Question

modeling real life in exercises 29 and 30, x is the horizontal distance (in feet) and y is the height (in feet). find and interpret the coordinates of the vertex. 29. the path of a basketball thrown at an angle of 45° can be modeled by y = -0.02x² + x + 6.

Explanation:

Step1: Identify the coefficients

For the quadratic function $y = ax^{2}+bx + c$, in $y=-0.02x^{2}+x + 6$, we have $a=-0.02$, $b = 1$, $c = 6$.

Step2: Find the x - coordinate of the vertex

The formula for the x - coordinate of the vertex of a quadratic function is $x=-\frac{b}{2a}$. Substitute $a=-0.02$ and $b = 1$ into the formula:
$x=-\frac{1}{2\times(-0.02)}=\frac{1}{0.04}=25$.

Step3: Find the y - coordinate of the vertex

Substitute $x = 25$ into the function $y=-0.02x^{2}+x + 6$.
$y=-0.02\times25^{2}+25 + 6=-0.02\times625+25 + 6=-12.5+25 + 6=18.5$.

Answer:

The coordinates of the vertex are $(25,18.5)$. This means that the horizontal distance of the basketball from the thrower when it reaches its maximum height is 25 feet, and the maximum height of the basketball is 18.5 feet.