1. **State the problem:** Match each quadratic function to its graph based on their properties. 2. **Given functions:** - $f(x) = 2x^2 - 12x + 16 = 2(x - 2)(x - 4)$ - $g(x) = -2x^2 - 4x - 2 = -2(x + 1)^2$ 3. **Analyze $g(x)$:** - The vertex form is $g(x) = -2(x + 1)^2$. - Vertex is at $(-1, 0)$. - Since the coefficient of $x^2$ is negative ($-2$), the parabola opens downward. 4. **Analyze $f(x)$:** - The factored form is $f(x) = 2(x - 2)(x - 4)$. - The vertex can be found by averaging the roots: $x = \frac{2 + 4}{2} = 3$. - Substitute $x=3$ into $f(x)$: $$f(3) = 2(3)^2 - 12(3) + 16 = 18 - 36 + 16 = -2$$ - So vertex is at $(3, -2)$. - Since the coefficient of $x^2$ is positive ($2$), the parabola opens upward. 5. **Match to graphs:** - Left graph: downward-opening parabola with vertex at $(-1, 0)$ matches $g(x)$. - Right graph: upward-opening parabola with vertex at $(3, -2)$ matches $f(x)$. **Final answer:** - Left graph corresponds to $g(x) = -2(x + 1)^2$. - Right graph corresponds to $f(x) = 2(x - 2)(x - 4)$.