Question 25·Hard·Advanced Math

The axis of symmetry is the parabola's vertical mirror line, halfway between its two x-intercepts. Type $\frac{-1+11}{2}$; Desmos shows $5$. So the maximum occurs at input $5$.

An x-intercept is where $f(x)=0$. In factored form, each intercept gives a factor that becomes $0$: $x+1$ at $-1$, and $x-11$ at $11$. So $f(x)=a(x+1)(x-11)$, where $a$ is an unknown multiplier. Check the signs by setting each factor equal to $0$.

The maximum is an output, not an input. Since it occurs at input $5$, the vertex, or highest point, is $(5,72)$. In function notation, that means $f(5)=72$.

Put the vertex input and output into the factored rule: $72=a(5+1)(5-11)$. Type $72\sim a(5+1)(5-11)$. The $\sim$ tells Desmos to fit the unknown multiplier; under PARAMETERS, it shows $a=-2$. The negative sign fits a parabola with a maximum.

In $g(4)$, replace $x$ with $4$ in the whole input $2x+1$. Type $2(4)+1$; Desmos shows $9$. So the given rule becomes $g(4)=f(9)-7$. Don't put $4$ straight into $f$.

Define $f(x)=a(x+1)(x-11)$ using the $a$ Desmos found, then type $f(9)$. Desmos shows $40$. That's $f$'s output before subtracting $7$, not yet the value of $g(4)$.

Type $g(x)=f(2x+1)-7$, then $g(4)$; Desmos shows $33$. The outside $-7$ changes the output from $40$ to $40-7=33$. Find the inside input before applying the outside change. So $g(4)$ is $33$. Choice C.