Question

modeling with mathematics the path of a basketball thrown at an angle of 45° can be modeled by ( y = - 0.02 x ^ { 2 } + x + 6 ), where ( x ) is the horizontal distance (in feet) and ( y ) is the vertical distance (in feet). find and interpret the coordinates of the vertex. the coordinates of the vertex are ( (square,square) ). when the basketball is at its highest point, it is ( square ) feet from its starting point and ( square ) feet off the ground.

Explanation:

Step1: Find the x - coordinate of the vertex

For a quadratic function \(y = ax^{2}+bx + c\), the formula for the x - coordinate of the vertex is \(x=-\frac{b}{2a}\).
In the given function \(y=-0.02x^{2}+x + 6\), \(a=-0.02\) and \(b = 1\).

$$x=-\frac{1}{2\times(-0.02)}=\frac{-1}{-0.04}=25$$

Step2: Find the y - coordinate of the vertex

Substitute \(x = 25\) into the function \(y=-0.02x^{2}+x + 6\).

$$y=-0.02\times(25)^{2}+25 + 6$$

$$y=-0.02\times625+25 + 6$$

$$y=-12.5+25 + 6$$

$$y=18.5$$

Answer:

The coordinates of the vertex are \((25,18.5)\). When the basketball is at its highest point, it is \(25\) feet from its starting point and \(18.5\) feet off the ground.