Answer:
[tex]h(t)=-6(t-2)^2+24[/tex]
Step-by-step explanation:
Given function:
[tex]h(t)=-6t^2+24t[/tex]
where:
- h = height (in feet)
- t = times (in seconds)
Rewrite in the form [tex]h(t)=a(t-h)^2+k[/tex]
The quickest way to do this is to expand [tex]h(t)=a(t-h)^2+k[/tex] and compare coefficients with the original function.
[tex]\begin{aligned}h(t)& =a(t-h)^2+k\\ & = at^2-2aht+ah^2+k\end{aligned}[/tex]
Comparing coefficients with [tex]h(t)=-6t^2+24t[/tex]
[tex]\begin{aligned}\implies at^2& =-6t^2\\ a &=-6\end{aligned}[/tex]
[tex]\begin{aligned}\implies -2aht & =24t\\ -2(-6)ht& =24t\\ 12ht & = 24t\\ 12h & = 24\\ h & = 2\end{aligned}[/tex]
[tex]\begin{aligned}\implies ah^2+k & = 0\\(-6)(2)^2+k &=0\\ -24+k & =0\\ k &=24 \end{aligned}[/tex]
Therefore, substituting the found values into the equation:
[tex]h(t)=-6(t-2)^2+24[/tex]
