1. The problem is to match each quadratic equation to its corresponding graph. 2. The general form of a quadratic equation is $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. 3. Important rules: - The coefficient $a$ determines the parabola's direction (up if $a>0$). - The vertex's $x$-coordinate is given by $$x = -\frac{b}{2a}$$. - The $y$-intercept is the constant term $c$. 4. Analyze each equation: - For $$y = x^2 - 3$$: - $a=1$, $b=0$, $c=-3$ - Vertex at $$x = -\frac{0}{2 \times 1} = 0$$ - Vertex point is $(0, -3)$ - Opens upward, shifted down by 3 units - For $$y = x^2 + 3x - 6$$: - $a=1$, $b=3$, $c=-6$ - Vertex at $$x = -\frac{3}{2 \times 1} = -\frac{3}{2} = -1.5$$ - Calculate vertex $y$: $$y = (-1.5)^2 + 3(-1.5) - 6 = 2.25 - 4.5 - 6 = -8.25$$ - Vertex at $(-1.5, -8.25)$ - For $$y = x^2 + 4x + 5$$: - $a=1$, $b=4$, $c=5$ - Vertex at $$x = -\frac{4}{2 \times 1} = -2$$ - Calculate vertex $y$: $$y = (-2)^2 + 4(-2) + 5 = 4 - 8 + 5 = 1$$ - Vertex at $(-2, 1)$ 5. Match to graphs: - Left graph: $y = x^2 - 3$ (vertex at $(0,-3)$, opens upward) - Middle graph: $y = x^2 + 3x - 6$ (vertex at $(-1.5,-8.25)$, opens upward) - Right graph: $y = x^2 + 4x + 5$ (vertex at $(-2,1)$, opens upward) Final answer: - Left graph matches $$y = x^2 - 3$$ - Middle graph matches $$y = x^2 + 3x - 6$$ - Right graph matches $$y = x^2 + 4x + 5$$