Finding the vertex of a prabola

Question

We saw how the function h(t)=-16t^2+64t+80, which gives the height in feet of a projectile t seconds after it is launched. Find the vertex of its parabola, which gives the maximum height of the projectile, and use that to identify the range.

Answer 1

First to complete the square factor -16 out of the first two terms

h(t)=-16(t²-4t ) +80

Divide the -4 by 2

h(t) = -16(t-2)² +144

it is 144 because we square the -2 which is 4 then multiply that by -16. That is -64 we had to add to complete the square. To balance the equation, we would need to subtract that same amount from 80.

80-(-64)=144

Now we can see the vertex is at (2,144)

The function begins at 80 which is the y-intercept, reaches its highest point at 144, then will hit the ground y=0. So the range is from its lowest point of 0 to its highest point of 144 denotes by [0,144]