 Now that we have discussed how to graph different transformations, we are going to work on the opposite skill. So we are going to be given a graph of one of our parent functions and just from the graph we should be able to come up with the equation. So the first thing that you'll always want to do when finding an equation, of course, is identify what your parent function is. So if you just look at the shape of the graph, you should be able to figure that out. Here we have a parabola and so we know that our parent function will just be an x squared function. So once we have identified our parent function, what we want to look for is any shifting that has occurred. So if you think of your parent function for our parabola, it typically begins right at the origin for its vertex and then goes up from there. So if you look at this parabola here, you'll notice that its vertex is down here at the point 0 negative 4. So you want to look to see if it has shifted to the right or left at all, but here the right and left shift is none. It doesn't exist. And then you also want to look to see if it has shifted up or down. I'll use arrows instead. And here you'll notice that this graph has shifted down 4. So think about what we talked about when learning our basic transformations. A shift down 4 is going to be something that occurs outside of our typical function. So this shift down 4 is going to translate our function. The x squared will stay the same, but to shift down 4 we have to subtract 4 at the end. And if all that was happening was that this graph was shifting down 4, then we would be all set. But you'll notice that it's not a direct mapping that would take place. So our final step when finding equations is going to be to look for any stretching or compressing or even reflecting that occurs. And the way that you can do this is just by finding another point on the graph. And then we can plug that in in order to see if there really should be a leading coefficient here in front of our x squared. So looking at my graph I like to find it has to be a point other than that vertex that you used. So I'm looking for one that crosses perfectly on here. We would hope that I made that the case. It looks like up here is the first one that really does cross perfectly from what I'm seeing. So this is the point 5, 6. What I'm going to do is think of this as the function y equals a x squared minus 4. I'm going to plug that point 5, 6 in for my x and my y in that equation. So my x will get replaced with the 5 and my y will be replaced with the 6. If I solve this I'll simplify the 5 squared. I get 25a minus 4 equals 6 and then if I add 4 to each side get 10 equals 25a and then if I were to solve that for a 10 divided by 25 you can leave it as a fraction but simplify the fraction or you can do it as a decimal but I would get a equals point 4. So what that means is my final equation in this case would be y equals 0.4x squared minus 4. I think that went off the screen. I'll do that on the next page. I'll write that out. But what you have to see is this point 4 means that we would have a vertical compression and that is what's happening here. It's all compressing downward from where it would have been before. So that would be our final equation. Let's look at one more example just to practice finding that stretch compression or reflection number. Okay we're going to look now at the graph here given below and this graph here you should be able to tell by looking at it that the parent function for this is our square root function. So if we didn't have any transformations we would have the function y equals the square root of x. If you want to you can graph what that looks like or what that would have looked like on here just so you can kind of see how that transformation has changed it but it typically starts at the point zero zero and then shifts from there. So as I said before the first thing you want to look at is you want to check to see if it has shifted at all. So if you look here our graph typically starts at zero zero and you'll notice that it has shifted to the left three and it hasn't shifted up or down at all. So we just have a shift to the left three. If you need to look back through your notes you can examine how to show a shift left three but it's a horizontal shift and so that means right by our x value we're going to put a plus three. So all of that is going to be underneath our square root and if this just was a perfect shift three then that would be our equation but you'll notice like if we shifted this point left three it would have been right through this point and our graph doesn't quite go through that. So we have to now see if there is a compression or a stretch or a reflection. This one doesn't look like there's a reflection at all but we have to check for that and the way we check for that is we find our I always call it our a value and a is this leading coefficient that goes in here and so we will go about finding that. I'll use a different color down at the bottom but the way to find your a value is find another point on the new graph that you have and you want it to be one that crosses perfectly so I'm going to use this point here one three that it goes through. So what I'll do is I will replace the x with one and I will replace the y with three and what I get when I do that is three equals a times a square root of four which is just two so three equals two a and if I solve that for a a is equal to three over two or one point five and that I can see will mean I have a bit of a vertical stretch which makes sense because my point was supposed to be right here and it's stretched up a little bit so that makes sense with what the graph looks like and so my final equation here that a goes in front of the square root and then that plus three accounts for my horizontal shift to the left and that would be my final equation in this case