SAT Question b3b07f0c: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. According to the table, $f(24)=-8$, $f(30)=-8$, and $f(32)=8$. Let the quadratic function $g$ be defined by $g(x)=f(x)+8$. Therefore, $g(24)=f(24)+8$, $g(30)=f(30)+8$, and $g(32)=f(32)+8$. Substituting $-8$ for $f(24)$ in the equation $g(24)=f(24)+8$ yields $g(24)=-8+8$, or $g(24)=0$. Substituting $-8$ for $f(30)$ in the equation $g(30)=f(30)+8$ yields $g(30)=-8+8$, or $g(30)=0$. Substituting $8$ for $f(32)$ in the equation $g(32)=f(32)+8$ yields $g(32)=8+8$, or $g(32)=16$. By the factor theorem, if $g(24)=0$ and $g(30)=0$, then $g(x)=k(x-24)(x-30)$, where $k$ is a constant. Since $g(32)=16$, substituting $32$ for $x$ and $16$ for $g(x)$ in the equation $g(x)=k(x-24)(x-30)$ yields $16=k(32-24)(32-30)$, or $16=k(8)(2)$, which is equivalent to $16=16k$. Dividing both sides of this equation by $16$ yields $1=k$. Substituting $1$ for $k$ in the equation $g(x)=k(x-24)(x-30)$ yields $g(x)=(x-24)(x-30)$. Since $g(x)=f(x)+8$, it follows that $g(x)-8=f(x)$. Subtracting $8$ from both sides of the equation $g(x)=(x-24)(x-30)$ yields $g(x)-8=(x-24)(x-30)-8$. Substituting $f(x)$ for $g(x)-8$ in this equation yields $f(x)=(x-24)(x-30)-8$.