1. **State the problem:** We are given the height function of a basketball as a function of time: $$h(t) = -4.75t^2 + 8.75t + 1.5$$ for $$t \geq 0$$. We want to analyze this quadratic function to understand the basketball's height over time. 2. **Identify the type of function:** This is a quadratic function of the form $$h(t) = at^2 + bt + c$$ where $$a = -4.75$$, $$b = 8.75$$, and $$c = 1.5$$. Since $$a < 0$$, the parabola opens downward, meaning the basketball reaches a maximum height at the vertex. 3. **Find the vertex (maximum height):** The time at which the maximum height occurs is given by the vertex formula: $$t = -\frac{b}{2a} = -\frac{8.75}{2 \times -4.75} = -\frac{8.75}{-9.5} = \frac{8.75}{9.5}$$ 4. **Simplify the fraction:** $$\frac{8.75}{9.5} = \frac{\cancel{8.75}}{\cancel{9.5}} = 0.9211$$ (approximate decimal) 5. **Calculate the maximum height:** Substitute $$t = 0.9211$$ into the height function: $$h(0.9211) = -4.75(0.9211)^2 + 8.75(0.9211) + 1.5$$ 6. **Calculate each term:** $$-4.75 \times 0.8484 = -4.0309$$ $$8.75 \times 0.9211 = 8.0596$$ 7. **Sum all terms:** $$h(0.9211) = -4.0309 + 8.0596 + 1.5 = 5.5287$$ meters (approximate) 8. **Find the intercepts:** - At $$t=0$$, $$h(0) = 1.5$$ meters (initial height). - To find when the basketball hits the ground, solve $$h(t) = 0$$: $$-4.75t^2 + 8.75t + 1.5 = 0$$ 9. **Use quadratic formula:** $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8.75 \pm \sqrt{8.75^2 - 4(-4.75)(1.5)}}{2(-4.75)}$$ 10. **Calculate discriminant:** $$8.75^2 = 76.5625$$ $$4 \times -4.75 \times 1.5 = -28.5$$ $$\sqrt{76.5625 - (-28.5)} = \sqrt{105.0625} = 10.25$$ 11. **Calculate roots:** $$t = \frac{-8.75 \pm 10.25}{-9.5}$$ 12. **Calculate each root:** $$t_1 = \frac{-8.75 + 10.25}{-9.5} = \frac{1.5}{-9.5} = -0.1579$$ (discard since $$t \geq 0$$) $$t_2 = \frac{-8.75 - 10.25}{-9.5} = \frac{-19}{-9.5} = 2$$ seconds 13. **Interpretation:** The basketball starts at 1.5 meters, reaches a maximum height of approximately 5.53 meters at about 0.92 seconds, and hits the ground at 2 seconds. **Final answer:** - Maximum height: $$5.53$$ meters at $$t = 0.92$$ seconds - Hits ground at $$t = 2$$ seconds