 This is a video about the transformation of graphs. You can see here that I've drawn a graph of y equals f of x. And over on the right I've shown the coordinates of some of the points on this graph. You can see that if x is minus ten, y is equal to five. If x is minus five, y is zero. If x is zero, y equals three. If x is five, y is zero. And if x is ten, y equals seven. Now what we're interested in is what will happen to the graph if we make a slight change to what we're drawing? If instead of drawing y equals f of x, we draw something slightly different from that, how will that affect the graph? The first thing I'd like to look at is y equals f of x plus two. What will that graph look like? How will it be different from the original graph that you can see already? Okay so in order to think about this we need to draw a function diagram. So as always we start with x and the first thing that happens to x here is it gets fed into f. After that whatever comes out of f, two is added in order to give us the y-coordinate. Now the key thing here is that f hasn't changed. It's still doing the same operations as above. So what that means is that if we make the x-coordinate minus ten, then the output of f is still five. And if the x-coordinate is minus five, the output of f is zero and so on. Exactly the same as before. What's different this time is that after f outputs a number, two is added on. So this time the y-coordinate instead of five will be seven. Instead of zero will be two, instead of three, five, and so on. Okay now if you compare this with the green picture at the top right, you can see that the x-coordinates are exactly the same as before, but the y-coordinates are all two units more. So if you draw this graph it will look something like this. So you can see that what we've got here is a translation. The graph has moved two units up, two units parallel to the y-axis. Okay I'd like to look at a different transformation now. We were looking at y equals f of x plus two. I now want to look at something that sounds similar if you read it out, but is actually crucially different. I now want to look at f of x plus two. Okay so we'll draw the function diagram again. This time things happen in a different order. We start off with x, but this time the first thing that happens is that we add two to x. And it's then that we put the number into f in order to get the y-coordinate. Now the key thing is the same as before. f hasn't changed, so if you put the same numbers into f as we did above the same numbers will come out. That means that if we have minus ten at this stage the y-coordinate will end up being five. And if we have minus five at this stage the y-coordinate will end up being zero and so on. What we have to ask ourselves though is what would the x-coordinate need to be so that we put minus ten into f? And the answer is that the x-coordinate would need to be minus twelve. Likewise what would the x-coordinate need to be so that minus five is put into f? And the answer is minus seven. And you can see that the x-coordinates here are going to have to be minus two or three or eight. Okay so this time if we compare the red diagram with the green one you can see that the y-coordinates are all the same as before. Instead it's the x-coordinates which have changed and the x-coordinates are all two less. So let's draw the graph. The graph looks something like this. You can see that again it's a translation but this time it's a translation in a different direction. The graph has moved two units to the left and that's something that we really need to pay attention to. It can seem a little surprising because we had f equals x plus two but what's happened is that the graph has moved two units in the negative direction. It's moved to the left. Okay let's move on. I'd like to look at another pair of transformations now. We've just been looking at what happens when we add two different positions. Let's look now at what happens if we multiply. So first of all let's look at y equals twice f of x. Here's the function diagram for that. We start off with x. That gets fed into our function f and then we double the output of that and the thinking is quite similar to before. If we start off with the same x-coordinates as above f does the same thing to them. So minus ten becomes five, minus five becomes zero, zero becomes three, five becomes zero and ten becomes seven. What happens now though is that all those numbers get doubled. So eventually five becomes ten, zero stays at zero, three becomes six, zero stays at zero, seven becomes fourteen. So if you have to draw this graph it will look something like this. That doesn't quite fit on my picture. Okay so this time what's happened is that the graph has been stretched out parallel to the y-axis. The stretch scale factor two parallel to the y-axis. Okay let's compare this now with something related. Let's compare that transformation with this one. Y equals f of 2x. So again there's some doubling happening here but at a different stage. What's this transformation? Well let's draw the function diagram. We start off with x. And the first thing that happens is that it's doubled. The answer is then fed into f and then we get our y-coordinate. Okay again always the key thing to remember is that f is still the same thing as above. So if we give f the same inputs we must get the same outputs. So if we give f an input of minus ten the output will be five. If we give f an input of minus five the output will be zero and so on. Exactly as before. Now the question is what must the x-coordinate be so that those are the inputs to f? Okay well if the input to f is minus ten then before the doubling operation we must have had five. So x needs to be minus five minus ten is the input to f. Again in order to put minus five into f then the original x-coordinate before the doubling must have been minus two point five. If five is fed into f then the original x-coordinate must be positive two point five and if ten is the input to f then the original x-coordinate must have been five. Okay if we compare this with the diagram above you can see that the y-coordinates are all the same but this time the x-coordinates are all half what they used to be. So if you have to draw this picture it would look like this. Actually of course it would carry on there would be more of it but we can't tell how it would carry on because there isn't enough of the green graph in order to tell so I'll just leave it like that. Okay so what's happened this time is that it has been squashed parallel to the x-axis. It's been squashed scale factor two or if you like it's been stretched scale factor a half and as before that can seem a bit surprising because remember that what we did was we said that we were going to double x and this doubling has actually resulted in a squashing rather than a stretching but the reason for that you can see back in our function diagram you can see that the x-coordinates all needed to be a half of what they were before so that we would have the same input to f. Okay now we've done a couple of these you might be able to see a little bit of a pattern you can see that when we made a change to the output from f like this one where we doubled the output then we get a transformation parallel to the y-axis remember that this one it ended up being a stretch scale factor two parallel to the y-axis but when we make a change to the input of f then we get a change parallel to the x-axis and there's a sense in which it's the opposite of the change that you might have expected it's what's needed to undo the change that's been made so here x is doubled before feeding it into f and therefore the changes that we need to half the x-coordinates so that the input to f is the same as it was before Okay there's one more pair of transformations that I'd like to look at what happens if we have minus signs around so first of all we'll have a look at this y equals minus f of x what will this graph look like in comparison to the original one so here's the function diagram for that first of all the x-coordinate it's made the input to f and then we take the answer to that whatever it is and make it negative and that gives us the y-coordinate so same thinking as before if we make x10 or minus 10 or minus 5 or 0 or 5 or 10 we'll get the same outputs as before so if we make the x-coordinate minus 10 then f outputs 5 if we make x minus 5 then f outputs 0 and so on the difference this time is that that number whatever it is is then made negative so 5 turns into minus 5 3 turns into minus 3 and 7 turns into minus 7 if we have to draw this graph then it will look something like this again it doesn't quite fit on but you can see what's happening this time the graph is being reflected in the line in the x-axis and the line y equals 0 so the transformation this time is a reflection in the x-axis let's look at one last transformation let's look at what happens if we have y equals f of minus x again we'll draw the function diagram this time things happen in a different order the first thing that happens is we make the x-coordinate negative or we put a minus sign in front of it then we feed the answer into f and finally we get our y-coordinate ok so same thinking as before if we put the same inputs to f we get the same outputs so if we have minus 10 at this stage we get 5 out if we have minus 5 at this stage we get 0 out and so on but now we must ask what would the x-coordinate need to be if minus 10 is fed into f and the answer is that the x-coordinate would have to be 10 if we start with 10 we make it negative we get minus 10 if minus 5 is meant to be the input to f then the x-coordinate would need to be 5 and so on if the input to f is positive 5 then x would need to be minus 5 because if we start with minus 5 and we put a minus in front of it we get minus minus 5 and if the x-coordinate is minus 10 then 10 will be the input to f ok you can see that this time the y-coordinates haven't changed the x-coordinates have they're all negative what they were originally so if we want to draw this graph it looks something like this you can see that what's happened this time is a reflection in the y-axis so the graph that we drew before was a reflection in the x-axis and the new graph that we've just drawn is a reflection in the y-axis ok thank you for listening I hope that's helped you to understand the transformations of graphs