to the 𝑦-axis towards the 𝑥-axis.

So let’s summarise everything we’ve learned then. So if we’ve got a function called 𝑓 of 𝑥 and we multiply that by 𝑎, we’ve transformed that function. So if we say 𝑔 of 𝑥 is the transformed function of 𝑓 of 𝑥, then if 𝑎 is bigger than one, remember that’s gonna stretch the 𝑓 of 𝑥 function times 𝑎 parallel to the 𝑦-axis. Points that were on the 𝑥-axis stay where they are, but everything else gets stretched further away from it. Then, if 𝑎 was equal to one, we’d be doing a stretch times one. We’ll be multiplying all the 𝑦-coordinates by one. So basically, the function is just transformed onto itself. It’ll look exactly the same. And if zero is less than 𝑎 is less than one, so 𝑎 is between zero and one, then this is gonna have the effect of squashing the curve of that function towards the 𝑥-axis. We’re multiplying all the 𝑦-coordinates by 𝑎. And because 𝑎 is between zero and one, the magnitude of all those 𝑦-coordinates is gonna get smaller. Each of them, if they’re positive, they’re gonna become smaller and get closer to zero. If they’re negative, they’re gonna be halved. And they’re going to get closer to zero.

And the most extreme example of that, I suppose, is when we put 𝑎 equals to zero. So all of the 𝑦-coordinates are multiplied by zero. They all become zero. That function is squashed so much. Its squash is transformed completely onto the 𝑥-axis. Then, if 𝑎 is between zero and negative one, we get— because it’s negative, we’re getting this reflection in the 𝑥-axis of the function. And then, we squash it towards the 𝑥-axis. So all of the 𝑦-coordinates multiplied by 𝑎, by a fractional amount, is moving closer to that 𝑥-axis. If 𝑎 was equal to negative one, that’s just a straight reflection of the function in the 𝑥-axis. All the 𝑦-coordinates are gonna stay the same distance away from the 𝑥-axis. They’re just gonna be the other side of it. And then, finally, if 𝑎 is less than negative one, we’ve got a reflection in the 𝑥-axis because that 𝑎 value is negative. But then, because we’re multiplying all the coordinates by a number which is less than negative one. We’re stretching that curve away from the 𝑥-axis. Okay, just before we go, here is a question for you to try.

Given the graph 𝑦 equals 𝑓 of 𝑥 below, sketch the curve 𝑦 equals 𝑔 of 𝑥, where 𝑔 of 𝑥 is equal to two times 𝑓 of 𝑥.

So I’d like you to pause the video, have a go at this question, and then come back. And I’ll show you how to do it.

So let’s just have a look at this curve then. So we’ve got an uppie-downie curve around the 𝑥-axis here. When I put various values into this function, I get these 𝑦-coordinate values out. So the value of the function we’re using is the 𝑦-coordinate in each of these. Now, we’re gonna put the same 𝑥-values into the 𝑔 function as we did into the 𝑓 function. But for the 𝑔 function, we’re gonna take double the 𝑓-values. So now, we’re gonna take twice the value of the 𝑦-coordinates in each case. Now these points here on the curve, the 𝑦-coordinate was equal to zero. And if I double zero, I still get zero. So those points on the 𝑓 function are gonna map to still the same points on the 𝑔 function. So I can sketch those in now.

And the points on the 𝑓 of 𝑥 function which have a 𝑦-coordinate of one, they’re gonna have that 𝑦-coordinate doubled to map onto the 𝑔 of 𝑥 function. So this point here is gonna map up to here. This point here is gonna be doubled up to two up here. And this point here is gonna be doubled to make two up here. And likewise, for the points that have a 𝑦-coordinate of negative one in our 𝑓 of 𝑥 function to turn it into our 𝑔 of 𝑥 function, to transform into the 𝑔 of 𝑥 function, we’re gonna be doubling those 𝑦-coordinates, two times 𝑓 of 𝑥. So we’re gonna be doubling negative one to make negative two. And that means that this point here is gonna be doubled to come down here. It’s gonna have its 𝑦-coordinate doubled. This point here is gonna have the same 𝑥-input. But it’s gonna have its 𝑦-coordinate doubled down here. I’ve just drawn it slightly in the wrong place. So that’s gonna come down to here.

So what we’re gonna do now is join up those dots to make our curve. And for all of these points in between the points that we mapped here, they’re gonna have their 𝑦-coordinates doubled as well. So we’re gonna get a curve that’s coming down, getting closer to zero, still gonna end up at zero. Likewise, in this section here, we’re doubling all of those 𝑦-coordinates. In this section here, doubling all of the 𝑦-coordinates. It’s gonna look not quite as sharp as that. But it’s gonna look something like that. In this section here, we’re gonna get a curve that comes up. It goes like that. This section here is gonna come down like that. And this section here, the 𝑦-coordinates are gonna come down here. And in this section here, it’s gonna look something like this. And this section here, it’s gonna look something like this.

Well, I am so happy, and I’ve checked my sketch there. Then, I’ve just filled it in to make it stand out nice and clearly on the graph. And in some questions, they might ask you to describe that transformation. So something like a stretch times two parallel to the 𝑦-axis away from the 𝑥-axis would be a description of that transformation. And it’s also worth remembering that anything that was on the 𝑥-axis, because it had a 𝑦-coordinate of zero. It doesn’t matter what you multiply that by. It’s always gonna stay where it is, at zero. So those points on the 𝑥-axis are going to stay there. They’re anchored to the 𝑥-axis.