Nonlinear equations in one variable and systems of equations in two variables SAT question

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, $y=a{(x-h)}^2+k$, the vertex of its graph in the xy-plane is $(h,k)$. The given quadratic equation, $y=-x^2+9x-100$, can be rewritten as $y=-(x^2-2(\frac{9}{2})x+{(\frac{9}{2})}^2)+{(\frac{9}{2})}^2-100$, or $y=-{(x-\frac{9}{2})}^2+(-\frac{319}{4})$. Thus, the value of $c$ is equal to $-\frac{319}{4}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.