a ball was tossed up into the air. the height of the ball as a function of the time the ball is in the air…

Explanation:

Step1: Write the vertex - form of a quadratic function

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. The vertex of the ball's height - time function is $(h,k)=(1.25,15.3125)$. So the function is $y=a(x - 1.25)^2+15.3125$. When $x = 0$, $y=7.5$. Substitute these values into the function: $7.5=a(0 - 1.25)^2+15.3125$

Step2: Solve for $a$

First, expand $(0 - 1.25)^2=(-1.25)^2 = 1.5625$. Then the equation becomes $7.5 = 1.5625a+15.3125$. Subtract 15.3125 from both sides: $7.5-15.3125=1.5625a$, so $- 7.8125 = 1.5625a$. Divide both sides by 1.5625: $a=\frac{-7.8125}{1.5625}=-5$. So the quadratic function is $y=-5(x - 1.25)^2+15.3125$.

Step3: Find the height at $x = 2$

Substitute $x = 2$ into the function: $y=-5(2 - 1.25)^2+15.3125$. First, calculate $(2 - 1.25)^2=(0.75)^2 = 0.5625$. Then $y=-5\times0.5625+15.3125$. $y=-2.8125 + 15.3125$. $y = 12.5$.

Answer:

$12.5$