Graph y=5/3sin(-(2pi)/3x)

$y = \frac{5}{3} \sin (- \frac{2 π}{3} x)$

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

$a = - \frac{5}{3}$

$b = \frac{2 π}{3}$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{5}{3}$

Find the period of $- \frac{5 \sin (\frac{2 x π}{3})}{3}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Cancel the common factor of $2 π$ .

Cancel the common factor.

$2 π \frac{3}{2 π}$

Rewrite the expression.

$3$

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: $\frac{0}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{2 π})$

Multiply $0$ by $\frac{3}{2 π}$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{5}{3}$

Period: $3$

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.

$f (0) = - \frac{5 \sin (2 (0) \cdot \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Multiply $2$ by $0$ .

$f (0) = - \frac{5 \sin (0 \cdot \frac{π}{3})}{3}$

Multiply $0$ by $\frac{π}{3}$ .

$f (0) = - \frac{5 \sin (0)}{3}$

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

$f (0) = - \frac{5 \cdot 0}{3}$

Simplify the expression.

Multiply $5$ by $0$ .

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

Divide $0$ by $3$ .

$f (0) = - 0$

Multiply $- 1$ by $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

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

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

$f (\frac{3}{4}) = - \frac{5 \sin (2 (\frac{3}{4}) \cdot \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{3}{4}) = - \frac{5 \sin (2 (\frac{3}{2 (2)}) \cdot \frac{π}{3})}{3}$

Cancel the common factor.

$f (\frac{3}{4}) = - \frac{5 \sin (2 (\frac{3}{2 \cdot 2}) \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{3}{4}) = - \frac{5 \sin (\frac{3}{2} \cdot \frac{π}{3})}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{3}{4}) = - \frac{5 \sin (\frac{3}{2} \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{3}{4}) = - \frac{5 \sin (\frac{1}{2} \cdot π)}{3}$

Combine $\frac{1}{2}$ and $π$ .

$f (\frac{3}{4}) = - \frac{5 \sin (\frac{π}{2})}{3}$

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

$f (\frac{3}{4}) = - \frac{5 \cdot 1}{3}$

Multiply $5$ by $1$ .

$f (\frac{3}{4}) = - \frac{5}{3}$

The final answer is $- \frac{5}{3}$ .

$- \frac{5}{3}$

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

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

$f (\frac{3}{2}) = - \frac{5 \sin (2 (\frac{3}{2}) \cdot \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{3}{2}) = - \frac{5 \sin (2 (\frac{3}{2}) \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{3}{2}) = - \frac{5 \sin (3 \cdot \frac{π}{3})}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{3}{2}) = - \frac{5 \sin (3 \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{3}{2}) = - \frac{5 \sin (π)}{3}$

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

$f (\frac{3}{2}) = - \frac{5 \sin (0)}{3}$

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

$f (\frac{3}{2}) = - \frac{5 \cdot 0}{3}$

Simplify the expression.

Multiply $5$ by $0$ .

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

Divide $0$ by $3$ .

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

Multiply $- 1$ by $0$ .

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

The final answer is $0$ .

$0$

Find the point at $x = \frac{9}{4}$ .

Replace the variable $x$ with $\frac{9}{4}$ in the expression.

$f (\frac{9}{4}) = - \frac{5 \sin (2 (\frac{9}{4}) \cdot \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{9}{4}) = - \frac{5 \sin (2 (\frac{9}{2 (2)}) \cdot \frac{π}{3})}{3}$

Cancel the common factor.

$f (\frac{9}{4}) = - \frac{5 \sin (2 (\frac{9}{2 \cdot 2}) \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{9}{4}) = - \frac{5 \sin (\frac{9}{2} \cdot \frac{π}{3})}{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$f (\frac{9}{4}) = - \frac{5 \sin (\frac{3 (3)}{2} \cdot \frac{π}{3})}{3}$

Cancel the common factor.

$f (\frac{9}{4}) = - \frac{5 \sin (\frac{3 \cdot 3}{2} \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{9}{4}) = - \frac{5 \sin (\frac{3}{2} \cdot π)}{3}$

Combine $\frac{3}{2}$ and $π$ .

$f (\frac{9}{4}) = - \frac{5 \sin (\frac{3 π}{2})}{3}$

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.

$f (\frac{9}{4}) = - \frac{5 (- \sin (\frac{π}{2}))}{3}$

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

$f (\frac{9}{4}) = - \frac{5 (- 1 \cdot 1)}{3}$

Multiply $- 1$ by $1$ .

$f (\frac{9}{4}) = - \frac{5 \cdot - 1}{3}$

Simplify the expression.

Multiply $5$ by $- 1$ .

$f (\frac{9}{4}) = - \frac{- 5}{3}$

Move the negative in front of the fraction.

$f (\frac{9}{4}) = \frac{5}{3}$

The final answer is $\frac{5}{3}$ .

$\frac{5}{3}$

Find the point at $x = 3$ .

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

$f (3) = - \frac{5 \sin (2 (3) \cdot \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Multiply $2$ by $3$ .

$f (3) = - \frac{5 \sin (6 \cdot \frac{π}{3})}{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$f (3) = - \frac{5 \sin (3 (2) \cdot \frac{π}{3})}{3}$

Cancel the common factor.

$f (3) = - \frac{5 \sin (3 \cdot 2 \cdot \frac{π}{3})}{3}$

Rewrite the expression.

$f (3) = - \frac{5 \sin (2 \cdot π)}{3}$

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

$f (3) = - \frac{5 \sin (0)}{3}$

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

$f (3) = - \frac{5 \cdot 0}{3}$

Simplify the expression.

Multiply $5$ by $0$ .

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

Divide $0$ by $3$ .

$f (3) = - 0$

Multiply $- 1$ by $0$ .

$f (3) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3}{4} & - \frac{5}{3} \\ \frac{3}{2} & 0 \\ \frac{9}{4} & \frac{5}{3} \\ 3 & 0 \end{array}$

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

Amplitude: $\frac{5}{3}$

Period: $3$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{3}{4} & - \frac{5}{3} \\ \frac{3}{2} & 0 \\ \frac{9}{4} & \frac{5}{3} \\ 3 & 0 \end{array}$

Graph y=5/3sin(-(2pi)/3x)

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