Find Maximum of a Quadratic Function

Example on how to find the maximum of a quadratic function.

Example 1


A ball thrown from an initial height of $4$ feet at an initial velocity of $32$ feet per second has its height, as a function of time, given by the equation $h=−16 t^2+32 t+4$ , where $h$ is the height of the ball in feet and $t$ is the time in seconds after the ball was thrown. After how many seconds will the ball reach its maximum height?

  1. $1$
  2. $2$
  3. $3$
  4. $4$
  5. $5$

Solution


  1. If the height $h$ given above is plotted against time $t$, we obtain the graph of a quadratic function with a maximum point (vertex) at which the height is maximum. Since the height $h$ has the form $h(t)=a t^2+b t+c$, the time $T$, in seconds, at which the height $h$ is maximum is the $t$-coordinate of the vertex and is given by $-\dfrac{b}{2 a}$. Hence
    $T=-\dfrac{32}{2\cdot (-16)} = \dfrac{-32}{-32}=1$

    Answer A