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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) = x^2 + 3x + \frac{1}{4} $

Vertex: $\left(\frac{-3}{2},-2\right)$Axis of symmetry: $x=\frac{3}{2}$$x$ -intercepts: $\left(\frac{-3}{2} \pm \sqrt{2}, 0\right)$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Campbell University

Baylor University

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so to graft this problem. Let's go ahead and write this in the vortex farm and we'LL see why this will help us. So is take this three in front of the X, divide that by two and then square it. However, since we've just added this in, we'LL also subtracted. So it's a nine over four. Now, if I look inside these Prentice's, I could go ahead and rewrite this X plus three halfs square and then minus two. So if we start off in red with the graph of let's say g of X equals x square, then how do we get to the graph of F first you add three half City X and then after that you shifted down by two. So this first step here, this is a shift three half units to the left and then this one down here is two units down. Also a shift. So in red will start here with the graph of ex Claire one, two, three, four. So he was just a rough sketch of the graph of ex players and then for our breath. Let's go ahead and do this in green F. So the first step was is to go three half units to the left and then we go down by two. So if we go three half So the left that'LL put us over here and negative one and a half and then you go down too. This point over here one one. So go back Negative one point five and then down to same for this point over here. Negative one one. Go back one and a half and then down to and we can give these other points as well. So we have two and four. So the two that goes back to the point point five and then bring that down by two. This over here Was that negative too? So that'LL go too negative. Three and a half And then that also goes down too. And so here we are. So I should keep going up here and there we go. And there's our graph. So a few more things to do here, Identify the birth, Tex. Well, well, you already have the birth text form here so we can see that the birth tex is the point Negative three over to common Negative too. And that's down here for up, Avi, for Vertex assesses imagery. The access is always given by you. Just take X and set that equal to the first corner of the Vertex. So that will be this folk align over here going right through the Vertex and also X intercepts. We see that autograph has X intercepts. This is going and find out what those are. And since I am running out of room here, let me go ahead and do this on the next page. So start I'LL go ahead and said this vertex form equal to zero and then saw for X. All right, so add that to over to both sides, take the square room and then go ahead and subtract three halfs from both sides. So for the ex intercepts, since these air points, we could technically right, thesis of order pairs. So the first one will be, And then the second one will be when we use the plus sign on the square root

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