**Transformations of the sine function**
**Amplitude**Multiplying $\sin(x)$ by a positive constant $A$ causes its graph to oscillate between $A$ and $-A$ instead of between $1$ and $-1$. For instance, the following graph shows $2\sin(x)$ (plotted in red).

$y = \sin(x) \qquad \qquad \qquad$$y = 2\sin(x)$

For this reason we say that the function $2\sin(x)$ of $x$ has

**amplitude $2$.**
**Vertical shift**Adding a quantity $C$ to $\sin(x)$ causes its graph to shift upwards by $C$. (If $C$ is negative, this is understood to mean shifting

*downward* by $|C|$.) For instance, the following graph shows $1/2 + \sin(x)$ (plotted in red).

$y = \sin(x) \qquad \qquad \qquad$$y = 1/2 + \sin(x)$

**Horizontal shift**If $a$ is positive, then replacing $x$ by $x-a$ causes the graph of $\sin(x)$ to shift right by $a$ units, and replacing $x$ by $x+a$ causes the graph to shift

*left* by $a$ units. For instance, the following graph shows $\sin(x-\pi/4)$ (plotted in red).

$y = \sin(x) \qquad \qquad \qquad$$y = \sin (x-\pi/4)$

**Angular frequency**If $\omega$ is positive, then replacing $x$ by $\omega x$ causes the graph of $\sin(x)$ to oscillate $\omega$ times as fast. For instance, the following graph shows $\sin(2x)$ (plotted in red).

$y = \sin(x) \qquad \qquad \qquad$$y = \sin(2x)$

For this reason we say that the function $\sin(2x)$ of $x$ has

**angular frequency $2$.**