We explored the two cases of stretching/compression the Sine function. Now, we will turn to shifting the Sine function. 

Vertical Shift

Let's consider \(y=\sin(\theta)+C\):

Notice that shifting \(C\) up or down simply moves the graph up and down. We like to keep track of the midline of the graph, which will always correspond to exactly \(y=C\). This also happens to be the average value of the function! Looking back at the circle, the vertical shift has the effect of moving the circle up or down by \(C\).

One final point to note is that when the function shifts up or down, the amplitude remains the same; this is because the amplitude is the distance from either max or min to the midline. Since the midline and max/min move up/down the same amount with the transformation, the amplitude remains constant.

Horizontal Shift

The horizontal shift for sinusoidal functions is usually called the phase shift (as it happens to relate to many phenomena from physics and this language carries over into mathematics) and is represented with the symbol \(\omega\).

Similar to the vertical shift, the circle is simply moved by the phase shift, which will have the impact of shifting horizontal intercepts by \(\pm \omega\). There are some oddities you may notice, however. The graph is programmed to allow you to change \(\omega\), but when you do so, you should also see that \(\theta\) will change. This is because we are fixing the rightmost endpoint and keeping the graph going to that point. By adjusting \(\omega\), you are changing "how far" the \(\theta\) needs to go to get to the same endpoint!