1. **Problem statement:** Given the function $f(x) = -x^2 - 2x + 6$, find the vertex, intercepts with coordinate axes, domain, range, and describe the graph including its growth direction. 2. **Formula and concepts:** This is a quadratic function of the form $f(x) = ax^2 + bx + c$ with $a = -1$, $b = -2$, and $c = 6$. - The vertex of a parabola is at $x = -\frac{b}{2a}$. - The parabola opens upwards if $a > 0$ and downwards if $a < 0$. - The domain of any quadratic function is all real numbers $(-\infty, \infty)$. - The range depends on the vertex and the direction of the parabola. 3. **Find the vertex:** $$x_v = -\frac{b}{2a} = -\frac{-2}{2 \times -1} = -\frac{-2}{-2} = -1$$ Calculate $f(x_v)$: $$f(-1) = -(-1)^2 - 2(-1) + 6 = -1 + 2 + 6 = 7$$ So the vertex is at $(-1, 7)$. 4. **Find the intercepts:** - **Y-intercept:** Set $x=0$: $$f(0) = -0 - 0 + 6 = 6$$ So the y-intercept is $(0, 6)$. - **X-intercepts:** Set $f(x) = 0$: $$-x^2 - 2x + 6 = 0$$ Multiply both sides by $-1$ to simplify: $$\cancel{-}x^2 \cancel{-} 2x + \cancel{6} = \cancel{0}$$ $$x^2 + 2x - 6 = 0$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-6)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 24}}{2} = \frac{-2 \pm \sqrt{28}}{2}$$ Simplify $\sqrt{28} = 2\sqrt{7}$: $$x = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7}$$ So the x-intercepts are: $$\left(-1 - \sqrt{7}, 0\right) \quad \text{and} \quad \left(-1 + \sqrt{7}, 0\right)$$ 5. **Domain:** All real numbers, $(-\infty, \infty)$. 6. **Range:** Since $a = -1 < 0$, the parabola opens downward, so the vertex is a maximum point. Range is: $$(-\infty, 7]$$ 7. **Graph behavior:** The parabola opens downward, so it grows (increases) to the left of the vertex and decreases to the right of the vertex. --- **Final answer:** - Vertex: $(-1, 7)$ - X-intercepts: $\left(-1 - \sqrt{7}, 0\right)$ and $\left(-1 + \sqrt{7}, 0\right)$ - Y-intercept: $(0, 6)$ - Domain: $(-\infty, \infty)$ - Range: $(-\infty, 7]$ - The parabola opens downward, increasing on $(-\infty, -1)$ and decreasing on $(-1, \infty)$.