Problem

In Exercises 13-16, graph each function. Compare …

04:14

Problem 13 Medium Difficulty

In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $.

(a) $ f(x) = \frac{1}{2} x^2 $ (b) $ g(x) = -\frac{1}{8} x^2 $
(c) $ h(x) = \frac{3}{2} x^2 $ (d) $ k(x) = -3x^2 $

Answer

a) vertically compressed by a factor of $\frac{1}{2}$
b) Vertical shrink and reflection over the $x$ -axis.
c) Vertical stretch
d) Vertical stretch and reflection over the $x$ -axis

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Video Transcript

let's go ahead and grab each function and then compare his function with the graph of Explorer. Suffer part, eh? Well, let's go ahead and just grab X squared for so plugging a few points there. So this isn't our graph right here. But this will be the graph of X squared, and we can use a transformation to obtain the graft that we want. So this is just a vertical compression by a factor of two or a vertical stretched by a factor of one half. So in that case is it will on ly affect our graph vertically. And as you can see here, well, just multiply all around. Why values by half? So over here, why equal zero if you multiply that by a half, still get zero over here, X equals one. You see the Y values one. So multiply that by a half same thing for a negative one and then when we plugged into instead of getting four will get half of that same holds for minus two. And we have enough points here to give a decent looking graphs. So it's still a problem, but it's it was very clean, compressed by a factor of two. So this is one half X clear. Put a rose on those graphs. All right, Next one, maranto G. This one's part B. So this is the graph of G. So negative one over eight. So first Well, go ahead and just graph so that'Ll be over here and be As you can see, my axes aren't the best looking here. And be so let me just fix this a bit and then we can go ahead and just give a rough sketch of the proble X squared again. And then now we'll do some operations here to obtain the graph we like. So you see what the negative does The negative flips your graph upside down and then that one over eight makes a small supply. Are y values by one over eight? So over here when we plug in So here we see why a zero If you multiply that by negative on, eh? Still get zero. Now, if you plug in one instead of getting one is before we'LL get negative one of her eight times that such is negative one over eight. And then how about X equals two? Well before the Wye Valley was for but nonverbal, supplying that by negative one over eight. So that puts us that negative one half. So this is negative one. It's negative a half. And then this one was the negative one over eight. So let's go ahead and just give a rough looking sketch there. It shouldn't be much flatter than the original because of the one over eight. So she really welcome or something like that? And then over here, China, emphasize that it that it is flatter nearer near the origin. So that's our graph of B Now on to the next one. That's h So this one is three over two x square. So just before we'Ll just keep graphing this x squared so we could compare the graph So rough sketch of the problem here Very rough sketch. Now we'Ll multiply each of these y values by three halves So this y value down here zero multiply that by three house still zero when x is one on X squared You see why is one But we'Ll supply that by three halves and you'LL get three halves back And then when you plumped into x square to get four and Now you want to multiply that four by three half to give you six so now in excess to will have six. So's we can see. It just took the original graphic square and it vertically stretched it by a factor of three house. So now for this one again, I'll have to just extend This wax is a little bit stalking of a rough rough sketch of the X squared graph. Okay, so it's a rough sketch there than now going on to Kay, So that's negative. Three. So we'LL multiply each of these y values by negative three. So start with zero. Multiply that you still zero Dennis X is one that X squared is one. So multiply that by minus three and you get minus three back and then if you plug into swear that you get for it won't supply led by minus three. That gives you minus twelve. So this is kind of similar apart be because of the negative sign. But this time, instead of shrinking like it didn't be, this one is vertically stretched by a factor of three. So there's a rough sketch of our travelin and for convenience, we've labeled the grafts A, B C and that our final graph here D So let's just go back in and label this graph the blue one of the end negative three x squared and that's our final answer.

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