Question

the position of a ball after it is kicked can be determined by using the function $f(x) = -0.11x^2 + 2.2x + 1$, 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?
the height of the ball when it was kicked was 1 foot(feet).
(type an integer or a decimal.)
the maximum height the ball reaches is
(type an integer or a decimal.)
foot(feet)
square foot(feet)
cubic foot(feet)

Explanation:

Step1: Identify the function type

The function \( f(x) = -0.11x^2 + 2.2x + 1 \) is a quadratic function in the form \( f(x)=ax^2 + bx + c \), where \( a=-0.11 \), \( b = 2.2 \), and \( c = 1 \). Since \( a<0 \), the parabola opens downward, and the vertex represents the maximum point.

Step2: Find the x - coordinate of the vertex

The formula for the x - coordinate of the vertex of a quadratic function \( f(x)=ax^2+bx + c \) is \( x=-\frac{b}{2a} \).
Substitute \( a=-0.11 \) and \( b = 2.2 \) into the formula:
\( x=-\frac{2.2}{2\times(-0.11)}=-\frac{2.2}{-0.22} = 10 \)

Step3: Find the maximum height (y - coordinate of the vertex)

Substitute \( x = 10 \) into the function \( f(x)=-0.11x^2+2.2x + 1 \):
\( f(10)=-0.11\times(10)^2+2.2\times(10)+1 \)
\( f(10)=-0.11\times100 + 22+1 \)
\( f(10)=-11 + 22+1 \)
\( f(10)=12 \)

Answer:

The maximum height the ball reaches is \( 12 \) foot(feet).