SAT Question a54753ca: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Choice C is correct. In the xy-plane, the graph of the line $y=c$ is a horizontal line that crosses the y-axis at $y=c$ and the graph of the quadratic equation $y=-x^2+9x-100$ is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of $c$ should be equal to the y-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, $y=a{\left(x-h\right)}^2+k$, the vertex of its graph in the xy-plane is $\left(h,k\right)$. The given quadratic equation, $y=-x^2+9x-100$, can be rewritten as $y=-\left(x^2-2\left(\frac{9}{2}\right)x+{\left(\frac{9}{2}\right)}^2\right)+{\left(\frac{9}{2}\right)}^2-100$, or $y=-{\left(x-\frac{9}{2}\right)}^2+\left(-\frac{319}{4}\right)$. Thus, the value of $c$ is equal to $-\frac{319}{4}$.