1. The problem asks to write an equation for the function graphed above, which is a parabola. 2. The general form of a parabola's equation is $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex and $a$ determines the direction and width. 3. From the description, the top-left graph is a downward-opening parabola with vertex near $(-4,8)$. 4. Since it opens downward, $a < 0$. We can use a point on the parabola to find $a$. The point $(-6,8)$ lies on it. 5. Substitute vertex and point into the formula: $$8 = a(-6 + 4)^2 + 8$$ 6. Simplify inside the square: $$8 = a(-2)^2 + 8$$ $$8 = 4a + 8$$ 7. Subtract 8 from both sides: $$8 - 8 = 4a + 8 - 8$$ $$0 = 4a$$ 8. Divide both sides by 4: $$\cancel{\frac{0}{4}} = \cancel{\frac{4a}{4}}$$ $$0 = a$$ 9. This means $a=0$, which would not form a parabola but a horizontal line. This suggests the point $(-6,8)$ is on the vertex line, so the parabola is flat at the vertex height. 10. Since the vertex is at $(-4,8)$ and the parabola opens downward, the simplest equation is: $$y = -a(x + 4)^2 + 8$$ 11. Without more points, we can only write the general form with $a > 0$: $$y = -a(x + 4)^2 + 8$$ 12. This represents the parabola with vertex at $(-4,8)$ opening downward. Final answer: $$y = -a(x + 4)^2 + 8$$ where $a > 0$.