Graph y=-1/4sin(x/3)

$y = - \frac{1}{4} \sin (\frac{x}{3})$

Step 1

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

$a = - \frac{1}{4}$

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

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Step 3

Find the period of $- \frac{\sin (\frac{x}{3})}{4}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

Step 4

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: $\frac{0}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 3$

Multiply $0$ by $3$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $6 π$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

Find the point at $x = 0$ .

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

$f (0) = - \frac{\sin (\frac{0}{3})}{4}$

Simplify the result.

Cancel the common factor of $0$ and $3$ .

Factor $3$ out of $0$ .

$f (0) = - \frac{\sin (\frac{3 (0)}{3})}{4}$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (0) = - \frac{\sin (\frac{3 \cdot 0}{3 \cdot 1})}{4}$

Cancel the common factor.

$f (0) = - \frac{\sin (\frac{3 \cdot 0}{3 \cdot 1})}{4}$

Rewrite the expression.

$f (0) = - \frac{\sin (\frac{0}{1})}{4}$

Divide $0$ by $1$ .

$f (0) = - \frac{\sin (0)}{4}$

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

$f (0) = - \frac{0}{4}$

Divide $0$ by $4$ .

$f (0) = - 0$

Multiply $- 1$ by $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{3 π}{2}$ .

Replace the variable $x$ with $\frac{3 π}{2}$ in the expression.

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

Simplify the result.

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3 π}{2}) = - \frac{\sin (\frac{3 π}{2} \cdot \frac{1}{3})}{4}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

$f (\frac{3 π}{2}) = - \frac{\sin (\frac{3 (π)}{2} \cdot \frac{1}{3})}{4}$

Cancel the common factor.

$f (\frac{3 π}{2}) = - \frac{\sin (\frac{3 π}{2} \cdot \frac{1}{3})}{4}$

Rewrite the expression.

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

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

$f (\frac{3 π}{2}) = - \frac{1}{4}$

The final answer is $- \frac{1}{4}$ .

$- \frac{1}{4}$

Find the point at $x = 3 π$ .

Replace the variable $x$ with $3 π$ in the expression.

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

Simplify the result.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (3 π) = - \frac{\sin (π)}{4}$

Simplify the numerator.

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

$f (3 π) = - \frac{\sin (0)}{4}$

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

$f (3 π) = - \frac{0}{4}$

Simplify the expression.

Divide $0$ by $4$ .

$f (3 π) = - 0$

Multiply $- 1$ by $0$ .

$f (3 π) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{9 π}{2}$ .

Replace the variable $x$ with $\frac{9 π}{2}$ in the expression.

$f (\frac{9 π}{2}) = - \frac{\sin (\frac{\frac{9 π}{2}}{3})}{4}$

Simplify the result.

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 π}{2}) = - \frac{\sin (\frac{9 π}{2} \cdot \frac{1}{3})}{4}$

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

$f (\frac{9 π}{2}) = - \frac{\sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{4}$

Cancel the common factor.

$f (\frac{9 π}{2}) = - \frac{\sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{4}$

Rewrite the expression.

$f (\frac{9 π}{2}) = - \frac{\sin (\frac{3 π}{2})}{4}$

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.

$f (\frac{9 π}{2}) = - \frac{- \sin (\frac{π}{2})}{4}$

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

$f (\frac{9 π}{2}) = - \frac{- 1 \cdot 1}{4}$

Multiply $- 1$ by $1$ .

$f (\frac{9 π}{2}) = - \frac{- 1}{4}$

Move the negative in front of the fraction.

$f (\frac{9 π}{2}) = \frac{1}{4}$

The final answer is $\frac{1}{4}$ .

$\frac{1}{4}$

Find the point at $x = 6 π$ .

Replace the variable $x$ with $6 π$ in the expression.

$f (6 π) = - \frac{\sin (\frac{6 π}{3})}{4}$

Simplify the result.

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 π$ .

$f (6 π) = - \frac{\sin (\frac{3 (2 π)}{3})}{4}$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (6 π) = - \frac{\sin (\frac{3 (2 π)}{3 (1)})}{4}$

Cancel the common factor.

$f (6 π) = - \frac{\sin (\frac{3 (2 π)}{3 \cdot 1})}{4}$

Rewrite the expression.

$f (6 π) = - \frac{\sin (\frac{2 π}{1})}{4}$

Divide $2 π$ by $1$ .

$f (6 π) = - \frac{\sin (2 π)}{4}$

Simplify the numerator.

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

$f (6 π) = - \frac{\sin (0)}{4}$

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

$f (6 π) = - \frac{0}{4}$

Simplify the expression.

Divide $0$ by $4$ .

$f (6 π) = - 0$

Multiply $- 1$ by $0$ .

$f (6 π) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3 π}{2} & - \frac{1}{4} \\ 3 π & 0 \\ \frac{9 π}{2} & \frac{1}{4} \\ 6 π & 0 \end{array}$

Step 7

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

Amplitude: $\frac{1}{4}$

Period: $6 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3 π}{2} & - \frac{1}{4} \\ 3 π & 0 \\ \frac{9 π}{2} & \frac{1}{4} \\ 6 π & 0 \end{array}$

Step 8

Graph y=-1/4sin(x/3)

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