To evaluate the height of a falling object using a polynomial function, substitute the given time into the equation. For example, the function $h(t) = -16t^2 + 175$ models the height in feet of a ball $t$ seconds after being dropped from a $175$-foot bridge. To find the ball's height after $3$ seconds, evaluate the function at $t = 3$. Substitute $3$ for $t$ to obtain $h(3) = -16(3)^2 + 175$. First, square the $3$ to get $9$. Next, multiply $-16$ by $9$ to get $-144$. Finally, add $-144$ and $175$ to arrive at $31$. Therefore, the height of the ball after $3$ seconds is $31$ feet.