4. The position of a ball after it is kicked can be determined by using the function \( f(x)=-0.11 x^{2}+2.2 x+1 \), where \( f(x) \) is the height, in feet, above the ground and \( x \) is the horizontal distance, in feet, of the ball from the point at which it was kicked. What is the height of the ball when it is kicked? What is the highest point of the ball in the air?

To find the height of the ball when it is kicked, we evaluate the function at \( x = 0 \). Plugging in the value gives us \( f(0) = -0.11(0^2) + 2.2(0) + 1 = 1 \) foot. So, the ball is 1 foot above the ground when it is kicked. To determine the highest point, we need to find the vertex of the parabola described by the function. The \( x \)-coordinate of the vertex can be found using \( x = -\frac{b}{2a} \), where \( a = -0.11 \) and \( b = 2.2 \). This results in \( x = -\frac{2.2}{2(-0.11)} \approx 10.00 \) feet. Plugging this back into the function yields \( f(10) = -0.11(10^2) + 2.2(10) + 1 = 12 \) feet. Thus, the highest point of the ball is 12 feet above the ground.