Graph y=1/3sin(2x)

$y = \frac{1}{3} \sin (2 x)$

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = \frac{1}{3}$

$b = 2$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{3}$

Find the period of $\frac{\sin (2 x)}{3}$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{2}$

Divide $0$ by $2$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{3}$

Period: $π$

Phase Shift: None

Vertical Shift: None

Select a few points to graph.

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{\sin (2 (0))}{3}$

Simplify the result.

Simplify the numerator.

Multiply $2$ by $0$ .

$f (0) = \frac{\sin (0)}{3}$

The exact value of $\sin (0)$ is $0$ .

$f (0) = \frac{0}{3}$

Divide $0$ by $3$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{π}{4}$ .

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

$f (\frac{π}{4}) = \frac{\sin (2 (\frac{π}{4}))}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = \frac{\sin (2 (\frac{π}{2 (2)}))}{3}$

Cancel the common factor.

$f (\frac{π}{4}) = \frac{\sin (2 (\frac{π}{2 \cdot 2}))}{3}$

Rewrite the expression.

$f (\frac{π}{4}) = \frac{\sin (\frac{π}{2})}{3}$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{π}{4}) = \frac{1}{3}$

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Find the point at $x = \frac{π}{2}$ .

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

$f (\frac{π}{2}) = \frac{\sin (2 (\frac{π}{2}))}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{π}{2}) = \frac{\sin (2 (\frac{π}{2}))}{3}$

Rewrite the expression.

$f (\frac{π}{2}) = \frac{\sin (π)}{3}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\frac{π}{2}) = \frac{\sin (0)}{3}$

The exact value of $\sin (0)$ is $0$ .

$f (\frac{π}{2}) = \frac{0}{3}$

Divide $0$ by $3$ .

$f (\frac{π}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{3 π}{4}$ .

Replace the variable $x$ with $\frac{3 π}{4}$ in the expression.

$f (\frac{3 π}{4}) = \frac{\sin (2 (\frac{3 π}{4}))}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{3 π}{4}) = \frac{\sin (2 (\frac{3 π}{2 (2)}))}{3}$

Cancel the common factor.

$f (\frac{3 π}{4}) = \frac{\sin (2 (\frac{3 π}{2 \cdot 2}))}{3}$

Rewrite the expression.

$f (\frac{3 π}{4}) = \frac{\sin (\frac{3 π}{2})}{3}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{3 π}{4}) = \frac{- \sin (\frac{π}{2})}{3}$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{3 π}{4}) = \frac{- 1 \cdot 1}{3}$

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{4}) = \frac{- 1}{3}$

Move the negative in front of the fraction.

$f (\frac{3 π}{4}) = - \frac{1}{3}$

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Find the point at $x = π$ .

Replace the variable $x$ with $π$ in the expression.

$f (π) = \frac{\sin (2 (π))}{3}$

Simplify the result.

Simplify the numerator.

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (π) = \frac{\sin (0)}{3}$

The exact value of $\sin (0)$ is $0$ .

$f (π) = \frac{0}{3}$

Divide $0$ by $3$ .

$f (π) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{4} & \frac{1}{3} \\ \frac{π}{2} & 0 \\ \frac{3 π}{4} & - \frac{1}{3} \\ π & 0 \end{array}$

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $\frac{1}{3}$

Period: $π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{4} & \frac{1}{3} \\ \frac{π}{2} & 0 \\ \frac{3 π}{4} & - \frac{1}{3} \\ π & 0 \end{array}$

Graph y=1/3sin(2x)

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