Determine equation

Question

A soccer ball is kicked from a point 12 m to the left of the halfway line and lands at a point 18 m to the right of the halfway line. It reaches a maximum height of 15 m during i’d parabola flight. determine an equation to represent the path of the soccer ball if the halfway line passes through the origin.

Answer 1

Hi Grace,

 

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

Answer 2

The path is a concave-down parabola with a maximum value of 15, and zeros at -12 and 18.

 

y = a(x - h)² + k

 

k = 15 (given)

 

h is at the midpoint of the zeros, so

 

h = (-12 + 18)/2 = 3

 

y = a(x - 3)² + 15

 

At x = 18, y = 0

 

0 = a(18 - 3)² + 15

 

a = -15/15² = -1/15

 

y = (-1/15)(x - 3)² + 15

 

Test the answer.