1. **Problem statement:** Find the equation of a parabola in standard form that passes through points $(-5, 2)$ and $(-2, 2)$ and has a maximum value of 20. 2. **Recall the standard form of a parabola:** $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. 3. Since the parabola has a maximum value of 20, the vertex is at $(h, k) = (h, 20)$ and the parabola opens downward, so $a < 0$. 4. The vertex lies midway between the points $(-5, 2)$ and $(-2, 2)$ because these points have the same $y$-value and are symmetric about the vertex. Calculate the midpoint: $$h = \frac{-5 + (-2)}{2} = \frac{-7}{2} = -3.5$$ So the vertex is at $(-3.5, 20)$. 5. Substitute the vertex into the standard form: $$y = a(x + 3.5)^2 + 20$$ 6. Use one of the given points to find $a$. Using $(-5, 2)$: $$2 = a(-5 + 3.5)^2 + 20$$ $$2 = a(-1.5)^2 + 20$$ $$2 = a(2.25) + 20$$ 7. Solve for $a$: $$2 - 20 = 2.25a$$ $$-18 = 2.25a$$ $$a = \frac{-18}{2.25} = -8$$ 8. Write the equation: $$y = -8(x + 3.5)^2 + 20$$ 9. Expand to standard form $y = ax^2 + bx + c$: $$y = -8(x^2 + 7x + 12.25) + 20$$ $$y = -8x^2 - 56x - 98 + 20$$ $$y = -8x^2 - 56x - 78$$ 10. **Domain and range:** - Domain: All real numbers, $$(-\infty, \infty)$$ - Range: Since the parabola opens downward with maximum 20, range is $$(-\infty, 20]$$ **Final answer:** $$y = -8x^2 - 56x - 78$$ Domain: $$(-\infty, \infty)$$ Range: $$(-\infty, 20]$$