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 $y = \sin (x)$ is looks like this:

Properties of the Sine Function, $y = \sin (x)$

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

Range : $[- 1, 1]$ or $- 1 \leq y \leq 1$

$y$ -intercept : $(0, 0)$

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

Period: $2 π$

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

Symmetry: origin (odd function)

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

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

Amplitude and Period of a Since Function

The amplitude of the graph of $y = a \sin (b x)$ 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 = $\frac{2 π}{| b |}$

Example:

Sketch the graphs of $y = \sin (x)$ and $y = 2 \sin (x)$ . Compare the graphs.

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

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

Also see Trigonometric Functions .

Graphing Sine Function

Properties of the Sine Function, y = sin ( x )

Amplitude and Period of a Since Function

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Properties of the Sine Function, $y = \sin (x)$