### Video Transcript

In this video, weβre 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 π¦-axis. Weβll 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 π¦-axis.

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

Now, for points on the graph of π¦ equals π of π₯ that had a π¦-coordinate of two, weβre 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 π¦-coordinate in the π of π₯ function would have been full, weβll double that to make it eight. And we can join up all those points to see what our curve π¦ equals two π of π₯ would look like. And itβs something like this. Now, itβs important to remember what weβve done. Weβve doubled the π¦-coordinates. And instead of our π¦-coordinate going up to here, weβve doubled that distance to bring it up to here. Instead of this π¦-coordinate going up to here, weβve doubled it to come up to here.

Now, a quick visual inspection of that might make it look like weβve just squashed the curve towards the π¦-axis. But that really isnβt whatβs happened, remember. We have doubled all of our π¦-coordinates. The transformation two times π of π₯ has moved the π¦ equals π of π₯ curve twice as far away from the π₯-axis. Itβs been stretched times two in the π¦-direction.

Letβs look at another function now, π of π₯. In fact, π of π₯ is equal to a function called sign of π₯. But donβt worry about that for now. The point is itβs got positive and negative π¦-coordinates above and below the π₯-axis. Now, weβre gonna plot π¦ equals three times π of π₯, which means weβre going to multiply all the π¦-coordinates by three. Now, that means that whenever the π¦-coordinate on the original π of π₯ curve was zero β so here, here, here, and here β then weβre 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 π¦-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 π¦-coordinate tripled. So our function is going to look β 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 π¦-coordinates in that section will make it look something like that.

So now, letβs consider when π¦ 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 π¦-coordinates now is gonna look something like that and that. So carrying out the transformation π¦ equals three times π of π₯ takes all of the π¦-coordinates and multiplies them by three. So if they were positive, theyβre gonna get three times higher up the graph. And if they were negative, theyβre gonna get three times lower down the graph, further away from the π₯-axis. And itβs important to remember that any points that were on the π₯-axis are gonna stay exactly where they were before the transformation happened. So itβs like weβve locked down the π₯-axis and weβre stretching that curve away from the π₯-axis above and below.

Now, starting again, weβve got π¦ equals π of π₯. Now Iβve zoomed in on the π¦-axis. So the π¦-axis and π₯-axis have got a slightly different scale on them now. And if we want to plot π¦ equals a half times π of π₯, weβre gonna be multiplying all of our π¦-coordinates by a half. Now, any points that, on- for π of π₯, were on the π₯-axis that had a π¦-coordinate of zero, when we multiply that by a half, itβs still gonna be zero. So these points are gonna be transformed to themselves. Again, weβre locking down the π₯-axis. Any points that were on the π₯-axis are gonna stay there. And if our π¦-coordinate was one, so here and here, on the π of π₯ function, when we multiply that by a half, those π¦-coordinates are gonna be only a half. Theyβre gonna be squashed towards the π₯-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 π¦-coordinates in that section will give us a curve that looks like this. And halving all of those π¦-coordinates in that section from π of π₯ will give us a curve that looks like this. π¦ equals a half π of π₯. And again, letβs consider the points in π¦ equals π of π₯ which had a π¦-coordinate of negative one. Weβre 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 π¦-coordinates in this section of the graph, that part of the π¦ equals a half π of π₯ curve is gonna look like that. And that part of the π¦ equals a half π of π₯ curve is gonna look like that. So weβve locked off the π₯-axis again. All points on the π₯-axis stay where they are. But this time, weβre squashing our curve towards the π₯-axis. Weβre multiplying all of our π¦-coordinates by a half.

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

Here, weβve got a function π¦ equals π of π₯. So now, letβs define a function π of π₯ which is equal to negative one times π of π₯. So our π-value in the previous slide wouldβve been negative one. So now, letβs plot π¦ equals π of π₯. Well, in the π of π₯ function, if our π¦-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 π of π₯ function our π¦-coordinate was one, so here and here, thatβs gonna be multiplied by negative one. So thatβs gonna become negative one. And when the π¦-coordinate wouldβve been two in our π of π₯ function, those π¦-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 π¦ equals π of π₯ curve is gonna look like this. So this is a reflection in the π₯-axis. The π¦-coordinates are the same distance from the π₯-axis but theyβre on the other side. So when our π transformation here, our π-value, is negative one, we get a reflection in the π₯-axis.

Well, now, letβs think about when π is less than negative one. So, for example, when π is negative two. Letβs plot π¦ equals π times π of π₯ when π is negative two. Hereβs our π of π₯ curve. Now, when we multiply all of our π¦-coordinates by negative two, weβre getting that reflection in the π₯-axis. But all of the distances away from the π₯-axis are doubling. So weβre 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 π₯-axis and doubles. But again, any points that were on the π₯-axis, it doesnβt matter what you multiply it by. If it started off at zero, you multiply it by a number, itβs still gonna stay at zero. So the points that started on the π₯-axis stay on the π₯-axis. So this transformation of multiplying the π of π₯ function by negative two is creating a reflection in the π₯-axis followed by a stretch times two parallel to the π¦-axis.

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

So letβs summarise everything weβve learned then. So if weβve got a function called π of π₯ and we multiply that by π, weβve transformed that function. So if we say π of π₯ is the transformed function of π of π₯, then if π is bigger than one, remember thatβs gonna stretch the π of π₯ function times π parallel to the π¦-axis. Points that were on the π₯-axis stay where they are, but everything else gets stretched further away from it. Then, if π was equal to one, weβd be doing a stretch times one. Weβll be multiplying all the π¦-coordinates by one. So basically, the function is just transformed onto itself. Itβll look exactly the same. And if zero is less than π is less than one, so π is between zero and one, then this is gonna have the effect of squashing the curve of that function towards the π₯-axis. Weβre multiplying all the π¦-coordinates by π. And because π is between zero and one, the magnitude of all those π¦-coordinates is gonna get smaller. Each of them, if theyβre positive, theyβre gonna become smaller and get closer to zero. If theyβre negative, theyβre gonna be halved. And theyβre going to get closer to zero.

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

Given the graph π¦ equals π of π₯ below, sketch the curve π¦ equals π of π₯, where π of π₯ is equal to two times π of π₯.

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

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

And the points on the π of π₯ function which have a π¦-coordinate of one, theyβre gonna have that π¦-coordinate doubled to map onto the π of π₯ 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 π¦-coordinate of negative one in our π of π₯ function to turn it into our π of π₯ function, to transform into the π of π₯ function, weβre gonna be doubling those π¦-coordinates, two times π of π₯. So weβre gonna be doubling negative one to make negative two. And that means that this point here is gonna be doubled to come down here. Itβs gonna have its π¦-coordinate doubled. This point here is gonna have the same π₯-input. But itβs gonna have its π¦-coordinate doubled down here. Iβve just drawn it slightly in the wrong place. So thatβs gonna come down to here.

So what weβre 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, theyβre gonna have their π¦-coordinates doubled as well. So weβre gonna get a curve thatβs coming down, getting closer to zero, still gonna end up at zero. Likewise, in this section here, weβre doubling all of those π¦-coordinates. In this section here, doubling all of the π¦-coordinates. Itβs gonna look not quite as sharp as that. But itβs gonna look something like that. In this section here, weβre 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 π¦-coordinates are gonna come down here. And in this section here, itβs gonna look something like this. And this section here, itβs gonna look something like this.

Well, I am so happy, and Iβve checked my sketch there. Then, Iβve 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 π¦-axis away from the π₯-axis would be a description of that transformation. And itβs also worth remembering that anything that was on the π₯-axis, because it had a π¦-coordinate of zero. It doesnβt matter what you multiply that by. Itβs always gonna stay where it is, at zero. So those points on the π₯-axis are going to stay there. Theyβre anchored to the π₯-axis.