Question 820·Hard·Math

Set each factor to $0$ to find the zeros, the inputs where $f(x)=0$:

Solve each equation:

A parabola mirrors around its vertex, the turning point, so the vertex is halfway between $-1$ and $7$. Type $\frac{-1+7}{2}$ in Desmos; it shows $3$. The maximum occurs at $x=3$, halfway between the zeros.

The maximum $8$ is an output at input $3$, so the condition means:

$f(3)$ means put $3$ in for $x$ in the given function:

Type $(3+1)(3-7)$ in Desmos. It shows $-16$, so both factors together give:

Type $8\sim a(3+1)(3-7)$ in Desmos. The $\sim$ tells Desmos to find the coefficient that makes the two sides equal. Under PARAMETERS, it reports $a=-0.5$, or $-\frac{1}{2}$. The negative coefficient makes the parabola open downward, as a maximum requires. Choice A.