Question

An object was launched off the top of a building. The function

f(x) = -16x² + 16x + 192 represents the height of the object above the ground,
in feet, a seconds after being launched. Find and interpret the given function values
and determine an appropriate domain for the function.

Answer 1

Check the picture below.

`~~~~~~\textit{initial velocity in feet} \\\\ h(t) = -16t^2+v_ot+h_o \quad \begin{cases} v_o=\textit{initial velocity}&16~(ft)/(s)\\ \qquad \textit{of the object}\\ h_o=\textit{initial height}&192~ft\\ \qquad \textit{of the object}\\ h=\textit{object's height}&h\\ \qquad \textit{at`

now, keeping in mind that x-axis is the seconds, however a parabola could go into the 2nd, 3rd and 4th Quadrants, but for this case, we only need it in the 1st Quadrant, because the seconds are positive, it begins at 0 seconds and goes up, then down and hits the ground later on when f(x) = 0, hmmm when is that?

`\stackrel{f(x)}{0}=\stackrel{ \textit{hits the ground} }{-16x^2+16x+192}\implies 0=-16() \\\\\\ 0=x^2-x-12\implies 0=(x+3)(x-4)\implies x= \begin{cases} -3 ~~ \bigotimes\\\\ ~~ 4 ~~ \checkmark \end{cases}`

now, both values are valid for "x", however for this specific case, we can't have negative seconds, so the negative value won't work. So the appropriate domain or values for "x" are from 0 to 4, the object went up and down in 4 seconds flat.