Trigonometry Examples

$y = 4 \sin (- 3 x - π)$

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.

$a = 4$

$b = - 3$

$c = π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $4 \sin (- 3 x - π)$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

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

$\frac{2 π}{| - 3 |}$

Step 3.3

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

$\frac{2 π}{3}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

Phase Shift: $\frac{π}{- 3}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{3}$ ( $\frac{π}{3}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

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

Step 6.1.2.1.1.2

Factor $3$ out of $- 3$ .

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

Step 6.1.2.1.1.3

Cancel the common factor.

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

Step 6.1.2.1.1.4

Rewrite the expression.

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

Step 6.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.1.3

Multiply $π$ by $1$ .

$f (- \frac{π}{3}) = 4 \sin (π - π)$

Step 6.1.2.2

Subtract $π$ from $π$ .

$f (- \frac{π}{3}) = 4 \sin (0)$

Step 6.1.2.3

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

$f (- \frac{π}{3}) = 4 \cdot 0$

Step 6.1.2.4

Multiply $4$ by $0$ .

$f (- \frac{π}{3}) = 0$

Step 6.1.2.5

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = - \frac{π}{6}$ .

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Step 6.2.1

Replace the variable $x$ with $- \frac{π}{6}$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

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

Step 6.2.2.1.1.2

Factor $3$ out of $- 3$ .

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

Step 6.2.2.1.1.3

Factor $3$ out of $6$ .

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

Step 6.2.2.1.1.4

Cancel the common factor.

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

Step 6.2.2.1.1.5

Rewrite the expression.

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

Step 6.2.2.1.2

Move the negative in front of the fraction.

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

Step 6.2.2.1.3

Multiply $- 1 (- \frac{π}{2})$ .

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Step 6.2.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.2.1.3.2

Multiply $\frac{π}{2}$ by $1$ .

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

Step 6.2.2.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.2.3

Combine fractions.

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Step 6.2.2.3.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.2.2.3.2

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Simplify the numerator.

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Step 6.2.2.4.1

Multiply $2$ by $- 1$ .

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

Step 6.2.2.4.2

Subtract $2 π$ from $π$ .

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

Step 6.2.2.5

Move the negative in front of the fraction.

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

Step 6.2.2.6

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

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

Step 6.2.2.7

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.

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

Step 6.2.2.8

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

$f (- \frac{π}{6}) = 4 (- 1 \cdot 1)$

Step 6.2.2.9

Multiply $4 (- 1 \cdot 1)$ .

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Step 6.2.2.9.1

Multiply $- 1$ by $1$ .

$f (- \frac{π}{6}) = 4 \cdot - 1$

Step 6.2.2.9.2

Multiply $4$ by $- 1$ .

$f (- \frac{π}{6}) = - 4$

Step 6.2.2.10

The final answer is $- 4$ .

$- 4$

Step 6.3

Find the point at $x = 0$ .

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Step 6.3.1

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

$f (0) = 4 \sin (- 3 \cdot 0 - π)$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $- 3$ by $0$ .

$f (0) = 4 \sin (0 - π)$

Step 6.3.2.2

Subtract $π$ from $0$ .

$f (0) = 4 \sin (- π)$

Step 6.3.2.3

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

$f (0) = 4 \sin (0)$

Step 6.3.2.4

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

$f (0) = 4 \cdot 0$

Step 6.3.2.5

Multiply $4$ by $0$ .

$f (0) = 0$

Step 6.3.2.6

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{π}{6}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{π}{6}$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.1

Factor $3$ out of $- 3$ .

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

Step 6.4.2.1.1.2

Factor $3$ out of $6$ .

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

Step 6.4.2.1.1.3

Cancel the common factor.

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

Step 6.4.2.1.1.4

Rewrite the expression.

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

Step 6.4.2.1.2

Rewrite $- 1 \frac{π}{2}$ as $- \frac{π}{2}$ .

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

Step 6.4.2.2

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.4.2.3

Combine fractions.

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Step 6.4.2.3.1

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.4.2.3.2

Combine the numerators over the common denominator.

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

Step 6.4.2.4

Simplify the numerator.

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Step 6.4.2.4.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.4.2

Subtract $2 π$ from $- π$ .

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

Step 6.4.2.5

Move the negative in front of the fraction.

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

Step 6.4.2.6

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

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

Step 6.4.2.7

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

$f (\frac{π}{6}) = 4 \cdot 1$

Step 6.4.2.8

Multiply $4$ by $1$ .

$f (\frac{π}{6}) = 4$

Step 6.4.2.9

The final answer is $4$ .

$4$

Step 6.5

Find the point at $x = \frac{π}{3}$ .

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Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1.1

Factor $3$ out of $- 3$ .

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

Step 6.5.2.1.1.2

Cancel the common factor.

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

Step 6.5.2.1.1.3

Rewrite the expression.

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

Step 6.5.2.1.2

Rewrite $- 1 π$ as $- π$ .

$f (\frac{π}{3}) = 4 \sin (- π - π)$

Step 6.5.2.2

Subtract $π$ from $- π$ .

$f (\frac{π}{3}) = 4 \sin (- 2 π)$

Step 6.5.2.3

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

$f (\frac{π}{3}) = 4 \sin (0)$

Step 6.5.2.4

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

$f (\frac{π}{3}) = 4 \cdot 0$

Step 6.5.2.5

Multiply $4$ by $0$ .

$f (\frac{π}{3}) = 0$

Step 6.5.2.6

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $4$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{3}$ ( $\frac{π}{3}$ to the left)

Vertical Shift: None

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

Step 8