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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.)

$ ( 0, 0 ) $, $ ( 10, 0 ) $

$f(x)=a\left(x^{2}-10 x\right), a>0$$f(x)=a\left(x^{2}-10 x\right), a<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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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…

Let's find a quadratic that has thie given x intercepts. So let's try X minus zero and then x minus ten. And this method always works because you can check here if you plug in zero or if you plug in ten for X, then one of these two factors will be zero so that the whole function is there. So I guarantees that our function f does have the correct intercepts. And then if you just go ahead and simplify this and then distribute you see that leading coefficient is positive a positive one so that the graf opens upward. And similarly, we want a graph that also has the same X intercepts. But this time opens down work. So why don't we just take our original function over here and just multiply this by a negative one? And you could you should commit yourself that if you plug in zero or ten and two g using the same reasoning is before we'LL get out zero because if you plug in zero for energy, then you get his unit over here. But excuse me, that should've been a minus. I gotta fix this in blue. And similarly, if you plug in ten for X, the second termer zero. So the function is zero. Now, when we go ahead and multiply out the eggs, this time we get a negative coefficient in the front. And so this tells you that the Graf opens down work, including the negative sign there and your answer, and these two make up our final answer.

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