1. **Problem Statement:** We are given the quadratic function $$y = a(x + h)^2 + k$$ where $$a$$, $$h$$, and $$k$$ are constants. The graph is a parabola opening downwards with vertex at $$(-h, k)$$. We need to determine which of the statements I. $$h > 0$$, II. $$k > 0$$, and III. $$ah > 0$$ must be true. 2. **Recall the vertex form:** The vertex of the parabola is at $$(-h, k)$$. The parabola opens upwards if $$a > 0$$ and downwards if $$a < 0$$. 3. **Analyze the graph:** Since the parabola opens downwards, we know $$a < 0$$. 4. **Check statement I: $$h > 0$$** - The vertex is at $$x = -h$$. - The graph shows the vertex in the top-right quadrant (positive $$x$$-coordinate), so $$-h > 0$$ which implies $$h < 0$$. - Therefore, statement I is **false**. 5. **Check statement II: $$k > 0$$** - The vertex's $$y$$-coordinate is $$k$$. - The vertex is above the $$x$$-axis (top-right quadrant), so $$k > 0$$. - Statement II is **true**. 6. **Check statement III: $$ah > 0$$** - We know $$a < 0$$ and $$h < 0$$ from above. - The product $$ah$$ is $$a imes h$$, which is negative times negative, so $$ah > 0$$. - Statement III is **true**. 7. **Conclusion:** Statements II and III are true, so the correct answer is **C. II and III only**.