Question 97·Hard·Advanced Math

The graph touches the $x$-axis at $-2$, so its factor $x+2$ must repeat. A touch needs a squared factor at the least possible degree. The graph crosses at $1$ and $4$, so use one factor for each of those roots, or inputs with output $0$:

The number $a$ sets the graph’s vertical scale.

The point $(0,2)$ says input $0$ gives output $2$. Substitute both coordinates to make an equation for the missing multiplier:

Unlike a root, this point gives an equation that can determine $a$.

Type $2\sim a(0+2)^2(0-1)(0-4)$. The $\sim$ tells Desmos to find $a$ from the equation. Under PARAMETERS, Desmos shows $a=0.125$, or $\frac18$.

Type $f(x)=a(x+2)^2(x-1)(x-4)$, using the value of $a$ Desmos found. This keeps the squared factor at the touch; dropping that square would change the function.

Type $f(3)$, which means put $3$ in for $x$. Desmos displays $-6.25$; tap its fraction button to see $-\frac{25}{4}$. So the output at $x=3$ is $-\frac{25}{4}$. Choice A.