Question

a rock is thrown from a cliff up into the air from a height of 6.4 meters. after 1.4 seconds, the rock reaches a maximum height of 16.2 meters. it then begins to fall and hits the ground 3.2 seconds after it is thrown. let f(x) be the height (in meters) of the rock x seconds after it is thrown. then, the function f is quadratic (its graph is a parabola.) write an equation for the quadratic function f.

Explanation:

Step1: Recall vertex - form of quadratic function

The vertex - form of a quadratic function is $f(x)=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola. The vertex of the parabola representing the height of the rock is at the point where the rock reaches its maximum height. Given that the maximum height is reached at $x = 1.4$ seconds and $y=16.2$ meters, so $h = 1.4$ and $k = 16.2$. Then $f(x)=a(x - 1.4)^2+16.2$.

Step2: Use the initial - height condition

The rock is thrown from a height of 6.4 meters, so when $x = 0$, $f(0)=6.4$. Substitute $x = 0$ and $f(0)=6.4$ into $f(x)=a(x - 1.4)^2+16.2$:
\[

$$\begin{align*} 6.4&=a(0 - 1.4)^2+16.2\\ 6.4&=a\times1.96 + 16.2\\ a\times1.96&=6.4 - 16.2\\ a\times1.96&=- 9.8\\ a&=\frac{-9.8}{1.96}=-5 \end{align*}$$

\]

Step3: Write the quadratic function

Substitute $a=-5$ into $f(x)=a(x - 1.4)^2+16.2$. We get $f(x)=-5(x - 1.4)^2+16.2$.

Answer:

$f(x)=-5(x - 1.4)^2+16.2$