SAT Question f2d60b99: Nonlinear Functions answer and explanation with Desmos solution

Choice A is correct. The graph of a quadratic equation in the form $y=a{\left(x-h\right)}^2+k$, where $a$, $h$, and $k$ are positive constants, is a parabola that opens upward with vertex $\left(h,k\right)$. The given function $f\left(x\right)=\frac{1}{9}{\left(x-7\right)}^2+3$ is in the form $y=a{\left(x-h\right)}^2+k$, where $y=f\left(x\right)$, $a=\frac{1}{9}$, $h=7$, and $k=3$. Therefore, the graph of $y=f\left(x\right)$ is a parabola that opens upward with vertex $\left(7,3\right)$. Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the y-coordinate of the vertex of the graph of $y=f\left(x\right)$ is the minimum value of $f\left(x\right)$. Therefore, the minimum value of $f\left(x\right)$ is $3$. It's given that $f\left(x\right)=\frac{1}{9}{\left(x-7\right)}^2+3$ represents the metal ball's height above the ground, in inches, $x$ seconds after it started moving on a track. Therefore, the best interpretation of the vertex of the graph of $y=f\left(x\right)$ is that the metal ball's minimum height was $3$ inches above the ground.