They want you find a quadratic equation in vertex form

    h(t) = .    (1)


It is the general quadratic equation in vertex form.



Now, the problem says that the ball reaches its maximum height after 2 seconds.

It means that the vertex is at  t= ;  in other words, we should take  = 2 (seconds).

So, I write  

    h(t) = .    (2)



Next, the problem says that After 1 second the height of the ball is 15 feet off the ground.

It means that I can write the last equation (2) in the form

    15 = ,    (3)

substituting 15 ft on the left side and replacing "t" by 1 second in the right side.

It gives me 

    15 = a + b.    (4)



Finally, the problem says that the ball is kicked off the ground.

It means that at t= 0 the height is zero: h = 0.

So, I substitute t= 0 and h= 0 into equation (2), and I get then

    0 = ,

or

    0 = 4a + b.    (5)


From (5), I have  b = -4a and substitute it into equation (4).  I get then

    15 = a + (-4a) = -3a;

hence

    a = ,  or  a = -5.


Now I am ready to write the final equation from given data

    h(t) = .     <<<---=== 


      +---------------------------------------------+
      |    But now it is good time to discuss it.   |
      +---------------------------------------------+


First, having this equation in front of you, you can check that it SATISFIES the problem's conditions.


Second, it is good to re-write this equation in the standard form quadratic equation by opening parentheses

    h(t) = -5*t^2 + 10t + 0.    (6)


The zero (0) as a constant term is good, because it tells that the initial height is zero at t= 0.


Next, from the standard form (6) everybody, who is familiar with the subject, will say

    "HEY, they missed everything in the problem: the leading coefficient a= -5 says that the problem 
    should consider METERS instead of FEET.  For the Earth condition, a= -5 tells that the input numbers
    in the condition relate to METERS, not feet".


At this point, I complete my solution and my explanations.


From my post, you learn TWO THINGS:

    - first, you learn on HOW TO analyze and solve the problem, step-by-step.

    - second, you learn that the problem's formulation, as it is given in the post,
      is INCOORRECT and the word "feet" MUST BE replaced by the word "METERS" there.