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 function \( h = 8t - 4t^2 \) in the form \( h = a(t - p)(t - q) \), we’ll first factor the quadratic equation. Start by rewriting the equation as: \[ h = -4t^2 + 8t. \] Next, we can factor out \(-4\): \[ h = -4(t^2 - 2t). \] Now, we can complete the square for the quadratic \( t^2 - 2t \): \[ t^2 - 2t = (t - 1)^2 - 1. \] Substituting back gives: \[ h = -4((t - 1)^2 - 1) = -4(t - 1)^2 + 4. \] Thus, we have the function in vertex form but we need it in the product form. From \( h = -4(t - 1)^2 + 4 \), we can identify the roots as \( t = 0 \) and \( t = 2 \). Therefore, we can write: \[ h = -4(t - 0)(t - 2). \] Now it’s clear in the desired form: \[ h = -4(t)(t - 2). \] In this case, \( a = -4 \), \( p = 0 \), and \( q = 2 \).