Answer:
y = (-1/15)(x - 3)2 + 15
Step-by-step explanation:
The general equation of a parabola is
y = a(x - h)2 + k
If the halfway line if the origin, i.e. (x, y) = (0, 0), then the ball is kicked at (-12, 0) and it lands at (18, 0). By symmetry, the ball reaches its greatest height halfway between the starting and ending x values, or at x = 3. So, we know three points on this parabola: (-12, 0), (18, 0) and (3, 15).
Let us plug the first point into the generalized equation:
y = (x - h)2 + k
0 = (-12 - h)2 + k
So, k = -(-h - 12)2
or
k = -(h + 12)2, since we can factor out -1 and it becomes squared.
We can also plug (18, 0) into the generalized equation, and get:
0 = (18 - h)2 + k
So, k = -(18 - h)2
Since we have two expressions both equal to k, we can set them equal to each other and eliminate k:
-(h + 12)2 = -(18 - h)2
We can cancel out the negative signs and square out the terms on each side of the last equation.
h2 + 24h + 144 = h2 - 36h + 324
We can cancel out h2 and then solve for h.
24h + 144 = -36h + 324
60h = 180
h = 3
Therefore, the generalized equation becomes
y = a(x - 3)2 + k
We know that if x = 3, y = 15 from out third point.
15 = a(3 - 3)2 + k
So, k = 15
Thus, the equation becomes
y = a(x - 3)2 + 15
We can then pick one of the other points, plug in the corresponding x and y and then solve for a. I choose (18, 0).
0 = a(18 - 3)2 + 15
0 = 152a + 15
Divide this equation through by 15.
0 = 15a + 1
or
a = -1/15
Therefore, the generalized equation is
y = (-1/15)(x - 3)2 + 15
