SAT Question 16aa9758: Nonlinear Functions answer and explanation with Desmos solution

Choice C is correct. The given equation is quadratic. The maximum value of a function defined by a quadratic equation can be displayed as a constant in the vertex form of the equation, $f(x)=a(x-h{)}^2+k$, where the maximum value of the function is $k$, which occurs when $x=h$, and $a$ is a constant. The given equation can be rewritten in this form by completing the square. To complete the square, the given equation can be rewritten as $f(x)=-6(x^2-10x)-126$, which is equivalent to $f(x)=-6(x^2-10x+25)-126+6(25)$, or $f(x)=-6(x-5{)}^2+24$. This equation is in vertex form, where $a=-6$, $h=5$, and $k=24$. Therefore, $f(x)=-6(x-5{)}^2+24$ displays the maximum value, $24$, of the function as a constant.