Now that we have looked at the generation of the graph \(y=\sin(\theta)\), how can we go about graphing more general transformations of this graph? Moreover, what exactly do those transformations mean in the context of the original Unit Circle? 

Let's look at \(y=A\sin(\theta)\) first. This is a vertical stretch/compression by a factor of \(A\). The result of this would directly impact the range of the original Sine function, taking it from \([-1,1]\) to \([-A,A]\). How can we visualize this and relate it back to the Unit Circle? Quite simply, this is the result of generating the Sine graph from a non-Unit Circle! Take a look here (feel free to adjust the radius of the circle and see how things change):

Do you see how the radius of the circle determines the min/max of the Sine graph? We call the distance above/below the midline of the Sine graph the Amplitude of the graph. Note that Amplitude is a distance and should always be considered a positive value. The amplitude of the sine graph will always be exactly equal to the radius of the circle which generated it.

Next, let's look at what happens when we perform a horizontal stretch/compression with \(y=\sin(B\theta)\):

This results in changing the period of the sine graph, either causing the period to occur more frequently or less frequently. In fact, we say that the frequency of the graph is exactly determined by the reciprocal of the period: \(f = \frac{{1}}{{P}}\)  where \(f\) is frequency and \(P\) is period. Moreover, the value \(B\) is related to the frequency directly by \(B=2\pi f\). Using substitution, we can also write \(B\) related to the period: \(B=\frac{2\pi}{{P}}\).