 This is a video about transformations of graphs, and in particular how we can draw the graph of a complex function by starting with the graph of a much simpler function and transforming it. Here's my first example. We're going to draw the graph of y equals the square of x plus 2 plus 4 by starting with the graph of y equals x squared and transforming it. At the top right I've shown the x and y coordinates of some of the points on this graph to help with my explanation. Now the main thing we have to do is to work out what transformations to apply to y equals x squared in order to draw the graph of y equals the square of x plus 2 plus 4. And the way to do this is to draw a function diagram. So imagine we were trying to work out the y coordinates of some points on this graph. We'd start with x. The first thing that we would do is add 2. The second step is that we would square. And the third step is that we would add 4, and that would tell us the y coordinate. Now the most important thing to understand is that the original graph is illustrating this part of the function diagram. On the original graph, the x coordinates are the input to the square function and the y coordinates are the output from the square function. So when x was minus 6, y is 36. When x is minus 3, y is 9, and so on. I'm just copying these numbers from the table at the top of the page. Okay, but on the new graph, the one that we want to draw, both the x coordinates and the y coordinates need to change. That's because the x coordinate is no longer the input to the square function and the y coordinate is no longer the output from the square function. Instead, the y coordinates are all 4 more than they used to be. So where before we had 36, here we'll have 40. Where before we had 9, here we'll have 13, and so on. The x coordinates also need to change, and they need to be 2 less than they were before. So where before we had minus 6, here we've got minus 8. Where before we had minus 3, here we've got minus 5, and so on. They all need to be 2 less than they were before, so that when we add on 2, we get the numbers that are the input to the square function. Okay, so now we can see what transformations need to be performed. First of all, the y coordinates are all 4 more than they used to be. So the plus 4 here means that the entire graph moves 4 units up. That's a translation of 4 units parallel to the y axis. The x coordinates are all 2 less than they used to be. So the plus 2 here means that the graph shifts 2 units to the left. That's a translation parallel to the x axis. So now we can draw the graph. On the original graph, the minimum point is at the origin. But after it shifted 4 units up and 2 units to the left, the minimum point will move to here. And this is what the graph will look like. Okay, so here we've been able to draw the graph of y equals the square of x plus 2 plus 4 by starting with the graph of y equals x squared and making 2 transformations. Here's my second example. This time we're going to draw the graph of y equals sine 2x plus 1 by starting with the graph of y equals sine x and transforming that. Again, we need to draw a function diagram. So this time, starting with x, the first thing that happens is it's doubled. Then we put that into the sine function. Finally, we add 1 in order to get the y coordinate. Now just as before, the key thing to understand is that the original graph is illustrating this part of the function diagram. On the original graph, the x coordinates were the input to the sine function and the y coordinates were the output from the sine function. So when x was minus 180 degrees, y was 0. When x was minus 90 degrees, y was minus 1 and so on. Again, I'm just copying these values from the table at the top of the page. But again, the x and the y coordinates both need to change because on the new graph, the x coordinate isn't the input to the sine function and the y coordinate isn't the output from the sine function. The y coordinates are all 1 more than they used to be. So where before the y coordinate was 0, now it will be 1. Where before it was minus 1, now it will be 0 and so on. The x coordinates also need to change and they all need to be half what they were before. So where before the x coordinate was minus 180, now it will be minus 90. Where before it was minus 90, now it will be minus 45 and so on. The x coordinates all need to be half what they were before so that when they get doubled, we have the same inputs to the sine function. Okay, now we can see what transformations have to be applied. First of all, the y coordinates have all increased by 1. So the effect of this plus 1 here is to move the entire graph one unit upwards, one unit parallel to the y axis. The x coordinates have all been divided by 2. So the effect of this times 2 is to squash the whole graph parallel to the x axis scale factor 2. You might like to think of that as a stretch scale factor half. So let's draw this graph. The original graph had a minimum where x was minus 90 but now it's going to have its minimum when x is minus 45 and the minimum value isn't minus 1, it's 0. So the minimum will be there. The old graph had a maximum when x was 90 but now the maximum is when x is 45 and the maximum value is 2. We also know that it passes through these points here. So now we can draw this, it looks like this. And obviously it carries on so if I draw the rest we should get something like this. Okay, so this is the graph of y equals the sine of 2x plus 1 and we've obtained that from the graph of y equals sine x by applying two transformations. Here's my last example. This time we're going to draw y equals 4 take away the absolute value of x take away 6. By starting with the graph of y equals the absolute value of x and transforming that. Here's the function diagram. We start with x. The first thing that happens is we subtract 6 from x. Then we find the modulus of that. The next step is we need to make whatever we get negative before finally we add 4 in order to get the y coordinate. As with the previous examples the original graph is illustrating this part of the diagram. On the original graph the x coordinates were the input to the modulus function and the y coordinates were the output from the modulus function. So when x was minus 10, y is 10, when x is minus 5, y is positive 5 and so on. Now this time there are three changes to the coordinates. The y coordinate changes twice and the x coordinate changes once. To get the new y coordinates from the old y coordinates first of all we have to make them negative. So that where we had 10 before now we have minus 10. Where before we had 5 now we have minus 5 and so on. And then we add 4 so that minus 10 becomes minus 6. Minus 5 becomes minus 1 and so on. The x coordinates also change. This time they need to be 6 more than they used to be. That's so that when we subtract 6 we then get the right inputs to the modulus function. So where before we had minus 10 now we have minus 4. And where we had minus 5 now we've got 1. And instead of 0 we need 6. Instead of 5 we need 11. And instead of 10 we need 16. Okay so now we can see what transformations we need to apply. The only thing is that this time we have to be very careful about what order to do things in. Because with the y coordinate it's essential that the first step is to make the y coordinates negative. So the first step is a reflection in the x axis. The second step is that the y coordinates are all increased by 4. So that the entire graph is moved 4 units upwards. That's a translation parallel to the y axis. It's very important to get those transformations in the right order. If you do them the wrong way around we will get a different graph and the answer would be quite wrong. The third transformation is that the entire graph needs to move 6 units to the right. The x coordinates have all increased by 6. So we need to translate the graph 6 units parallel to the x axis. Okay now because there are 3 transformations this time and it's a bit more complex than before I'll do the steps one at a time. So first of all I'll do the transformation which reflects the entire graph in the x axis. The minus there means that the whole graph is reflected and ends up looking like that. The second step is it moves 4 units upwards and when we do that we end up with this. The last step is that we need to move the graph 6 units to the right. 6 units parallel to the x axis. When we do that we get this. So this is the graph of y equals 4 take away the modulus of x minus 6. We've obtained it by starting with the graph of y equals the modulus of x and performing 3 separate transformations. Okay I hope this video has helped you to understand how we can draw the graph of a complex function by starting with the graph of a much simpler function and transforming it.