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

be approximated by the function:

y = -12⋅x^2 + 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

y = -12 x^2 + 48 x

complete square to find vertex
-y/12 = x^2 - 4 x
-y/12 + 4 = x^2 - 4 x + 4
-y/12 + 4 = (x-2)^2
vertex, max height, when x = 2
then y/12 = 4 or y = 48 feet
y = 0 again when x = 4 seconds

To find the time it takes the soccer ball to reach its maximum height, we need to find the vertex of the quadratic function y = -12x^2 + 48x. The vertex of a quadratic function is given by the formula x = -b/2a, where a, b, and c are the coefficients in the quadratic equation ax^2 + bx + c = 0.

For our equation y = -12x^2 + 48x, the coefficient of x^2 is -12 and the coefficient of x is 48. Plugging these values into the formula, we get x = -48/(2*(-12)) = -48/(-24) = 2.

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

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

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

Therefore, the soccer ball's maximum height is 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 quadratic equation y = -12x^2 + 48x. The roots of a quadratic equation represent the points where the graph intersects the x-axis, which in this case, represents the moment the ball hits the ground.

To find the roots, we set y = 0 and solve for x:

-12x^2 + 48x = 0.

Taking out the common factor of 12x, we get:

12x(-x + 4) = 0.

This equation is satisfied when either 12x = 0 or -x + 4 = 0.

From the first equation, we find that x = 0, which represents the moment the ball is kicked.

From the second equation, we find that -x + 4 = 0, which gives x = 4. This represents the moment the ball returns to the ground.

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