1. **Problem Statement:** We are given the quadratic function $$y = x^2 + 2x - 7$$ and need to find: - The range of values of $x$ for which the function is positive. - The range of values of $x$ for which the function is decreasing positively. - The range of values of $x$ for which the function is increasing negatively. 2. **Formula and Important Rules:** - The quadratic function is in the form $$y = ax^2 + bx + c$$ with $a=1$, $b=2$, and $c=-7$. - The vertex of the parabola is at $$x = -\frac{b}{2a} = -\frac{2}{2 \times 1} = -1$$. - The parabola opens upwards since $a=1 > 0$. - The function is decreasing on the interval $$(-\infty, -1)$$ and increasing on $$(-1, \infty)$$. - The roots (where $y=0$) are found by solving $$x^2 + 2x - 7 = 0$$. 3. **Find the roots:** Using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{4 + 28}}{2} = \frac{-2 \pm \sqrt{32}}{2} = \frac{-2 \pm 4\sqrt{2}}{2} = -1 \pm 2\sqrt{2}$$ So the roots are: $$x_1 = -1 - 2\sqrt{2} \approx -3.828$$ $$x_2 = -1 + 2\sqrt{2} \approx 1.828$$ 4. **Range of $x$ for which the function is positive:** Since the parabola opens upwards, $y > 0$ outside the roots: $$x < -3.828 \quad \text{or} \quad x > 1.828$$ 5. **Range of $x$ for which the function is decreasing positively:** The function is decreasing on $$(-\infty, -1)$$. "Decreasing positively" means the function values are positive and decreasing. Since the function is positive only for $$x < -3.828$$ or $$x > 1.828$$, the intersection with decreasing interval is: $$x < -3.828$$ 6. **Range of $x$ for which the function is increasing negatively:** The function is increasing on $$(-1, \infty)$$. "Increasing negatively" means the function values are negative and increasing. The function is negative between the roots: $$-3.828 < x < 1.828$$ The intersection with increasing interval is: $$-1 < x < 1.828$$ **Final answers:** - Function positive: $$x < -3.828 \quad \text{or} \quad x > 1.828$$ - Function decreasing positively: $$x < -3.828$$ - Function increasing negatively: $$-1 < x < 1.828$$