8 Pr ectile Motion. A soccer ball is kic ed from the ground. The height, \( h \), of the ball above the ground time \( t \) seconds is given by \( h=8 t-4 t^{2} \). (i) Express the function in the form \( h=a(t-p)(t-q) \).

To express the height function \( h = 8t - 4t^2 \) in the form \( h = a(t - p)(t - q) \), follow these steps: 1. **Rearrange the Equation:** Begin by rewriting the equation in standard quadratic form: \[ h = -4t^2 + 8t \] 2. **Factor Out the Common Term:** Factor out the coefficient of \( t^2 \), which is \(-4\): \[ h = -4(t^2 - 2t) \] 3. **Factor the Quadratic Expression:** Now, factor the quadratic expression inside the parentheses: \[ t^2 - 2t = t(t - 2) \] 4. **Combine the Factors:** Substitute the factored form back into the equation: \[ h = -4(t)(t - 2) \] 5. **Final Form:** This gives the function in the desired form: \[ h = -4(t - 0)(t - 2) \] Simplifying further, it can be written as: \[ h = -4t(t - 2) \] **Final Answer:** \[ h = -4\,t\,\left(t\ -\ 2\,\right) \]