The height of a soccer ball that is kicked from the ground can


be approximated by the function:

y = -12⋅x2 + 48⋅x


where y is the height of the soccer ball in feet x seconds after it is kicked.


Find the time it takes the soccer ball to reach its maximum height: _____ seconds

What is the soccer ball’s maximum height:___ feet

Find the time it takes from the moment the soccer ball is kicked until it returns to the ground: ____ seconds

To find the time it takes for the soccer ball to reach its maximum height, we can find the vertex of the parabolic function. The vertex of a quadratic function in the form y = ax^2 + bx + c can be found using the formula x = -b / (2a).

In this case, the quadratic function is y = -12x^2 + 48x. Comparing it to the general form y = ax^2 + bx + c, we can see that a = -12 and b = 48.

Using the formula x = -b / (2a), we can plug in the values of a and b:

x = -48 / (2 * -12)
x = -48 / -24
x = 2

So, it takes the soccer ball 2 seconds to reach its maximum height.

To find the maximum height, we can substitute the value of x = 2 into the function y = -12x^2 + 48x:

y = -12(2)^2 + 48(2)
y = -12(4) + 96
y = -48 + 96
y = 48

Therefore, the soccer ball reaches a maximum height of 48 feet.

To find the time it takes from the moment the soccer ball is kicked until it returns to the ground, we need to find the roots of the equation y = -12x^2 + 48x. The roots represent the times when the ball reaches a height of 0.

Setting y = 0, we have:

0 = -12x^2 + 48x

Dividing both sides by -12, we get:

0 = x^2 - 4x

Factoring out x, we have:

0 = x(x - 4)

Setting each factor equal to zero, we find the two possible values for x:

x = 0
x - 4 = 0

Solving for x in each equation gives us:

x = 0
x = 4

Since negative values of x are not meaningful in this context, we can ignore x = 0.

Therefore, it takes the soccer ball 4 seconds from the moment it is kicked until it returns to the ground.

To find the time it takes for the soccer ball to reach its maximum height, we need to find the vertex of the quadratic function y = -12x^2 + 48x. The vertex is the highest point on the parabola, which corresponds to the maximum height of the soccer ball.

The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -12 and b = 48. Substituting the values into the formula, we have:

x = -48 / (2*(-12))
x = -48 / (-24)
x = 2

Therefore, the time it takes the soccer ball to reach its maximum height is 2 seconds.

To find the maximum height, we substitute the value of x = 2 seconds into the function y = -12x^2 + 48x:

y = -12(2)^2 + 48(2)
y = -12(4) + 96
y = -48 + 96
y = 48

Therefore, the soccer ball's maximum height is 48 feet.

To find the time it takes for the soccer ball to return to the ground, we need to find the x-intercepts of the function. The x-intercepts are the points where the height of the soccer ball is 0. In other words, we need to solve the equation -12x^2 + 48x = 0.

Factoring out x, we have:

x(-12x + 48) = 0

Setting each factor to 0, we get:

x = 0 or -12x + 48 = 0

Solving the second equation, we have:

-12x + 48 = 0
-12x = -48
x = -48 / -12
x = 4

Therefore, the time it takes from the moment the soccer ball is kicked until it returns to the ground is 4 seconds.

Since the parabola opens downward, the vertex is the max. point on the curve.

Y = -12x^2 + 48x

1. h = Xv = -B/2A = -48/-24 = 2 Seconds.

2. k = Yv = -12*2^2 + 48*2 = 48 Ft.

V(x,y)= (2,48).

3. Tr = Tf = 2 s.
T = 2 + 2 = 4 s.

Tr = Rise time.
Tf = Fall time.