How to transform the graph $\sin^{-1}(3x+1)$? So there is this question in which you have to transform the graph of $\sin^{-1}(3x+1)$ and so is the following:
$\sin^{-1}(3(x+\frac{1}{3}))$ with $D: -1 \leq x \leq 1$
However, if one applies the transformation of moving the graph $\frac{1}{3}$ to the left, it results in $\frac{2}{3}$. Now, you apply the transformation of horizontally compressing it by a factor of $\frac{1}{3}$, to get $\frac{2}{9}$. However, using graphing technology, the graph domain is $-\frac{2}{3} \leq x \leq 0$ which isn't known how to get. 
Thanks. 
 A: (Own-answer)
Okay, so there is order of transformations, and the stretches and compresses come first (horizontally) then the horizontal shifts. Or you can do the vertical stretches and compresses and then the vertical shifts then the horizontal transformations (with stretches and compresses first, then shifts). 
A: One easy way to transform the graph is to see what are the characteristic points. These occur when $3x+1\in\{-1,0,1\}$ and the corresponding values of $y = \sin^{-1}(3x+1)$ are $\{-\pi/2,0,\pi/2\}$. That is, the characteristic points are $A(-2/3,-\pi/2)$, $B(-1/3,0)$ and $C(0,\pi/2)$. All in all, it gives us:

A: Here's why ...
$$\begin{align}
y &= \sin^{-1}\bigg(3(x-(-\frac{1}{3}))\bigg)\\
\sin y &= 3(x-(-\frac{1}{3}))\\
\frac{1}{3}\sin y &= x-(-\frac{1}{3})\\
\frac{1}{3}\sin y + (-\frac{1}{3}) &= x\\
\end{align}$$
so after algebraically flipping the graph $45^{\circ}$ we can easily see that we have a compression along the x-axis by a factor of three and a shift along the x-axis of magnitude $-\frac{1}{3}$. (I avoided the terms 'vertical' and 'horizontal' intentionally here to avoid confusion). 
So, if $\sin y$ ranges over $[-1, 1]$, then $\frac{1}{3}\sin y$ ranges over $[-\frac{1}{3}, \frac{1}{3}]$, and $\frac{1}{3}\sin y + (-\frac{1}{3})$ ranges across $[-\frac{2}{3},0]$, but this is just the domain of $x$.
A: I teach this a bit differently than most. Perform "PEMDAS" in reverse for horizontal (inside) transformations, performing the opposite operation as indicated. Then perform "PEMDAS" as indicated directly with vertical transformations.
See colors:
$$\sin^{-1}({\color{red} 3}x{\color{blue}{+ 1}})$$


*

*$\text{${\color{black}{\text{Graph the parent function: $\sin^{-1}(x)$}}}$}$ 

*$\text{${\color{blue}{\text{Shift left 1 unit.}}}$}$ (Subtract 1 from x-values)

*$\text{${\color{red}{\text{Compress horizontally by a factor of 3.}}}$}$ (Divide x-values by three)


The $\text{${\color{red}{\text{Red}}}$}$ graph indicates the final transformation

