### Video Transcript

In this video, weโre 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 ๐ฆ-axis. Weโll 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 ๐ฆ-axis.

First then, hereโs a graph of the function ๐ of ๐ฅ is equal to ๐ฅ squared. Letโs think about what would happen to that graph if we plotted ๐ฆ equals two times ๐ of ๐ฅ. Well, weโd be doubling all of the ๐ฆ-coordinates from the graph ๐ฆ equals ๐ of ๐ฅ. So two times all of the ๐ฆ-coordinates from ๐ of ๐ฅ. So in the original ๐ of ๐ฅ function, if the ๐ฆ-coordinate was zero, two times zero is zero. So that point would stay where it was. And if the ๐ฆ-coordinate had been one, we would double that to make it two. So this point here doubles its ๐ฆ-coordinate and ends up here. And again, at this point, the ๐ฆ-coordinate is going to double from one to two and end up here.

Now, for points on the graph of ๐ฆ equals ๐ of ๐ฅ that had a ๐ฆ-coordinate of two, weโre 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 ๐ฆ-coordinate in the ๐ of ๐ฅ function would have been full, weโll double that to make it eight. And we can join up all those points to see what our curve ๐ฆ equals two ๐ of ๐ฅ would look like. And itโs something like this. Now, itโs important to remember what weโve done. Weโve doubled the ๐ฆ-coordinates. And instead of our ๐ฆ-coordinate going up to here, weโve doubled that distance to bring it up to here. Instead of this ๐ฆ-coordinate going up to here, weโve doubled it to come up to here.

Now, a quick visual inspection of that might make it look like weโve just squashed the curve towards the ๐ฆ-axis. But that really isnโt whatโs happened, remember. We have doubled all of our ๐ฆ-coordinates. The transformation two times ๐ of ๐ฅ has moved the ๐ฆ equals ๐ of ๐ฅ curve twice as far away from the ๐ฅ-axis. Itโs been stretched times two in the ๐ฆ-direction.

Letโs look at another function now, ๐ of ๐ฅ. In fact, ๐ of ๐ฅ is equal to a function called sign of ๐ฅ. But donโt worry about that for now. The point is itโs got positive and negative ๐ฆ-coordinates above and below the ๐ฅ-axis. Now, weโre gonna plot ๐ฆ equals three times ๐ of ๐ฅ, which means weโre going to multiply all the ๐ฆ-coordinates by three. Now, that means that whenever the ๐ฆ-coordinate on the original ๐ of ๐ฅ curve was zero โ so here, here, here, and here โ then weโre 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 ๐ฆ-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 ๐ฆ-coordinate tripled. So our function is going to look โ 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 ๐ฆ-coordinates in that section will make it look something like that.

So now, letโs consider when ๐ฆ 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 ๐ฆ-coordinates now is gonna look something like that and that. So carrying out the transformation ๐ฆ equals three times ๐ of ๐ฅ takes all of the ๐ฆ-coordinates and multiplies them by three. So if they were positive, theyโre gonna get three times higher up the graph. And if they were negative, theyโre gonna get three times lower down the graph, further away from the ๐ฅ-axis. And itโs important to remember that any points that were on the ๐ฅ-axis are gonna stay exactly where they were before the transformation happened. So itโs like weโve locked down the ๐ฅ-axis and weโre stretching that curve away from the ๐ฅ-axis above and below.

Now, starting again, weโve got ๐ฆ equals ๐ of ๐ฅ. Now Iโve zoomed in on the ๐ฆ-axis. So the ๐ฆ-axis and ๐ฅ-axis have got a slightly different scale on them now. And if we want to plot ๐ฆ equals a half times ๐ of ๐ฅ, weโre gonna be multiplying all of our ๐ฆ-coordinates by a half. Now, any points that, on- for ๐ of ๐ฅ, were on the ๐ฅ-axis that had a ๐ฆ-coordinate of zero, when we multiply that by a half, itโs still gonna be zero. So these points are gonna be transformed to themselves. Again, weโre locking down the ๐ฅ-axis. Any points that were on the ๐ฅ-axis are gonna stay there. And if our ๐ฆ-coordinate was one, so here and here, on the ๐ of ๐ฅ function, when we multiply that by a half, those ๐ฆ-coordinates are gonna be only a half. Theyโre gonna be squashed towards the ๐ฅ-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 ๐ฆ-coordinates in that section will give us a curve that looks like this. And halving all of those ๐ฆ-coordinates in that section from ๐ of ๐ฅ will give us a curve that looks like this. ๐ฆ equals a half ๐ of ๐ฅ. And again, letโs consider the points in ๐ฆ equals ๐ of ๐ฅ which had a ๐ฆ-coordinate of negative one. Weโre 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 ๐ฆ-coordinates in this section of the graph, that part of the ๐ฆ equals a half ๐ of ๐ฅ curve is gonna look like that. And that part of the ๐ฆ equals a half ๐ of ๐ฅ curve is gonna look like that. So weโve locked off the ๐ฅ-axis again. All points on the ๐ฅ-axis stay where they are. But this time, weโre squashing our curve towards the ๐ฅ-axis. Weโre multiplying all of our ๐ฆ-coordinates by a half.

So weโve been looking at transforming a function by multiplying it by a constant. For example, mapping ๐ฆ equals ๐ of ๐ฅ onto ๐ฆ equals ๐ times ๐ of ๐ฅ. And as weโve seen, that ends up with just multiplying all of the ๐ฆ-coordinates by the constant ๐. And we saw that when ๐ is greater than one, then the function gets stretched away from the ๐ฅ-axis. And when ๐ is between zero and one, then ๐ of ๐ฅ gets squashed towards the ๐ฅ-axis. So what do you think would happen if ๐ was equal to one? Well, then, weโd be multiplying all of our ๐ฆ-coordinates by one. So ๐ of ๐ฅ would stay exactly where it is. Okay then, what about if ๐ was negative? Well, letโs try one with ๐ is equal to negative one.

Here, weโve got a function ๐ฆ equals ๐ of ๐ฅ. So now, letโs define a function ๐ of ๐ฅ which is equal to negative one times ๐ of ๐ฅ. So our ๐-value in the previous slide wouldโve been negative one. So now, letโs plot ๐ฆ equals ๐ of ๐ฅ. Well, in the ๐ of ๐ฅ function, if our ๐ฆ-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 ๐ of ๐ฅ function our ๐ฆ-coordinate was one, so here and here, thatโs gonna be multiplied by negative one. So thatโs gonna become negative one. And when the ๐ฆ-coordinate wouldโve been two in our ๐ of ๐ฅ function, those ๐ฆ-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 ๐ฆ equals ๐ of ๐ฅ curve is gonna look like this. So this is a reflection in the ๐ฅ-axis. The ๐ฆ-coordinates are the same distance from the ๐ฅ-axis but theyโre on the other side. So when our ๐ transformation here, our ๐-value, is negative one, we get a reflection in the ๐ฅ-axis.

Well, now, letโs think about when ๐ is less than negative one. So, for example, when ๐ is negative two. Letโs plot ๐ฆ equals ๐ times ๐ of ๐ฅ when ๐ is negative two. Hereโs our ๐ of ๐ฅ curve. Now, when we multiply all of our ๐ฆ-coordinates by negative two, weโre getting that reflection in the ๐ฅ-axis. But all of the distances away from the ๐ฅ-axis are doubling. So weโre 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 ๐ฅ-axis and doubles. But again, any points that were on the ๐ฅ-axis, it doesnโt matter what you multiply it by. If it started off at zero, you multiply it by a number, itโs still gonna stay at zero. So the points that started on the ๐ฅ-axis stay on the ๐ฅ-axis. So this transformation of multiplying the ๐ of ๐ฅ function by negative two is creating a reflection in the ๐ฅ-axis followed by a stretch times two parallel to the ๐ฆ-axis.

So finally, letโs think about when negative one is less than ๐ is less than zero. So ๐ is between zero and negative one. So specifically, letโs try when ๐ is equal to negative a half. So weโve got to plot ๐ฆ equals ๐ times ๐ of ๐ฅ, where ๐ is negative a half. So weโre plotting ๐ฆ equals negative a half times ๐ of ๐ฅ. Now, weโre given the graph of the ๐ of ๐ฅ function there. So weโre just gonna go through and work out what happens to all our ๐ฆ-coordinates. Again, if your ๐ฆ-coordinate started off as being zero, itโs gonna be multiplied by negative a half. Itโs still gonna be zero. So thatโs gonna stay where it is. And now, multiplying all of our ๐ฆ-coordinates by negative a half, theyโre going to the opposite side of the ๐ฅ-axis. But theyโre being halved. Their distance away from the ๐ฅ-axis is half of what it was for the ๐ of ๐ฅ function. So the curve is being squashed towards the ๐ฅ-axis and any points that were on the ๐ฅ-axis stay there. But weโve also got this reflection in the ๐ฅ-axis going on. So thatโs reflected in the ๐ฅ-axis and then squashed times a half parallel to the ๐ฆ-axis towards the ๐ฅ-axis.

So letโs summarise everything weโve learned then. So if weโve got a function called ๐ of ๐ฅ and we multiply that by ๐, weโve transformed that function. So if we say ๐ of ๐ฅ is the transformed function of ๐ of ๐ฅ, then if ๐ is bigger than one, remember thatโs gonna stretch the ๐ of ๐ฅ function times ๐ parallel to the ๐ฆ-axis. Points that were on the ๐ฅ-axis stay where they are, but everything else gets stretched further away from it. Then, if ๐ was equal to one, weโd be doing a stretch times one. Weโll be multiplying all the ๐ฆ-coordinates by one. So basically, the function is just transformed onto itself. Itโll look exactly the same. And if zero is less than ๐ is less than one, so ๐ is between zero and one, then this is gonna have the effect of squashing the curve of that function towards the ๐ฅ-axis. Weโre multiplying all the ๐ฆ-coordinates by ๐. And because ๐ is between zero and one, the magnitude of all those ๐ฆ-coordinates is gonna get smaller. Each of them, if theyโre positive, theyโre gonna become smaller and get closer to zero. If theyโre negative, theyโre gonna be halved. And theyโre going to get closer to zero.

And the most extreme example of that, I suppose, is when we put ๐ equals to zero. So all of the ๐ฆ-coordinates are multiplied by zero. They all become zero. That function is squashed so much. Its squash is transformed completely onto the ๐ฅ-axis. Then, if ๐ is between zero and negative one, we getโ because itโs negative, weโre getting this reflection in the ๐ฅ-axis of the function. And then, we squash it towards the ๐ฅ-axis. So all of the ๐ฆ-coordinates multiplied by ๐, by a fractional amount, is moving closer to that ๐ฅ-axis. If ๐ was equal to negative one, thatโs just a straight reflection of the function in the ๐ฅ-axis. All the ๐ฆ-coordinates are gonna stay the same distance away from the ๐ฅ-axis. Theyโre just gonna be the other side of it. And then, finally, if ๐ is less than negative one, weโve got a reflection in the ๐ฅ-axis because that ๐ value is negative. But then, because weโre multiplying all the coordinates by a number which is less than negative one. Weโre stretching that curve away from the ๐ฅ-axis. Okay, just before we go, here is a question for you to try.

Given the graph ๐ฆ equals ๐ of ๐ฅ below, sketch the curve ๐ฆ equals ๐ of ๐ฅ, where ๐ of ๐ฅ is equal to two times ๐ of ๐ฅ.

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

So letโs just have a look at this curve then. So weโve got an uppie-downie curve around the ๐ฅ-axis here. When I put various values into this function, I get these ๐ฆ-coordinate values out. So the value of the function weโre using is the ๐ฆ-coordinate in each of these. Now, weโre gonna put the same ๐ฅ-values into the ๐ function as we did into the ๐ function. But for the ๐ function, weโre gonna take double the ๐-values. So now, weโre gonna take twice the value of the ๐ฆ-coordinates in each case. Now these points here on the curve, the ๐ฆ-coordinate was equal to zero. And if I double zero, I still get zero. So those points on the ๐ function are gonna map to still the same points on the ๐ function. So I can sketch those in now.

And the points on the ๐ of ๐ฅ function which have a ๐ฆ-coordinate of one, theyโre gonna have that ๐ฆ-coordinate doubled to map onto the ๐ of ๐ฅ 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 ๐ฆ-coordinate of negative one in our ๐ of ๐ฅ function to turn it into our ๐ of ๐ฅ function, to transform into the ๐ of ๐ฅ function, weโre gonna be doubling those ๐ฆ-coordinates, two times ๐ of ๐ฅ. So weโre gonna be doubling negative one to make negative two. And that means that this point here is gonna be doubled to come down here. Itโs gonna have its ๐ฆ-coordinate doubled. This point here is gonna have the same ๐ฅ-input. But itโs gonna have its ๐ฆ-coordinate doubled down here. Iโve just drawn it slightly in the wrong place. So thatโs gonna come down to here.

So what weโre 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, theyโre gonna have their ๐ฆ-coordinates doubled as well. So weโre gonna get a curve thatโs coming down, getting closer to zero, still gonna end up at zero. Likewise, in this section here, weโre doubling all of those ๐ฆ-coordinates. In this section here, doubling all of the ๐ฆ-coordinates. Itโs gonna look not quite as sharp as that. But itโs gonna look something like that. In this section here, weโre 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 ๐ฆ-coordinates are gonna come down here. And in this section here, itโs gonna look something like this. And this section here, itโs gonna look something like this.

Well, I am so happy, and Iโve checked my sketch there. Then, Iโve 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 ๐ฆ-axis away from the ๐ฅ-axis would be a description of that transformation. And itโs also worth remembering that anything that was on the ๐ฅ-axis, because it had a ๐ฆ-coordinate of zero. It doesnโt matter what you multiply that by. Itโs always gonna stay where it is, at zero. So those points on the ๐ฅ-axis are going to stay there. Theyโre anchored to the ๐ฅ-axis.