How do transformations change the **values** of circular functions? E.g. I know that for the base function sin(x), when you dilate it for 2 units from the x-axis and dilate it for 1/3 units from the y-axis, you have the transformed function y=2sin(3x), but in terms of the values of x and y, how are they different? What do I change? How do I change it?

Sorry for not being clear. My question is about how any given point on the graph moves after a sequence of transformations are applied (specifically circular functions).

The neat thing about simple transformations (dilations, reflections, translations) is that their actions are the same on every relation.

A dilation by factor \(a\) from the \(x\)-axis will map any point \((x,\,y)\) on a graph to the point \((x,\,ay)\). That is, the \(y\)-coordinate of every point on a graph will be multiplied by \(a\). For example, if \(a=2\), every \(y\)-coordinate is doubled. Shown below are the graphs of \(y=\sin(x)\) and \(y=2\sin(x)\) for \(x\in[0,\,2\pi]\).

Now, take a dilation by factor \(b\) from the \(y\)-axis. This will map any point \((x,\,y)\) on a graph to the point \((bx,\,y)\). That is, the \(x\)-coordinate of every point on a graph will be multiplied by \(b\). Given the graph of \(y=\sin(x)\), where \(x\in[0,\,2\pi]\), could you draw the image relation after a dilation by factor \(2\) from the \(y\)-axis?

**In a nutshell**(1) Dilation by factor \(a\) from the \(x\)-axis

\[(x',\,y')=(x,\,ay)\quad \text{and}\quad y=f(x)\overset{(1)}{\longrightarrow}y=a\,f(x)\]

(2) Dilation by factor \(b\) from the \(y\)-axis

\[(x',\,y')=(bx,\,y)\quad \text{and}\quad y=f(x)\overset{(2)}{\longrightarrow}y=f\left(\frac{x}{b}\right)\]

(3) Reflection in the \(x\)-axis

\[(x',\,y')=(x,\,-y)\quad \text{and}\quad y=f(x)\overset{(3)}{\longrightarrow}y=-f(x)\]

(4) Reflection in the \(y\)-axis

\[(x',\,y')=(-x,\,y)\quad \text{and}\quad y=f(x)\overset{(4)}{\longrightarrow}y=f(-x)\]

(5) Translation of \(h\) units in the positive \(x\)-direction

\[(x',\,y')=(x+h,\,y)\quad \text{and}\quad y=f(x)\overset{(5)}{\longrightarrow}y=f(x-h)\]

(6) Translation of \(k\) units in the positive \(y\)-direction

\[(x',\,y')=(x,\,y+k)\quad \text{and}\quad y=f(x)\overset{(6)}{\longrightarrow}y=f(x)+k\]

(7) Reflection in the line \(y=x\)

\[(x',\,y')=(y,\,x)\quad \text{and}\quad y=f(x)\overset{(7)}{\longrightarrow}x=f(y)\]