8.1 Graphs of the sine and cosine functions  (Page 3/13)

Analyzing graphs of variations of y = sin x And y = cos x

Now that we understand how $A$ and $B$ relate to the general form equation for the sine and cosine functions, we will explore the variables $C$ and $D.$ Recall the general form:

The value $\frac{C}{B}$ for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If $C>0,$ the graph shifts to the right. If $C<0,$ the graph shifts to the left. The greater the value of $\left|C\right|,$ the more the graph is shifted. [link] shows that the graph of $f(x)=\sin \left(x-\pi \right)$ shifts to the right by $\pi$ units, which is more than we see in the graph of $f(x)=\sin \left(x-\frac{\pi }{4}\right),$ which shifts to the right by $\frac{\pi }{4}$ units.

While $C$ relates to the horizontal shift, $D$ indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function $y=\cos \left(x\right)+D$ has its midline at $y=D.$

Any value of $D$ other than zero shifts the graph up or down. [link] compares $f(x)=\sin x$ with $f(x)=\sin x+2,$ which is shifted 2 units up on a graph.