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CCSS.Math:

here have drawn the most classic parabola y is equal to x squared and what I want to do is think about what happens if I were to if or how can I go about shifting this parabola and so let's think about a couple of examples so let's think about the graph of the curve this is y is equal to x squared let's think about what the curve of y minus K is equal to x squared what would this look like well right over here we see when X is equal to 0 x squared is equal to 0 that's the one this the CLO curve so x squared is equal to Y or Y is equal to x squared but for this one x squared isn't equal to Y it's equal to Y minus K so when x equals 0 and we square it 0 squared doesn't get us to Y it gets us to Y minus K so this is going to be K less than Y or another way of thinking about it this is 0 if it's K less than Y y must be at K wherever K might be so y must be at K right over there so at least for this point it had the effect of shifting up the Y the Y value by K and that's actually true for any of these values so let's think about X being right over here for this yellow curve you square this x value and you get it there and it's clearly not drawn the scale the way that I've done it right over here but you square this x value you get there but now for this curve right over here x squared doesn't cut it it only gets you to Y minus K so K must be or Y must be K higher than this so this is y minus K y must be K higher than this so y must be right over here so this curve is essentially this blue curve shifted up by K shifted up so making it Y minus K is equal x squared shifted it up by K whatever value this is shifted up by K this distance is a constant is a constant K the vertical distance between between these two parabolas I'll try to draw it as cleanly as cleanly as I can this vertical distance is a constant K now let's think about shifting in the horizontal direction let's think about what happens if I were to say Y is equal to not x squared but X minus H X minus H squared so let's think about it this is the value you get for y when you just square 0 you get Y is equal to 0 how do we get y equals 0 over here well this quantity right over here has to be 0 so X minus H has to be 0 or X has to be equal to H so let's say that H is right over here so X has to be equal to H so one way to think about it is whatever value you are squaring here to get your Y you now have to have an H higher value to square that same thing because you're going to subtract H from it just to get to 0 X has to even just 2 square is 0 just to square 0 X has to equal H here if you wanted to square one X just have to be equal to 1 so if your year let's just say for the sake of argument that this is the value this is X is equal to 1 and this is 1 squared clearly not drawn to scale so that would be 1 as well but now to square one we don't have to just get it x equals 1 X has to be H plus 1 it has to be one higher than H it has to be H plus 1 to get to that same point so you see the net effect is that by instead of squaring just X but squaring X minus H we shifted the curve to the right so the curve we do this in this purple color the curve this magenta color will look like this we shifted it we shifted it to the right and we shifted it to the right to the right by we shifted it to the right by H now let's think of another thought experiment let's imagine let's imagine that let's think about the curve Y is equal to negative x squared well now whatever the value of x squared is we're going to take the negative of it so here no matter what X we took we square it we got a positive value now we're going to we're always going to get a negative value once we multiply it times a negative one so it's going to look like this it's going to be a mirror image of y equals x squared reflected over the horizontal axis so it's going to look something like that so that's y is equal to negative x squared and now let's just imagine scaling it even more what would Y what would y equal y equal negative 2x squared well now actually let me do two things so what would Y equals 2x squared look like so let's just the positive version so y equals 2x squared well now as we square things we're going to multiply them by two so it's going to increase faster so it's going to look something like this it's going to be narrower and steeper so it might look something like this and once again I'm just giving you the idea haven't really drawn this to scale so this will this increasing it by a factor will make it increase faster if we did y equals negative 2x squared if we did y equals negative 2x squared well then it's going to get negative faster on either side so it's going to look something like this it's going to be the mirror image of what I just drew so it's going to be a narrower a narrower parabola just like that and similarly and I know that my diagram is getting really messy right now but just remember we started with y equals x squared which is this curve right over here what happens if we did y equals 1/2 x squared so let's move I'm running out of colors as well if we did y equals 1/2 x squared well then the thing is going to increase slower it's going to look the same but it's going to open up wider it's going to increase slower it's going to look something something like this so this hopefully gives you a sense of how we can shift parabolas around so for example so for example if I have if I have and I'm going to do it I'm doing a very rough drawing here to give you the general idea of what we're talking about so if this if this is y equals x squared so that's the graph of y equals x squared that's y is equal to x squared the graph of y let me do this in a color that I haven't used yet the graph of y minus K is equal to a times X minus H squared will look something like this instead of the vertex being at zero zero the vertex or the lowest or I guess you could say the minimum or the maximum point the extreme point in the parabola at this point right over here would be the maximum point for a downward-opening parabola a minimum point for an upward-opening parabola that's going to be shifted it's going to be shifted by H to the right H to the right and K up K up so its vertex it's going to be right over here and it's going to be scaled by a if a is greater than so if a is equal to one it's going to look the same it's going to have the same opening it's going to have the same opening so that's a equals one if a is greater than one it's going to it's going to it's going to be steeper like this if a is less than 1 but greater than zero it's just going to be wider opening like that if a is actually if a is zero then it just turns into a flat it turns into a flat line and then if a is negative but less than negative one it's kind of a broad opening things like that or I should say greater than negative one if it's between zero and negative one it'll be a broad opening thing like that at negative one it'll look like a reflection of our original curve and then if it's if a is less than negative one so it's even more negative than it's going to be even a steeper a steeper parabola that might look like that so hopefully that gives you a good way of how to shift and scale parabolas