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Describe the sequence of transformation from to given that $ f(x) = x^2 $ and $ g(x) = a(x - h)^2 + k $. (Assume a, h, and k are positive.)

Vertical stretch by a, shift right by h, shift up by k

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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in this problem. We describe the sequence of transformation that take us from the graph f equals X squared to this graph given by the equation involving G So we'LL do This is a sequence one step at a time. So the first step is just to go ahead and start off with our parable a X squared. So that's over here in the left. The first graph. So let's just give a rough, rough sketch of this applauding a few points. So again, this is just a rough sketch. That's our problem. Why equals X squared then? Next step is to go ahead and deal with this transformation. This horizontal shift. So now we'LL look at the graph Why equals and I will do X minus eight square. So this is a shift eight units to the right and the reason we're going to the right, it's because we're given that these are positive numbers. So now we take the graft that we started with any point on this graph for example, this point zero zero the corresponding point on their new graph. Well, we went eight units to the right, so that will become H zero. So that's our new Vertex. So here's a TSH, and then we can go ahead and plot those two other point still. So this point now, this was originally negative one and one. So when we add age to the ex component, this is a check minus one common one. There's that each minus one. There's our Vertex, and similarly, we have each plus one comma one. So let's go ahead and grab that. Yeah, again, He's a rough, rough sketches, so that's the first step in our sequence. Now we go onto the next one, and that will involve dealing with this, eh? So now let's look at the graph eight times our previous rough. So is a positive number again because this has given information. So this is a vertical stretch or compression, depending on whether it's bigger than one or less than one by a factor of a so we don't know whether is bigger than one or less than one. All we know is that it's bigger than zero so and this graph it, it'LL look like perhaps a CZ bigger than one. So how do we graph this new graph over here? Angry. We just take all the points all the Y values in our previous graph and multiply them by a So this point over here the vertex the point h zero If we take this y value zero and multiply it by a we'LL still get zero So this vertex will still be at the same place and each However, if we go Teo h minus one her age plus one As we saw in the previous graphs, the Y values were one And so now when we multiply that one by a of course finding why value on our green graph will be Danny. So again, just it looks like here I'Ll just take a to be bigger than one. That's what the graph will look like. Those are our points there You still have our vertex on the X axis and then we'LL go ahead and just grab that problem. So this stretches a vertically by a factor of a and then our last step to get to G. It's just to deal with this, plus que so the difference between the green and the blue is the green is adding a kay to the Y value. So this is our previous why here and I were adding Kato that so this is going to be a vertical shift by a factor of K So shift K units okay, upward. And the reason we're going up is by the assumption that kay is also a positive number. So we'Ll take each of the points on our previous problem green and we'LL just move them up by a factor of kay so a vertex The y value was previously zero So now we'LL add me to that and that puts us at our new high value for the vortexes that kay and then we'LL look at other two points h minus one x plus one. So those previously had why values given by a But now that will get mapped too because we're shifting upward by a factor of kay So we'LL add Kato both of those So in our case, we'LL have our off the ad k to these values and then the X values are H minus one and a tch plus one And there we go there's are three points Another ref sketch here for a problem and finally we followed the sequence and we got to the wrath of G So this is our final answer, and we've explained the sequence of transformations to go from F two g, so that's our final answer.

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