1. **State the problem:** Match each quadratic function to its corresponding graph. 2. **Given functions:** - $f(x) = -2x^2 - 16x - 32 = -2(x + 4)^2$ - $g(x) = -3x^2 + 42x - 144 = -3(x - 6)(x - 8)$ 3. **Analyze $f(x)$:** - The vertex form is $-2(x + 4)^2$ which means the vertex is at $(-4, 0)$. - The parabola opens downward because the coefficient $-2$ is negative. - This matches the left graph description: downward-opening parabola with vertex at $(-4, 0)$. 4. **Analyze $g(x)$:** - The factored form is $-3(x - 6)(x - 8)$, so the roots (x-intercepts) are at $x=6$ and $x=8$. - The parabola opens downward because the coefficient $-3$ is negative. - The vertex is at the midpoint of the roots: $x = \frac{6 + 8}{2} = 7$. - Substitute $x=7$ into $g(x)$ to find the vertex y-coordinate: $$g(7) = -3(7 - 6)(7 - 8) = -3(1)(-1) = 3$$ - So vertex is at $(7, 3)$. - This matches the right graph description: downward-opening parabola with x-intercepts at 6 and 8 and vertex at $(7, 3)$. 5. **Conclusion:** - $f(x)$ corresponds to the left graph. - $g(x)$ corresponds to the right graph.