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# Scaling & reflecting parabolas

## Video transcript

function G can be thought of as a scaled version of f of X is equal to x squared write the equation for G of X so like always pause this video and see if you can do it on your own alright now let's work through this together so the first thing that we might appreciate is that G seems not only to be flipped over the x-axis but then flipped over and then stretched wider so let's do these in steps so first let's flip over flip over the x-axis so if we were to do this visually it would look like this so instead when X is equal to 0 we'll struggle why is still going to be equal to 0 but when X is equal to negative 1 instead of Y being equal to 1 it now be equal to negative 1 when X is equal to 1 instead of squaring one and getting one you then take the negative of that to get to negative 1 so when you flip it it looks like this Y when X is equal to negative 2 instead of Y being equal to 4 it would now be equal to negative 4 so it would look like this so as we just talked through is we're trying to draw this flipped over version whatever Y value we were getting before for a given X we would now get the opposite of it or the negative of it so this green function right over here is going to be y is equal to the negative of f of X or we could say Y is equal to negative x squared whatever the X is you square it and then you take the negative of it whatever X is you square it and then you take the negative of it and you see that that will flip it over the x-axis but that by itself does not get us to G of X G of X also seems to be stretched in the horizontal direction and so let's think about can we multiply this times some scaling factor so that it does that stretching so that we can match up to G of X and the best way to do this is to pick a point that we know sits on G of X and they in fact give us one they show us right over here that the point two comma negative one sits on G of X when X is equal to two Y is equal to negative one on G of X so you could say G of 2 is negative 1 now on our green function when X is equal to 2 y is equal to negative 4 so let's see maybe we can just multiply this by 1/4 to get our G so let's see if we were to let's see if we scale by 1/4 do that does that do the trick scale by 1/4 so in that case we're going to have y is equal to not just negative x squared but negative 1/4 1/4 x squared and if you're saying hey so how did you get 1/4 well I looked at when X is equal to 2 on our green function when X is equal to 2 I get 2 negative 4 but we want that 2 when X is equal to 2 to be equal to negative 1 well negative 1 is 1/4 of negative 4 so that's why I said okay let's let's just see if we could take our green function if I multiply it by 1/4 that seems like it'll be it'll will match up with G of X and so let's let's verify that when X is equal to 0 well this is still all going to be equal to 0 so that makes sense when X is equal to 1 let me do this in another color when X is equal to when X is equal to 1 then 1 squared times negative 1/4 well that does indeed look like negative 1/4 right there when X is equal to 2 2 squared is 4 times negative 1/4 is indeed equal to negative 1 when let's try at this point here because it looks like this is sitting on our graph as well when X is equal to 4 4 squared is 16 16 times negative 1/4 is indeed equal to negative 4 and it does work also for the negative values of X as well so I'm feeling really good that this is the equation of G of X G of X is equal to negative 1/4 times x squared and so in general that when we're saying we're scaling it we're scaling it by negative value this is what flips it over the x-axis and then multiplying it by this this just frac that has an absolute value less than one this is actually stretching it wider if this value right over here it's absolute value was greater than one then it would stretch it vertically or make it thinner in the horizontal direction