1. **State the problem:** We need to determine which quadratic equation best matches the given graph. 2. **Recall the general form of a quadratic equation:** $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. 3. **Analyze the graph features:** - The parabola opens downward, so $a < 0$. - The vertex is near $(-3, 6)$. - The parabola crosses the x-axis near $x = -1$ and $x = -5$. 4. **Check each option:** - Option 1: $y = x^2 - 6x - 4$ - Here, $a = 1 > 0$, so parabola opens upward. This contradicts the graph. - Option 2: $y = -x^2 - 6x - 4$ - $a = -1 < 0$, parabola opens downward. - Find vertex: $$x = -\frac{b}{2a} = -\frac{-6}{2(-1)} = -\frac{-6}{-2} = -3$$ - Calculate $y$ at vertex: $$y = -(-3)^2 - 6(-3) - 4 = -9 + 18 - 4 = 5$$ - Vertex is $(-3, 5)$, close to graph's $(-3, 6)$. - Find roots by solving $-x^2 - 6x - 4 = 0$: Multiply both sides by $-1$: $$\cancel{-}x^2 - 6x - 4 = 0 \Rightarrow x^2 + 6x + 4 = 0$$ Use quadratic formula: $$x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 4}}{2} = \frac{-6 \pm \sqrt{36 - 16}}{2} = \frac{-6 \pm \sqrt{20}}{2} = \frac{-6 \pm 2\sqrt{5}}{2} = -3 \pm \sqrt{5}$$ Approximate roots: $-3 + 2.236 = -0.764$, $-3 - 2.236 = -5.236$, close to $-1$ and $-5$. - Option 3: $y = -x^2 - 6x + 5$ - Vertex $x = -\frac{-6}{2(-1)} = -3$ - $y$ at vertex: $$y = -(-3)^2 - 6(-3) + 5 = -9 + 18 + 5 = 14$$ - Vertex is $(-3, 14)$, too high compared to graph. - Option 4: $y = x^2 - 6x + 5$ - $a = 1 > 0$, parabola opens upward, contradicts graph. 5. **Conclusion:** Option 2 best matches the graph. **Final answer:** $$y = -x^2 - 6x - 4$$