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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.

$ f(x) = - (x^2 + x - 30) $

Vertex $\left(-\frac{1}{2}, \frac{121}{4}\right)$axis of symmetry $x=-\frac{1}{2}$$x$ -intercepts: $(-6,0),(5,0)$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

Campbell University

McMaster University

University of Michigan - Ann Arbor

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let's use a graphing utility to grab this quadratic. So here's the graphing calculator. You have our graph f here and by looking at the graph, we identify the Vertex and the ex intercepts. And also we know the access of symmetry will be the vertical line passing through the Vertex. So that should be X equals negative one half. Now let's go back to the White Board and record what we saw on the graph access of symmetry. It's always X equals And then it's always this value right here from the Vertex. And we also saw the ex intercepts those points for negative six and five. So now we want to verify our results or true elder bravely and we will use the standard form. So the standard form of a parabola is this form right here where the vortex is the point each comic, eh? So let's take our function have here and I'LL complete the square. So I way see the coefficient in front of the ex is the one. So you take half of that and then you square and you add that in. And now since I want to keep this equation true, I added one over force. All subtracted as well. Then, using these first returns, we complete the square there. So this is X plus one half square and then minus thirty point two five and then just distribute this minus sign outside. So we see that our age is negative one half and that rk that's our value of K. That's the Vertex, and that agrees with our point over here from the graph. So that's correct. The access of symmetry, always the vertical line passing through the vortex. So that's true. And now for the ex intercepts, let's find these a swell. So in this case, we can just take H and said it equal to zero. Excuse me, F so thirty points. Who? Five. We'LL take a square room here, so it's going to be easier to write this as a fraction. That way, When I take the square root here on both sides, I can just take this square on twenty one and the square before, So X is negative one half plus or minus eleven over two, and the two values that you get here are negative six and five, and that agrees with our ex intercepts over here, and that resolves the problem

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