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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).

$ f(x) = \frac{1}{2} x^2 - 4 $

Vertex: $\quad(0,-4)$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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Let's graph this problem my first thirty with the graph of X Square and then we'LL multiply that by a half And then finally we'LL subtract by four. So we're using transformations of the problem X Square. So let's go ahead and give a rough sketch and red. So our usual Graff X squared. Let's go ahead and put that in it with a dotted line. So this is not our final answer. Take a step back here in a little sloppy, some more or less a rough sketch of your proble X squared. Next step in blue. It's the good and multiply that by a half. So this means you take each point on red, take the Y value and multiply it by a half. So over here we see the Y values for multiply That by a half, gives you two. Here we see one multiplying that by a half, gives you a half zero times a half a zero. One times a half is one half and once again, four times a half is to So this is the bluegrass one half X squared and out. Finally, we're ready for the final graph f and so we just take the blue graph and we do a minus for So this means we take each choir value on the blue graft and shifted down for So this point right here is too and subtracting for from that will put you at minus two. No, this point here was zero. Subtracting four puts you at minus four. Here we have two again subtracting for and this point was a half subtract for from that Excuse me there. And that should give you negative seven over too. Let's say three point five. And of course, we could go ahead and plug it. More points to get a more accurate looking graph. For example, if you plug in X equals three or minus three. So if you plug in three year minus three, you get a half. So let's just go ahead and for more detail, which is plug in that last point. Okay, so here's FX and Black. So that's the first task. Sketch the raft. Now let's identify the Vertex. Here we see the vortex is the point zero minus four and therefore the access of symmetry is always you just said x equal to the X value of the Vertex so that'LL just be the y axis. This is the access of symmetry and finally, the ex intercepts. So these we obtained by setting f of X equals zero Lisa room over here to do the algebra. So now you have one half X squared minus four equals zero So that gives you x square. So add the four over and then multiply by two to get X squared is eight and then finally X equals plus or minus run of a which is plus or minus too radical, too. So that will give us the ex intercepts Elice, potties and green And then over here, the minus one.

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