The function g(x) = -4x^2 + 16x determines the height at which Hamad kicks the ball after x seconds.
To determine where the function is increasing or decreasing, we need to find the vertex of the parabola. We can find the vertex using the formula x = -b/2a, where a = -4 and b = 16.
x = -b/2a = -16/(2*(-4)) = 2
So the vertex of the parabola is at x = 2.
When x < 2, the function is increasing because the ball is going up. When x > 2, the function is decreasing because the ball is coming back down.
To find how long the ball is in the air, we need to find the roots of the function (when the height is zero). We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
a = -4, b = 16, c = 0
x = (-16 ± √(16^2 - 4(-4)(0))) / 2(-4) = 2
So the ball is in the air for 2 seconds.
The domain of the function is all real numbers, since we can plug in any value for x and get a valid output (height).
I hope it will be of assistance to you.