 This is a video about transformations of graphs, and in particular how to find the equation of a transformed graph by starting with the equation of the original graph and making changes to it. Here's an example. I've started with the graph of y equals x squared and transformed it. The question is what's the equation of the transformed graph? Now there are several ways to produce the transformed graph, starting with y equals x squared. One way is to start by reflecting it in the x-axis. And if we do that, this is what we get. As a second step, we can translate this five units to the right, five units parallel to the x-axis. And if we do that, we get this. In this case, the third step would now be to translate this four units upwards, four units parallel to the y-axis. Okay, so let's write down those steps. The first was to reflect in the x-axis. The second was to translate five units to the right, five units parallel to the x-axis. So that's a translation with vector five zero. The final step was to translate parallel to the y-axis. So that's a translation with the vector naught four. Okay, now we've identified some transformations. We need to think about what changes we need to make to the function we're graphing. We know that when we reflect something in the x-axis, we change the function f of x. Into the function minus f of x. We know that when we translate something parallel to the x-axis with vector five zero, then we translate the function f of x into f of x minus five. And finally, we know that when we translate something parallel to the y-axis with vector naught four, we turn f of x into f of x plus four. Now this tells us the changes that we need to make to a general function f. Now we need to think about the changes that we need to make in terms of the specific function y equals x squared. The first change is to take whatever function we had before and put a minus sign in front of it. So where before we had y equals x squared, now we need to have y equals minus x squared. The second step says that where before x was the input to our function, now it needs to be x minus five. So we need to take x and replace it with x minus five. I've put the x minus five in brackets because we need the squared to apply to the whole of x minus five and not just to the five. The last step says to take whatever function we had before and add on four. So that means that we take whatever we just had and we add four onto the end. Of course, we would normally write that in a more elegant way. Okay, so we've discovered that an equation of the transformed graph is why it was for take away the square of x minus five. We found that out by finding some transformations that turn the original graph of y equals x squared into the new graph. Understanding those in terms of a general change to a function f and then applying those changes to the specific function y equals x squared. And this is the method for finding the equation of a transformed graph from the equation of the original graph.