Question

ryan throws a tennis ball straight up into the air. the ball reaches its maximum height at 2 seconds. the approximate height of the ball x seconds after being thrown is shown in the table. motion of tennis ball which equation models the motion of the ball? y = -17(x)(x - 4) y = -16(x)(x - 4) y = -16(x - 2)^2 + 68 y = -17(x - 2)^2 + 68

Explanation:

Step1: Recall 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. Since the ball reaches its maximum height at \(x = 2\) seconds and the maximum height is \(y = 68\) feet, the vertex of the parabola representing the height of the ball as a function of time is \((h,k)=(2,68)\). So the equation of the parabola is of the form \(y=a(x - 2)^{2}+68\).

Step2: Find the value of \(a\)

We can use one of the points from the table. Let's use the point \((0,4)\). Substitute \(x = 0\) and \(y = 4\) into \(y=a(x - 2)^{2}+68\). We get \(4=a(0 - 2)^{2}+68\).
\[

$$\begin{align*} 4&=a(-2)^{2}+68\\ 4&=4a + 68\\ 4a&=4 - 68\\ 4a&=- 64\\ a&=-16 \end{align*}$$

\]

Step3: Write the equation

The equation of the parabola is \(y=-16(x - 2)^{2}+68\).

Answer:

\(y=-16(x - 2)^{2}+68\)