The height of h (in feet) of a baseball t seconds after it is hit is given by the function y = −16t 2 + 96t + 3. Find the maximum height of the baseball.

Question

Hint- convert to vertex form

Answer 1

Hi Hannah!

The first thing to recognize here is that the graph of your function is a parabola. You can tell it opens downward, as the coefficient in-front for the t^2 term is negative. That means our parabola's vertex will be the maximum y-value (vs the vertex being the minimum y-value if the parabola opened up). This is what we are trying to solve - the maximum height of the ball.

For a parabola, y = ax^2 + bx + c, the vertex at point (h,k), is first found by the equation h = -b / 2a, and then plugging that point into the equation to find k.

So let's solve for h in this problem. Given the equation y = -16t^2 + 96t + 3, we know a = -16, b = 96, c = 3.

h = -b/2a

h = -(96) / 2(-16)

h = -96 / -32

h = 3 seconds

This means the the ball will reach it's maximum height after 3 seconds. But the problem is asking is for what that height is. So for our final step, let's plug in t=3 to our equation:

y = -16t^2 + 96t + 3

y = -16*(3)^2 + 96*(3)+ 3

y = -144 + 288 + 3

y = 147 feet

We have now found the vertex is (3,147), which means that the final answer is that the maximum height of the ball is 147 feet.

I hope this helps - let me know if you have any questions!

Wishing you the best,

Stephanie