1. **State the problem:** We need to match each quadratic function to its corresponding graph based on vertex and intercept information. 2. **Recall the quadratic form and vertex:** A quadratic function is generally $f(x) = ax^2 + bx + c$. - The vertex is at $x = -\frac{b}{2a}$. - The roots (x-intercepts) come from factoring or using the quadratic formula. 3. **Analyze each function:** - $f(x) = x^2 + 8x + 12 = (x + 6)(x + 2)$: Opens upward ($a=1>0$), roots at $x=-6$ and $x=-2$, vertex at $x = -\frac{8}{2} = -4$. - $g(x) = -x^2 - 17x - 72 = -(x + 9)(x + 8)$: Opens downward ($a=-1<0$), roots at $x=-9$ and $x=-8$, vertex at $x = -\frac{-17}{2(-1)} = -\frac{-17}{-2} = 8.5$ (check carefully: $x = -\frac{b}{2a} = -\frac{-17}{2(-1)} = -\frac{-17}{-2} = -8.5$). - $h(x) = -x^2 - 3$: Opens downward, vertex at $x=0$ (since no $x$ term), vertex point $(0,-3)$. - $k(x) = -2x^2 - 28x - 98 = -2(x + 7)^2$: Opens downward, vertex at $x=-7$, vertex value $k(-7) = -2(0)^2 = 0$. 4. **Match to graphs:** - Top-left: Upward parabola, vertex $(-4,-4)$, roots $(-6,0)$ and $(-2,0)$ matches $f(x)$. - Top-right: Downward parabola, vertex $(-7,0)$, roots near $(-8,0)$ and $(-6,0)$ matches $k(x)$. - Bottom-left: Downward parabola, vertex $(-8,0)$, roots near $(-9,0)$ and $(-7,0)$ matches $g(x)$. - Bottom-right: Downward parabola, vertex $(0,-3)$ matches $h(x)$. **Final matching:** - Top-left: $f(x) = x^2 + 8x + 12$ - Top-right: $k(x) = -2x^2 - 28x - 98$ - Bottom-left: $g(x) = -x^2 - 17x - 72$ - Bottom-right: $h(x) = -x^2 - 3$