![\bf ~~~~~~~~~~~~\textit{function transformations} \\\\\\ % templates f(x)=Asin(Bx+C)+D \\\\ f(x)=Acos(Bx+C)+D\\\\ f(x)=Atan(Bx+C)+D \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \bullet \textit{ stretches or shrinks}\\ ~~~~~~\textit{horizontally by amplitude } A\cdot B\\\\ \bullet \textit{ flips it upside-down if }A\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }B\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis}](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bfunction%20transformations%7D%20%5C%5C%5C%5C%5C%5C%20%25%20templates%20f%28x%29%3DAsin%28Bx%2BC%29%2BD%20%5C%5C%5C%5C%20f%28x%29%3DAcos%28Bx%2BC%29%2BD%5C%5C%5C%5C%20f%28x%29%3DAtan%28Bx%2BC%29%2BD%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20%5Cbullet%20%5Ctextit%7B%20stretches%20or%20shrinks%7D%5C%5C%20~~~~~~%5Ctextit%7Bhorizontally%20by%20amplitude%20%7D%20A%5Ccdot%20B%5C%5C%5C%5C%20%5Cbullet%20%5Ctextit%7B%20flips%20it%20upside-down%20if%20%7DA%5Ctextit%7B%20is%20negative%7D%5C%5C%20~~~~~~%5Ctextit%7Breflection%20over%20the%20x-axis%7D%20%5C%5C%5C%5C%20%5Cbullet%20%5Ctextit%7B%20flips%20it%20sideways%20if%20%7DB%5Ctextit%7B%20is%20negative%7D%5C%5C%20~~~~~~%5Ctextit%7Breflection%20over%20the%20y-axis%7D)
with that template in mind, let's take a peek
![\bf f(x)=2sin(x+\pi )-4\implies f(x)=\stackrel{A}{2}sin(\stackrel{B}{1}x\stackrel{C}{+\pi })\stackrel{D}{-4} \\\\[-0.35em] ~\dotfill\\\\ \textit{Amplitude}\implies 2 \\\\\\ \stackrel{phase}{\textit{Horizontal Shift}}\implies \cfrac{C}{B}\implies \cfrac{+\pi }{1}\implies +\pi \impliedby \pi \textit{ units to the left} \\\\\\ \textit{Period}\implies \cfrac{2\pi }{B}\implies \cfrac{2\pi }{1}\implies 2\pi \\\\\\ \textit{Vertical Shift}\implies D\implies -4\impliedby \textit{4 units downwards}](https://tex.z-dn.net/?f=%5Cbf%20f%28x%29%3D2sin%28x%2B%5Cpi%20%29-4%5Cimplies%20f%28x%29%3D%5Cstackrel%7BA%7D%7B2%7Dsin%28%5Cstackrel%7BB%7D%7B1%7Dx%5Cstackrel%7BC%7D%7B%2B%5Cpi%20%7D%29%5Cstackrel%7BD%7D%7B-4%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Ctextit%7BAmplitude%7D%5Cimplies%202%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7Bphase%7D%7B%5Ctextit%7BHorizontal%20Shift%7D%7D%5Cimplies%20%5Ccfrac%7BC%7D%7BB%7D%5Cimplies%20%5Ccfrac%7B%2B%5Cpi%20%7D%7B1%7D%5Cimplies%20%2B%5Cpi%20%5Cimpliedby%20%5Cpi%20%5Ctextit%7B%20units%20to%20the%20left%7D%20%5C%5C%5C%5C%5C%5C%20%5Ctextit%7BPeriod%7D%5Cimplies%20%5Ccfrac%7B2%5Cpi%20%7D%7BB%7D%5Cimplies%20%5Ccfrac%7B2%5Cpi%20%7D%7B1%7D%5Cimplies%202%5Cpi%20%5C%5C%5C%5C%5C%5C%20%5Ctextit%7BVertical%20Shift%7D%5Cimplies%20D%5Cimplies%20-4%5Cimpliedby%20%5Ctextit%7B4%20units%20downwards%7D)
now, the midline for the parent function of sin(x) is simply the x-axis, namely y = 0.
this shifted/transformed version of it, has a vertical shift of 4 units down, so the midline moved from y = 0, to y = -4.
Answer:
2 minutes
Step-by-step explanation:
The front of the train enters the tunnel and travels 1 km before the rear of the train enters the tunnel. Thus, the total distance traveled by the front of the train is 2 km, and this is covered in a certain number of hours:
Since distance = rate times time, time = distance / rate. In this particular case we have:
2 km
time = distance / rate = ------------------- = (1/30) hr
60 km/hr
(1/30) hr converts to 2 minutes, since 1 hr = 60 minutes
To get rid of the 3, you multiply the x/3 by 3 to get x by itself. But if you multiply one side you have to multiply the other. So x = 99
