Trigonometry Examples

$y = - 2 \sin (\frac{x}{3} + π) + 2$

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.

$a = - 2$

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

$c = - π$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

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

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

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

$\frac{2 π}{| \frac{1}{3} |}$

Step 3.1.3

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

$\frac{2 π}{\frac{1}{3}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 3.1.5

Multiply $3$ by $2$ .

$6 π$

Step 3.2

Find the period of $2$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

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

$\frac{2 π}{| \frac{1}{3} |}$

Step 3.2.3

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

$\frac{2 π}{\frac{1}{3}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 3.2.5

Multiply $3$ by $2$ .

$6 π$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$6 π$

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: $\frac{- π}{\frac{1}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- π \cdot 3$

Step 4.4

Multiply $3$ by $- 1$ .

Phase Shift: $- 3 π$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $6 π$

Phase Shift: $- 3 π$ ( $3 π$ to the left)

Vertical Shift: $2$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - 3 π$ .

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Step 6.1.1

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

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

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 $- 3$ and $3$ .

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Step 6.1.2.1.1.1

Factor $3$ out of $- 3 π$ .

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

Step 6.1.2.1.1.2

Cancel the common factors.

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Step 6.1.2.1.1.2.1

Factor $3$ out of $3$ .

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

Step 6.1.2.1.1.2.2

Cancel the common factor.

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

Step 6.1.2.1.1.2.3

Rewrite the expression.

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

Step 6.1.2.1.1.2.4

Divide $- π$ by $1$ .

$f (- 3 π) = - 2 \sin (- π + π) + 2$

Step 6.1.2.1.2

Add $- π$ and $π$ .

$f (- 3 π) = - 2 \sin (0) + 2$

Step 6.1.2.1.3

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

$f (- 3 π) = - 2 \cdot 0 + 2$

Step 6.1.2.1.4

Multiply $- 2$ by $0$ .

$f (- 3 π) = 0 + 2$

Step 6.1.2.2

Add $0$ and $2$ .

$f (- 3 π) = 2$

Step 6.1.2.3

The final answer is $2$ .

$2$

Step 6.2

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

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Step 6.2.1

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

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

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

Simplify each term.

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Step 6.2.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.2.1

Move the leading negative in $- \frac{3 π}{2}$ into the numerator.

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

Step 6.2.2.1.1.2.2

Factor $3$ out of $- 3 π$ .

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

Step 6.2.2.1.1.2.3

Cancel the common factor.

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

Step 6.2.2.1.1.2.4

Rewrite the expression.

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

Step 6.2.2.1.1.3

Move the negative in front of the fraction.

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

Step 6.2.2.1.2

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

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

Step 6.2.2.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.2.1.4

Combine the numerators over the common denominator.

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

Step 6.2.2.1.5

Simplify the numerator.

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Step 6.2.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 6.2.2.1.5.2

Add $- π$ and $2 π$ .

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

Step 6.2.2.1.6

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

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

Step 6.2.2.1.7

Multiply $- 2$ by $1$ .

$f (- \frac{3 π}{2}) = - 2 + 2$

Step 6.2.2.2

Add $- 2$ and $2$ .

$f (- \frac{3 π}{2}) = 0$

Step 6.2.2.3

The final answer is $0$ .

$0$

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.

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Divide $0$ by $3$ .

$f (0) = - 2 \sin (0 + π) + 2$

Step 6.3.2.1.2

Add $0$ and $π$ .

$f (0) = - 2 \sin (π) + 2$

Step 6.3.2.1.3

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

$f (0) = - 2 \sin (0) + 2$

Step 6.3.2.1.4

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

$f (0) = - 2 \cdot 0 + 2$

Step 6.3.2.1.5

Multiply $- 2$ by $0$ .

$f (0) = 0 + 2$

Step 6.3.2.2

Add $0$ and $2$ .

$f (0) = 2$

Step 6.3.2.3

The final answer is $2$ .

$2$

Step 6.4

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

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Step 6.4.1

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

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

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

Simplify each term.

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Step 6.4.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.2.1

Factor $3$ out of $3 π$ .

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

Step 6.4.2.1.1.2.2

Cancel the common factor.

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

Step 6.4.2.1.1.2.3

Rewrite the expression.

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

Step 6.4.2.1.2

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

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

Step 6.4.2.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.4.2.1.4

Combine the numerators over the common denominator.

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

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 6.4.2.1.5.2

Add $π$ and $2 π$ .

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

Step 6.4.2.1.6

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.

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

Step 6.4.2.1.7

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

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

Step 6.4.2.1.8

Multiply $- 2 (- 1 \cdot 1)$ .

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Step 6.4.2.1.8.1

Multiply $- 1$ by $1$ .

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

Step 6.4.2.1.8.2

Multiply $- 2$ by $- 1$ .

$f (\frac{3 π}{2}) = 2 + 2$

Step 6.4.2.2

Add $2$ and $2$ .

$f (\frac{3 π}{2}) = 4$

Step 6.4.2.3

The final answer is $4$ .

$4$

Step 6.5

Find the point at $x = 3 π$ .

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Step 6.5.1

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

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

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

Cancel the common factor.

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

Step 6.5.2.1.1.2

Divide $π$ by $1$ .

$f (3 π) = - 2 \sin (π + π) + 2$

Step 6.5.2.1.2

Add $π$ and $π$ .

$f (3 π) = - 2 \sin (2 π) + 2$

Step 6.5.2.1.3

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

$f (3 π) = - 2 \sin (0) + 2$

Step 6.5.2.1.4

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

$f (3 π) = - 2 \cdot 0 + 2$

Step 6.5.2.1.5

Multiply $- 2$ by $0$ .

$f (3 π) = 0 + 2$

Step 6.5.2.2

Add $0$ and $2$ .

$f (3 π) = 2$

Step 6.5.2.3

The final answer is $2$ .

$2$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $2$

Period: $6 π$

Phase Shift: $- 3 π$ ( $3 π$ to the left)

Vertical Shift: $2$

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

Step 8