# Graphing Sine Function

The trigonometric ratios can also be considered as functions of a variable which is the measure of an angle. This angle measure can either be given in degrees or radians. Here, we will use radians.

The graph of a sine function $y=\mathrm{sin}\left(x\right)$ is looks like this:

### Properties of the Sine Function, $y=\mathrm{sin}\left(x\right)$

Domain: $\left(-\infty ,\infty \right)$

Range: $\left[-1,1\right]$ or $-1\le y\le 1$

$y$-intercept: $\left(0,1\right)$

$x$-intercept: $n\pi$, where $n$ is an integer.

Period: $2\pi$

Continuity: continuous on $\left(-\infty ,\infty \right)$

Symmetry: origin (odd function)

The maximum value of $y=\mathrm{sin}\left(x\right)$ occurs when $x=\frac{\pi }{2}+2n\pi$, where $n$ is an integer.

The minimum value of $y=\mathrm{sin}\left(x\right)$ occurs when $x=\frac{3\pi }{2}+2n\pi$, where $n$ is an integer.

### Amplitude and Period of a Since Function

The amplitude of the graph of $y=a\mathrm{sin}\left(bx\right)$ is the amount by which it varies above and below the $x$-axis.

Amplitude = | $a$|

The period of a sine function is the length of the shortest interval on the $x$-axis over which the graph repeats.

Period = $\frac{2\pi }{|b|}$

Example:

Sketch the graphs of $y=\mathrm{sin}\left(x\right)$ and $y=2\mathrm{sin}\left(x\right)$. Compare the graphs.

For the function $y=2\mathrm{sin}\left(x\right)$, the graph has an amplitude $2$. Since $b=1$, the graph has a period of $2\pi$. Thus, it cycles once from $0$ to $2\pi$ with one maximum of $2$, and one minimum of $-2$.

Observe the graphs of $y=\mathrm{sin}\left(x\right)$ and $y=2\mathrm{sin}\left(x\right)$. Each has the same $x$-intercepts, but $y=2\mathrm{sin}\left(x\right)$ has an amplitude that is twice the amplitude of $y=\mathrm{sin}\left(x\right)$.

Also see Trigonometric Functions.