1. **State the problem:** Match each quadratic equation with its corresponding graph based on vertex and shape. 2. **Recall the vertex formula:** For a quadratic $y = ax^2 + bx + c$, the vertex $x$-coordinate is given by $$x = -\frac{b}{2a}$$ 3. **Analyze each equation:** - For $y = x^2 - x - 2$: - $a=1$, $b=-1$, $c=-2$ - Vertex $x = -\frac{-1}{2 \times 1} = \frac{1}{2} = 0.5$ - Vertex $y = (0.5)^2 - 0.5 - 2 = 0.25 - 0.5 - 2 = -2.25$ - For $y = x^2 + x - 2$: - $a=1$, $b=1$, $c=-2$ - Vertex $x = -\frac{1}{2 \times 1} = -0.5$ - Vertex $y = (-0.5)^2 + (-0.5) - 2 = 0.25 - 0.5 - 2 = -2.25$ - For $y = x^2 - 2x + 1$: - $a=1$, $b=-2$, $c=1$ - Vertex $x = -\frac{-2}{2 \times 1} = 1$ - Vertex $y = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0$ 4. **Match with graphs:** - Top-left graph: vertex around $x=0.5$, $y \approx -2$ matches $y = x^2 - x - 2$ - Top-right graph: vertex around $x=1$, $y \approx -1$ does not exactly match any vertex calculated but closest is $y = x^2 - 2x + 1$ with vertex at $(1,0)$ - Center graph: vertex at $x=1$, $y=0$ matches $y = x^2 - 2x + 1$ Since the top-right graph vertex is about $-1$ and the vertex of $y = x^2 + x - 2$ is at $x=-0.5$, $y=-2.25$, the top-right graph corresponds to $y = x^2 + x - 2$. **Final matching:** - Top-left graph: $y = x^2 - x - 2$ - Top-right graph: $y = x^2 + x - 2$ - Center graph: $y = x^2 - 2x + 1$