1. The problem asks to analyze the function $h(d) = - \frac{1}{9} (d - 4)^2 + 4$ which models the height of a basketball ball as a function of horizontal distance $d$. 2. This is a quadratic function in vertex form: $$h(d) = a(d - h)^2 + k$$ where $(h,k)$ is the vertex and $a$ determines the parabola's opening direction. 3. Here, $a = -\frac{1}{9}$, $h = 4$, and $k = 4$. Since $a$ is negative, the parabola opens downward. 4. The vertex is at $(4,4)$, meaning the maximum height of the ball is 4 units at horizontal distance $d=4$. 5. The parabola represents the ball rising to height 4 at $d=4$ and then falling as $d$ moves away from 4. 6. Therefore, the answer is: The basketball reaches its maximum height of 4 units at horizontal distance 4, and the height is given by $$h(d) = - \frac{1}{9} (d - 4)^2 + 4$$ which opens downward indicating the ball rises and then falls.