Suraj watched a hovercraft take off from a platform. The height of the hovercraft (in meters above the ground) \( t \) minutes after takeoff is modeled by \( h(t)=-4 t^{2}+8 t+32 \) Suraj wants to know when the hovercraft will reach its highest point. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. \( h(t)= \)

To find the vertex form of the quadratic equation \( h(t) = -4t^2 + 8t + 32 \), we first complete the square. Starting with the equation, we factor out the \(-4\) from the first two terms: \[ h(t) = -4(t^2 - 2t) + 32 \] Next, we complete the square inside the parentheses. To do this, we take half of the coefficient of \( t \) (which is \(-2\)), square it (getting \( 1 \)), and add and subtract it inside the parentheses: \[ h(t) = -4 \left( t^2 - 2t + 1 - 1 \right) + 32 \] \[ h(t) = -4 \left( (t - 1)^2 - 1 \right) + 32 \] \[ h(t) = -4(t - 1)^2 + 4 + 32 \] \[ h(t) = -4(t - 1)^2 + 36 \] So, the rewritten function in vertex form is \( h(t) = -4(t - 1)^2 + 36 \), indicating that the highest point occurs at \( t = 1 \) minute with a maximum height of \( 36 \) meters. --- Now, Suraj can see that the hovercraft reaches its highest point at \( t = 1 \) minute and will be \( h(1) = 36 \) meters above the ground!