1. The problem is to match each quadratic function to its respective graph based on the vertex and shape. 2. Recall the general form of a quadratic function: $$f(x) = ax^2 + bx + c$$ where: - If $a > 0$, the parabola opens upward. - If $a < 0$, the parabola opens downward. - The vertex form is useful: $$f(x) = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. 3. Analyze each function: - $f(x) = -x^2 - 5$: $a = -1 < 0$, opens downward, vertex at $(0, -5)$. - $g(x) = 2x^2 + 2$: $a = 2 > 0$, opens upward, vertex at $(0, 2)$. - $h(x) = -x^2 - 2x = -x(x + 2)$: $a = -1 < 0$, opens downward. Complete the square to find vertex: $$h(x) = -\left(x^2 + 2x\right) = -\left(x^2 + 2x + 1 - 1\right) = -\left((x + 1)^2 - 1\right) = -(x + 1)^2 + 1$$ Vertex at $(-1, 1)$. - $k(x) = -x^2 + 16x - 64 = -(x - 8)^2$: $a = -1 < 0$, opens downward, vertex at $(8, 0)$. 4. Match the graphs: - Bottom-left graph: upward-opening parabola with vertex at $(0, 2)$ matches $g(x) = 2x^2 + 2$. - Bottom-right graph: downward-opening parabola with vertex at $(0, -5)$ matches $f(x) = -x^2 - 5$. - Left graph (center-right area) downward-opening parabola with vertex at $(8, 0)$ matches $k(x) = -(x - 8)^2$. - Right graph (center area) downward-opening parabola with vertex at $(-1, 1)$ matches $h(x) = -(x + 1)^2 + 1$. Final matches: - $f(x)$ bottom-right - $g(x)$ bottom-left - $h(x)$ right graph - $k(x)$ left graph