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In Exercises 65-70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.)

$ ( -3, 0 ) $, $ \left(-\frac{1}{2}, 0 \right) $

Concave Up Equation: $x^{2}+\frac{7}{2} x+\frac{3}{2}$Concave Down Equation:$-x^{2}-\frac{7 x}{2}-\frac{3}{2}$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

University of Michigan - Ann Arbor

Idaho State University

Lectures

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In mathematics, the absolu…

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In Exercises 65-70, find t…

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Find two quadratic functio…

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For Exercises $67-70$, fin…

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In Exercises 59-64, use a …

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Graphical Reasoning In Exe…

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Let's find a quadratic function F that has given X intercepts. So what we could do here is look at X minus Negative three x minus Negative a half and let me just go ahead and rewrite this. Now, The reason for doing this is that you can see that if you do. If you plug in negative three in sex than that, don't make this factor zero so that the whole function is zero. And if you plug in negative one half that'LL make the second term zero so that the whole project is your own, and then to convince yourself that says this is a quadratic, you could go ahead and just multiply this so someone of foil this out and then just combined those X terms. And if you look at the leading coefficient, it's a positive one. This means that the Graf opens upwards. So this is one of our answers, and now we would like another graph another problem. But this one that opens downward. So what we can do is just take our original function over here. It just won't supplied by a negative one. This will ensure that the ex intercepts don't change because once again, if you plug in negative three or negative behalf and for X, then one of these two factors will be zero. And then to just convince yourself that this is ah, Paraiba below that opens down where you could supply foil this out and multiplied by negative one. We've already foiled over here, so we could just multiply this all by negative one. And due to the negative sign on the front, we see that this one opens our

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