Problem

In Exercises 35-42, use a graphing utility to gra…

05:18

Problem 41 Easy Difficulty

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

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

Answer

Vertex: $(-2,-3)$
Axis of symmetry: $x=-2$
$x$ -intercepts: $(-2 \pm \sqrt{6}, 0)$

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Video Transcript

Let's go ahead and graft this proble ji with the graphene utility. So here's a graph of G our problem and you can see that the Vertex is the point minus two minus three. And notice also that the access of symmetry is always the vertical line that passes through the birth. Thanks. So in our case, when I apply, X equals negative too. You can see that this is the Baroque align passing through the vortex. So the calculator gives this vortex and also accessed this imagery. And for the ex intercepts, we're getting negative. Four point four four nine and then positive zero point four four nine. The issue with eases they might just be approximations. Since we're on a calculator, they might not be the actual value And we'Ll find the salad momentarily when we find the X intercepts Algebraic Lee. So now Toa do this algebraic Lee, let's follow the instructions and find the standard form. So this would be representing G and the following. And if we can do that, we know that the Vertex is given by the point each calm. Okay, so here let me just take this G and I'LL go ahead and complete the square. So I have X square plus four x and then take half of four. You get two and square that you good for someone and foreign and to make up for it I'Ll subtract ful for us. Well, so these are equal because all I did was at and subtract the same number. So now completing the square and then just go ahead and distribute the one half. And now we see that we're on the Vertex form the standard form and here noticed the minus sign on the H. So in our case, our vertex becomes negative to negative three and that agrees with the graphing calculator. The access of symmetry is always given by X equals h where h is the x coordinate of the Vertex. So here's our age minus two. Look. So that gives us the axis Now finally, for the ex intercepts, well, go ahead and set our formula for G equal to zero and then solve for X. So let's add the three d the other side and then multiply both sides by two, even six, and then take the square root on both sides and then subtract two, two so I should be a plus or minus there in front of the radical and therefore X is equal to negative, too, plus Laura minus through six. And if you go ahead and actually plug these into the calculator and round off, you will obtain the two answers that we obtained in the calculator. So the final column these are the exact answers obtained. Algebraic Lee.

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