Aball is thrown vertically upward. after t seconds, its height h (in feet) is given by the function h(t)=52t-16t^2 . what is the maximum height that the ball will reach? do not round your answer.
height:

A quick way to do this problem is to use the vertex formula as if you were trying to graph the parabolic function. 
We know that in this case the vertex is at a maximum height because the leading coefficient is negative. If it were positive, then the vertex would be at the lowest point, in other words, they would ask you to solve for the lowest point. In this case we are solving for the highest point. 
Primarily, we can use the axis of symmetry formula to solve for the x-coordinate of the vertex which demonstrates the time the object takes to reach its apex. Once we find the t or the x-coordinate, we can plug it back into the equation to solve for the h or height. 
Axis of symmetry formula: -b/ 2a

(b) is 52 in this case. (a) is -16 in this case. Just plug the numbers in but be mindful of the signs included in the formula. 
-(52)/ 2(-16) = -52/-32
The quotient would be a terminating decimal: 1.625

Therefore, the time it takes for the ball to reach its highest point would be 1.625 seconds. Now you can plug this number back in as a substitution for the variable t and you can solve for h. 

h = -16(1.625)^2 + 52(1.625)
h = -16(2.640625) + 84.5
h= -42.25 + 84.5
h = 42.25 
Therefore, the maximum height that the ball will reach in 1.625 seconds is 42.25 feet. 

We find the vertex

a hack is that if we have h(t)=at²+bt+c, then the t value of the vertex is -b/(2a)
so

given
h(t)=-16t²+52t+0

-b/2a=-52/(2*-16)=-52/-32=26/16=13/8

sub back
h(13/8)=-16(13/8)²+52(13/8)
h(13/8)=-16(169/64)+(676/8)
h(13/8)=(-2704/64)+(169/2)
h(13/8)=-(169/4)+(169/2)
h(13/8)=-169/4+338/4
h(13/8)=169/4

max heigh is 169/4ft or aprox 42.25ft