# A soccer player kicks a ball from the ground to a maximum height of 12 m. The high point in the trajectory of the ball occurs at a distance of 18 m from the kicker. On the downward path, another player heads the ball at a height of 2.2 m from the ground.

Write a quadratic function that models the situation.

my ans -
y = a (x+2.2)2 + 12
y = a (0 +2.2)2 + 12
12 = a(2.2)2
a = 12/(2.2)2
a= 2.47..

## y(0) = 0 and the vertex is at x=18 (halfway to the other root) and the parabola opens downward, so

y = ax(36-x)

the vertex is at (18,12) so

18a*(18) = 12
x = 12/18^2 = 2/3*18 = 2/54

y = 2/54 x(36-x)

## To derive the quadratic function that models the situation, we can use the general form of a quadratic equation, which is y = ax^2 + bx + c.

Given the information in the problem, we know that the maximum height of the ball is 12 m, which occurs at a distance of 18 m from the kicker. This means that the vertex of the parabolic trajectory is at the point (18, 12).

To find the value of the quadratic coefficient, a, we can substitute the vertex coordinates into the equation:

12 = a(18)^2 + b(18) + c

Since the ball starts at ground level (y = 0) and reaches a maximum height of 12 m, we also know that the y-intercept is 0. We can substitute this information into the equation as well:

0 = a(0)^2 + b(0) + c

Now we have a system of equations:

1) 12 = a(18)^2 + b(18) + c
2) 0 = c

From equation 2, we can conclude that c = 0.

Substituting this value into equation 1, we have:

12 = a(18)^2 + b(18)

Simplifying further:

12 = 324a + 18b

Now we have a linear equation with two variables (a and b). Since we need another equation to solve the system, we can use the given information that another player heads the ball at a height of 2.2 m from the ground. This means that when y = 2.2, there is some corresponding value of x.

Using the vertex form of a parabolic equation, we know that the x-coordinate of the vertex (h) is given by:

h = -b/2a

In this case, the x-coordinate of the vertex (h) is 18. Substituting this value into the equation:

18 = -b/2a

We can rearrange the equation as:

b = -36a

Now we have two equations:

1) 12 = 324a + 18b
2) b = -36a

Substituting equation 2 into equation 1:

12 = 324a + 18(-36a)

Simplifying:

12 = 324a - 648a

Combining like terms:

12 + 648a = 324a

648a - 324a = 12

324a = 12

a = 12/324

a = 0.037... (approximately)

Now that we have the value of a, we can substitute it back into equation 2 to find b:

b = -36(0.037...)

b = -1.33... (approximately)

Finally, substituting the values of a and b into the general quadratic equation form, we have:

y = 0.037...x^2 - 1.33...x

Therefore, the quadratic function that models the situation is:

y = 0.037...x^2 - 1.33...x