1. **State the problem:** We are given the quadratic function $$f(x) = -x^2 + 20x - 96$$ which models the height of a soccer ball in meters as a function of horizontal distance $x$ in meters. We want to find the maximum height of the ball and the distance it traveled through the air. 2. **Formula and rules:** The quadratic function is in the form $$ax^2 + bx + c$$ with $$a = -1$$, $$b = 20$$, and $$c = -96$$. Since $$a < 0$$, the parabola opens downward, so the vertex represents the maximum point. The vertex $x$-coordinate is given by $$x = -\frac{b}{2a}$$. 3. **Find the vertex:** $$x = -\frac{20}{2 \times -1} = -\frac{20}{-2} = 10$$ 4. **Find the maximum height by evaluating $$f(10)$$:** $$f(10) = -(10)^2 + 20 \times 10 - 96 = -100 + 200 - 96 = 4$$ So, the maximum height is 4 meters. 5. **Find the x-intercepts (roots) to find the distance traveled:** Solve $$-x^2 + 20x - 96 = 0$$. Multiply both sides by $$-1$$ to simplify: $$\cancel{-}x^2 + 20x - 96 = 0 \Rightarrow x^2 - 20x + 96 = 0$$ 6. **Factor the quadratic:** $$x^2 - 20x + 96 = (x - 12)(x - 8) = 0$$ 7. **Find roots:** $$x - 12 = 0 \Rightarrow x = 12$$ $$x - 8 = 0 \Rightarrow x = 8$$ 8. **Calculate the distance traveled:** Distance = $$12 - 8 = 4$$ meters. **Final answers:** - Maximum height of the soccer ball is $$4$$ meters. - The soccer ball traveled $$4$$ meters through the air.