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

(a) $ f(x) = x^2 + 1 $ (b) $ g(x) = x^2 - 1 $(c) $ h(x) = x^2 + 3 $ (d) $ k(x) = x^2 - 3 $

a) vertically shifted by 1 unitb) Vertical shift down by 1c) Vertical shift up by 3d) Vertical shift down by 3

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Campbell University

Oregon State University

Baylor University

Lectures

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for this problem. Let's go ahead and graph thes four problems a through D. So notice that these air all transformations of the function, why equals X squared? So all of these grafts will be actually vertical transformation, these air all going to shift the proble other down because we're adding to the explainer. So recall the graph of y equals X squared our basic problem. This will be our starting point. So for part a, let's do this in blue. So notice here. This is the shift of my one unit because we're adding a positive one to the X. We're so that will take every point on this red graph and move it up by one. So the origin goes up to the point zero one. The point one one goes toe one, too. And the point two four goes to two five and so on. And we could also plug in negative one. A negative to those points will also get shifted up by one. And there's enough points to go ahead and give a basic sketch of the problem. And there's a very rough sketch there. Suddenly go back and try a little better for that okay, that looks more like a problem. More or less so. That's our first graph, eh? Now, let's go on to green second graf This time we're going down by one unit because we're subtracting one. So going back to the red graph will take each point and then shift it down by one unit. So the origin goes down to negative one. The point one one goes down to one zero and the point two four goes two, two, three. And similarly, we could plug in negative one too negative, too. And that's enough points to go ahead and give a rough sketch so she have the same shape as the blue graph. But you shifted down a few units. That's our draft B now going on to read X squared plus three sanabria going up by three units. So we take each point on the Red original Grab X, where shifted up by three. So the origin goes upto the point zero three. The point one one goes to the point one four. So where were here? So one, and similarly, the point two four goes to the point two seven. So there's a rough sketch of the Graff, see? And then finally in black will goto graffitti. This is K and notice for subtracting three. So this is a shift down three units. So going back to our starting point y equals X squared. Each point goes down by three. So the origin ghost zero minus three. The point one one goes to one minus two in the point two four goes to the point to one, So deliver another rough sketch And there it is. There's graft e, and that solves the problem.

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