## Related questions with answers

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.) (4, 0), (8, 0)

Solutions

VerifiedThe $x$-intercepts are $(4, 0)$ and $(8, 0)$ This means that the solutions of the function are $x = 4$ and $x = 8$. Write these as factors of the function. Let $a$ be the opening of the parabola, whether upward or downward.

$\begin{aligned} f(x) = a(x - 4)(x - 8) \end{aligned}$

Let us find the quadratic functions whose graphs have the given $x$-intercepts:

$\begin{align*} f(x)&=a(x-s)(x-r) && \text{$s$ and $r$ are values of the coordinate $x$ of the $x$-intercepts.}\\ &=a(x-4)(x-8) && \text{We substituted $4$ and $8$ for $s$ and $r$.}\\ &=a(x^2-8x-4x+32) && \text{We multiplied.}\\ &=a(x^2-12x+32) && \text{We subtracted like terms.} \end{align*}$

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