Graph y=-2sin(3(x-pi/8))

$y = - 2 \sin (3 (x - \frac{π}{8}))$

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

$a = - 2$

$b = 3$

$c = \frac{3 π}{8}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $- 2 \sin (3 x - \frac{3 π}{8})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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

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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: $\frac{\frac{3 π}{8}}{3}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{8} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

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Factor $3$ out of $3 π$ .

Phase Shift: $\frac{3 (π)}{8} \cdot \frac{1}{3}$

Cancel the common factor.

Phase Shift: $\frac{3 π}{8} \cdot \frac{1}{3}$

Rewrite the expression.

Phase Shift: $\frac{π}{8}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{8}$ ( $\frac{π}{8}$ to the right)

Vertical Shift: $0$

Select a few points to graph.

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Find the point at $x = \frac{π}{8}$ .

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Replace the variable $x$ with $\frac{π}{8}$ in the expression.

$f (\frac{π}{8}) = - 2 \sin (3 (\frac{π}{8}) - \frac{3 π}{8})$

Simplify the result.

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Combine $3$ and $\frac{π}{8}$ .

$f (\frac{π}{8}) = - 2 \sin (\frac{3 π}{8} - \frac{3 π}{8})$

Combine the numerators over the common denominator.

$f (\frac{π}{8}) = - 2 \sin (\frac{3 π - 3 π}{8})$

Subtract $3 π$ from $3 π$ .

$f (\frac{π}{8}) = - 2 \sin (\frac{0}{8})$

Divide $0$ by $8$ .

$f (\frac{π}{8}) = - 2 \sin (0)$

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

$f (\frac{π}{8}) = - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$f (\frac{π}{8}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{7 π}{24}$ .

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Replace the variable $x$ with $\frac{7 π}{24}$ in the expression.

$f (\frac{7 π}{24}) = - 2 \sin (3 (\frac{7 π}{24}) - \frac{3 π}{8})$

Simplify the result.

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Cancel the common factor of $3$ .

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Factor $3$ out of $24$ .

$f (\frac{7 π}{24}) = - 2 \sin (3 (\frac{7 π}{3 (8)}) - \frac{3 π}{8})$

Cancel the common factor.

$f (\frac{7 π}{24}) = - 2 \sin (3 (\frac{7 π}{3 \cdot 8}) - \frac{3 π}{8})$

Rewrite the expression.

$f (\frac{7 π}{24}) = - 2 \sin (\frac{7 π}{8} - \frac{3 π}{8})$

Combine the numerators over the common denominator.

$f (\frac{7 π}{24}) = - 2 \sin (\frac{7 π - 3 π}{8})$

Subtract $3 π$ from $7 π$ .

$f (\frac{7 π}{24}) = - 2 \sin (\frac{4 π}{8})$

Cancel the common factor of $4$ and $8$ .

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Factor $4$ out of $4 π$ .

$f (\frac{7 π}{24}) = - 2 \sin (\frac{4 (π)}{8})$

Cancel the common factors.

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Factor $4$ out of $8$ .

$f (\frac{7 π}{24}) = - 2 \sin (\frac{4 π}{4 \cdot 2})$

Cancel the common factor.

$f (\frac{7 π}{24}) = - 2 \sin (\frac{4 π}{4 \cdot 2})$

Rewrite the expression.

$f (\frac{7 π}{24}) = - 2 \sin (\frac{π}{2})$

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

$f (\frac{7 π}{24}) = - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f (\frac{7 π}{24}) = - 2$

The final answer is $- 2$ .

$- 2$

Find the point at $x = \frac{11 π}{24}$ .

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Replace the variable $x$ with $\frac{11 π}{24}$ in the expression.

$f (\frac{11 π}{24}) = - 2 \sin (3 (\frac{11 π}{24}) - \frac{3 π}{8})$

Simplify the result.

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Cancel the common factor of $3$ .

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Factor $3$ out of $24$ .

$f (\frac{11 π}{24}) = - 2 \sin (3 (\frac{11 π}{3 (8)}) - \frac{3 π}{8})$

Cancel the common factor.

$f (\frac{11 π}{24}) = - 2 \sin (3 (\frac{11 π}{3 \cdot 8}) - \frac{3 π}{8})$

Rewrite the expression.

$f (\frac{11 π}{24}) = - 2 \sin (\frac{11 π}{8} - \frac{3 π}{8})$

Combine the numerators over the common denominator.

$f (\frac{11 π}{24}) = - 2 \sin (\frac{11 π - 3 π}{8})$

Subtract $3 π$ from $11 π$ .

$f (\frac{11 π}{24}) = - 2 \sin (\frac{8 π}{8})$

Cancel the common factor of $8$ .

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Cancel the common factor.

$f (\frac{11 π}{24}) = - 2 \sin (\frac{8 π}{8})$

Divide $π$ by $1$ .

$f (\frac{11 π}{24}) = - 2 \sin (π)$

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

$f (\frac{11 π}{24}) = - 2 \sin (0)$

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

$f (\frac{11 π}{24}) = - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$f (\frac{11 π}{24}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{5 π}{8}$ .

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Replace the variable $x$ with $\frac{5 π}{8}$ in the expression.

$f (\frac{5 π}{8}) = - 2 \sin (3 (\frac{5 π}{8}) - \frac{3 π}{8})$

Simplify the result.

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Multiply $3 (\frac{5 π}{8})$ .

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Combine $3$ and $\frac{5 π}{8}$ .

$f (\frac{5 π}{8}) = - 2 \sin (\frac{3 (5 π)}{8} - \frac{3 π}{8})$

Multiply $5$ by $3$ .

$f (\frac{5 π}{8}) = - 2 \sin (\frac{15 π}{8} - \frac{3 π}{8})$

Combine the numerators over the common denominator.

$f (\frac{5 π}{8}) = - 2 \sin (\frac{15 π - 3 π}{8})$

Subtract $3 π$ from $15 π$ .

$f (\frac{5 π}{8}) = - 2 \sin (\frac{12 π}{8})$

Cancel the common factor of $12$ and $8$ .

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Factor $4$ out of $12 π$ .

$f (\frac{5 π}{8}) = - 2 \sin (\frac{4 (3 π)}{8})$

Cancel the common factors.

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Factor $4$ out of $8$ .

$f (\frac{5 π}{8}) = - 2 \sin (\frac{4 (3 π)}{4 (2)})$

Cancel the common factor.

$f (\frac{5 π}{8}) = - 2 \sin (\frac{4 (3 π)}{4 \cdot 2})$

Rewrite the expression.

$f (\frac{5 π}{8}) = - 2 \sin (\frac{3 π}{2})$

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.

$f (\frac{5 π}{8}) = - 2 (- \sin (\frac{π}{2}))$

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

$f (\frac{5 π}{8}) = - 2 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (\frac{5 π}{8}) = - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$f (\frac{5 π}{8}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{19 π}{24}$ .

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Replace the variable $x$ with $\frac{19 π}{24}$ in the expression.

$f (\frac{19 π}{24}) = - 2 \sin (3 (\frac{19 π}{24}) - \frac{3 π}{8})$

Simplify the result.

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Cancel the common factor of $3$ .

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Factor $3$ out of $24$ .

$f (\frac{19 π}{24}) = - 2 \sin (3 (\frac{19 π}{3 (8)}) - \frac{3 π}{8})$

Cancel the common factor.

$f (\frac{19 π}{24}) = - 2 \sin (3 (\frac{19 π}{3 \cdot 8}) - \frac{3 π}{8})$

Rewrite the expression.

$f (\frac{19 π}{24}) = - 2 \sin (\frac{19 π}{8} - \frac{3 π}{8})$

Combine the numerators over the common denominator.

$f (\frac{19 π}{24}) = - 2 \sin (\frac{19 π - 3 π}{8})$

Subtract $3 π$ from $19 π$ .

$f (\frac{19 π}{24}) = - 2 \sin (\frac{16 π}{8})$

Cancel the common factor of $16$ and $8$ .

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Factor $8$ out of $16 π$ .

$f (\frac{19 π}{24}) = - 2 \sin (\frac{8 (2 π)}{8})$

Cancel the common factors.

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Factor $8$ out of $8$ .

$f (\frac{19 π}{24}) = - 2 \sin (\frac{8 (2 π)}{8 (1)})$

Cancel the common factor.

$f (\frac{19 π}{24}) = - 2 \sin (\frac{8 (2 π)}{8 \cdot 1})$

Rewrite the expression.

$f (\frac{19 π}{24}) = - 2 \sin (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (\frac{19 π}{24}) = - 2 \sin (2 π)$

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

$f (\frac{19 π}{24}) = - 2 \sin (0)$

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

$f (\frac{19 π}{24}) = - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$f (\frac{19 π}{24}) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{8} & 0 \\ \frac{7 π}{24} & - 2 \\ \frac{11 π}{24} & 0 \\ \frac{5 π}{8} & 2 \\ \frac{19 π}{24} & 0 \end{array}$

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

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{8}$ ( $\frac{π}{8}$ to the right)

Vertical Shift: $0$

$\begin{array}{c} x & f (x) \\ \frac{π}{8} & 0 \\ \frac{7 π}{24} & - 2 \\ \frac{11 π}{24} & 0 \\ \frac{5 π}{8} & 2 \\ \frac{19 π}{24} & 0 \end{array}$

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