1. **State the problem:** We are given the quadratic function $f(x) = -2x^2 - 12x - 7$ and need to analyze its properties. 2. **Formula and rules:** A quadratic function is generally written as $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. - The parabola opens downward if $a < 0$ and upward if $a > 0$. - The vertex (maximum or minimum point) is at $x = -\frac{b}{2a}$. - The axis of symmetry is the vertical line $x = -\frac{b}{2a}$. - The y-intercept is $f(0) = c$. 3. **Find the vertex:** $$x = -\frac{b}{2a} = -\frac{-12}{2 \times -2} = -\frac{-12}{-4} = -3$$ 4. **Find the y-coordinate of the vertex:** $$f(-3) = -2(-3)^2 - 12(-3) - 7 = -2(9) + 36 - 7 = -18 + 36 - 7 = 11$$ 5. **Vertex:** The vertex is at $(-3, 11)$, which is the maximum point since $a = -2 < 0$. 6. **Find the y-intercept:** $$f(0) = -7$$ 7. **Find the x-intercepts:** Solve $-2x^2 - 12x - 7 = 0$. Divide both sides by $-1$ to simplify: $$\cancel{-}2x^2 - 12x - 7 = 0 \Rightarrow 2x^2 + 12x + 7 = 0$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-12 \pm \sqrt{12^2 - 4 \times 2 \times 7}}{2 \times 2} = \frac{-12 \pm \sqrt{144 - 56}}{4} = \frac{-12 \pm \sqrt{88}}{4}$$ Simplify $\sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22}$: $$x = \frac{-12 \pm 2\sqrt{22}}{4} = \frac{\cancel{2}(-6 \pm \sqrt{22})}{\cancel{2}2} = \frac{-6 \pm \sqrt{22}}{2}$$ 8. **Final x-intercepts:** $$x = \frac{-6 + \sqrt{22}}{2} \quad \text{and} \quad x = \frac{-6 - \sqrt{22}}{2}$$ **Summary:** - The parabola opens downward. - Vertex at $(-3, 11)$ (maximum point). - Y-intercept at $(0, -7)$. - X-intercepts at $\left(\frac{-6 + \sqrt{22}}{2}, 0\right)$ and $\left(\frac{-6 - \sqrt{22}}{2}, 0\right)$. This completes the analysis of the quadratic function $f(x) = -2x^2 - 12x - 7$.