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CCSS.Math:

function G can be thought of as a translated or shifted version of f of X is equal to x squared write the equation for G of X now pause this video and see if you can work this out on your own alright so whenever I think about shifting a function and in this case we're shifting a parabola I like to look for a distinctive point and on a parabola the vertex is going to be our most distinctive point and if I focus on the vertex of F it looks like if I shift that to the right by 3 and then if I were to shift that down by 4 at least our vertices would overlap I would be able to shift the vertex to where the vertex of G is and it does look and will validate this at least visually in a little bit so I'm going to go minus 4 in the vertical direction that not only would it make the vertices overlap but would make the entire curve overlap so we're going to make we're going to first shift to the right by 3 and we're going to think about how would we change our equation so it's shift F to the right by 3 and then we're going to shift down by 4 shift down by 4 now some of you might already be familiar with this and I go into the intuition a lot more depth in other videos but in general when you shift to the right by some value in this case we're shifting to the right by 3 you would replace X with X minus 3 so one way to think about this would be y is equal to f of X minus 3 or Y is equal to instead of it being x squared you would replace X with X minus 3 so it'd be X minus 3 squared now when I first learned this this was counterintuitive I'm shifting to the right by 3 the x-coordinate of my vertex is increasing by 3 but I'm replacing X with X minus 3 why does this make sense well let's graph the shifted version just to get a little bit more intuition here once again I go into much more depth in other videos here this is more of a worked example so this is what the shifted curve is look like think about the behavior that we want right over here in X equals three we want the same value that we use to have when x equals zero when x equals zero for the original f zero squared with zero y equals zero we still want y equals zero well the way that we can do that is if we are squaring zero in the way that we're going to square zero is if we could if is if we subtract three from X and you can validate that at other points think about what happens now when x equals 4 4 minus 3 is 1 squared it does indeed equal 1 the same behavior that you used to get well at X is equal to 1 so it does look like we have indeed shifted to the right by 3 when we replace X with X minus 3 if you replaced X with X plus 3 it would have had the opposite effect you would have shifted to the left by 3 I encourage you to think about why that actually makes sense so now that we've shifted to the right by 3 the next step is to shift down by 4 and this one is a little bit more intuitive so let's start with our shifted to the right so that's y is equal to X minus 3 squared but now whatever Y value we were getting we want to get 4 less than that so when x equals 3 instead of getting y equals 0 we want to get y equals 4 or less or negative 4 when x equals 4 instead of getting 1 we want to get Y is equal to negative 3 so whatever Y value we were getting we want to now get or less than that so the shifting in the vertical direction is a little bit more intuitive if we shift down we subtract that amount if we shift up we add that amount so this right over here is the equation for G of X G of X is going to be equal to X minus 3 squared minus 4 and once again just to review replacing the X with X minus 3 on f of X that's what shifted shifted right by 3 by 3 and then subtracting the 4 that shifted us down by 4 shifted down by 4 to give us this the next graph and you can visualize or you can verify visually that if you shift each of these points exactly down by four we are we are indeed going to overlap on top of G of X