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In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).

$ g(x) = x^2 - 8 $

Vertex: $\quad(0,-8)$Axis of symmetry: $\quad x=0$$x$ -intercepts: $\quad( \pm 2 \sqrt{2}, 0)$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

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for this problem will graph those quadratic function given by the formula G over here. So to graph that, let's first Graff x clear traveler. So here we have our Vertex at the origin. If you plug in negative one or one tow X squared, your Y values one and then we can go ahead and just give a rough sketch there of the of the travel. Very rough sketch. So that's not Ari answer. That's the F. And now we could obtain our grab chief from F I just shifting downwards a units. So shift the graph of F the one that we drove her here on the left. We'LL shift that down eight units to get our phonograph. So that just means we take any point here any of the y values. And then we'LL go ahead and just shift their downward by eight units. So when we have the point, X equals minus one instead of the y value being one that will now become one minus eight. Negative seven. So this is not drawn to scale what you say negative eights around here somewhere. Then, as we seen at negative one, the Y value is negative seven. And similarly, if X is one before the Y value was one. But now we have to subtract a from that. So once again, we're getting negative seven. And over here, when x zero we see that, why zero? But when we subtract it, that puts us at minus eight. So that's our first checks. And then we'LL go ahead and just give a rough sketch here he wanted more accurate graph. You can go ahead and plug in maybe two more points or so, but it's always will be a rough sketch. So this is the graph, but we still have some more questions to answer. One of them's to identify the Vertex. And just by looking at the graph, we know originally the vortex for zero zero for X Square, and we just shifted that down by eight units. So the birthday cakes it's just the point zero minus eight, and in general, the vortex has always given by negative B over two a where in beer, the coefficients of the some of the coefficients of the problem. So this would be another way to find the Vertex and our problem. There's no ex term. There's just that X squared in a constant. So you can see in our case, that b a zero and is one. So he plugged these numbers into this formula over here. You under getting the same birth sex that we have. So that's the Vertex there. The next thing we need is the access of symmetry. Just by looking at the graph, the access of symmetry is a vertical line passing through the Vertex. So in our case, that will just be the y axis. It's got to just draw that in and green since it overlaps with in one of the axes. So here access of symmetry is in general, the formula is negative. B over two A. And as we saw being zero is one. So this is just equal to zero. So the equation of that line is X equals zero. And we have one more question here to answer what air? The ex intercepts. So we see on the graph there these points over here, so we'LL actually need to find the coordinates there. So to do that, we have to look at the equation. X square minus eight equals zero and then solve that for X so at the age of the other side, and then go ahead and take the square root. And then you could simplify that radical pull out a two. So that gives us the exit ourselves. These are points, so you should list the ex an ally co ordinate. The first one is negative. Too radical, too comma zero And the other one over here, too radical, too comma zero. So that gives thes answers over here Too radical, too. And this one x value negative. Too radical, too. So that's photograph. We listed the Vertex access of symmetry and the ex intercepts, so that's our answer.

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