Algebra Examples

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

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.

$a = 2$

$b = - 3$

$c = \frac{π}{2}$

$d = 2$

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 $2 \sin (- 3 x - \frac{π}{2})$ .

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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 $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

Step 3.1.3

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

$\frac{2 π}{3}$

Step 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 $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

Step 3.2.3

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

$\frac{2 π}{3}$

Step 3.3

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

$\frac{2 π}{3}$

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: $\frac{\frac{π}{2}}{- 3}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{- 3}$

Step 4.4

Move the negative in front of the fraction.

Phase Shift: $\frac{π}{2} \cdot (- \frac{1}{3})$

Step 4.5

Multiply $\frac{π}{2} (- \frac{1}{3})$ .

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Step 4.5.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

Phase Shift: $- \frac{π}{2 \cdot 3}$

Step 4.5.2

Multiply $2$ by $3$ .

Phase Shift: $- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ 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 $x = - \frac{π}{6}$ .

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Step 6.1.1

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

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

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

Simplify each term.

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Step 6.1.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1.1.1

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

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

Step 6.1.2.1.1.1.2

Factor $3$ out of $- 3$ .

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

Step 6.1.2.1.1.1.3

Factor $3$ out of $6$ .

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

Step 6.1.2.1.1.1.4

Cancel the common factor.

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

Step 6.1.2.1.1.1.5

Rewrite the expression.

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

Step 6.1.2.1.1.2

Move the negative in front of the fraction.

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

Step 6.1.2.1.1.3

Multiply $- 1 (- \frac{π}{2})$ .

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Step 6.1.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.1.1.3.2

Multiply $\frac{π}{2}$ by $1$ .

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

Step 6.1.2.1.2

Combine the numerators over the common denominator.

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

Step 6.1.2.1.3

Subtract $π$ from $π$ .

$f (- \frac{π}{6}) = 2 \sin (\frac{0}{2}) + 2$

Step 6.1.2.1.4

Divide $0$ by $2$ .

$f (- \frac{π}{6}) = 2 \sin (0) + 2$

Step 6.1.2.1.5

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

$f (- \frac{π}{6}) = 2 \cdot 0 + 2$

Step 6.1.2.1.6

Multiply $2$ by $0$ .

$f (- \frac{π}{6}) = 0 + 2$

Step 6.1.2.2

Add $0$ and $2$ .

$f (- \frac{π}{6}) = 2$

Step 6.1.2.3

The final answer is $2$ .

$2$

Step 6.2

Find the point at $x = 0$ .

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Step 6.2.1

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

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

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

Multiply $- 3$ by $0$ .

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

Step 6.2.2.1.2

Subtract $\frac{π}{2}$ from $0$ .

$f (0) = 2 \sin (- \frac{π}{2}) + 2$

Step 6.2.2.1.3

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

$f (0) = 2 \sin (\frac{3 π}{2}) + 2$

Step 6.2.2.1.4

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.

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

Step 6.2.2.1.5

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

$f (0) = 2 (- 1 \cdot 1) + 2$

Step 6.2.2.1.6

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

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Step 6.2.2.1.6.1

Multiply $- 1$ by $1$ .

$f (0) = 2 \cdot - 1 + 2$

Step 6.2.2.1.6.2

Multiply $2$ by $- 1$ .

$f (0) = - 2 + 2$

Step 6.2.2.2

Add $- 2$ and $2$ .

$f (0) = 0$

Step 6.2.2.3

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{π}{6}$ .

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Step 6.3.1

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

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

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

Simplify each term.

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Step 6.3.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.1.1

Factor $3$ out of $- 3$ .

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

Step 6.3.2.1.1.1.2

Factor $3$ out of $6$ .

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

Step 6.3.2.1.1.1.3

Cancel the common factor.

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

Step 6.3.2.1.1.1.4

Rewrite the expression.

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

Step 6.3.2.1.1.2

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

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

Step 6.3.2.1.2

Combine the numerators over the common denominator.

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

Step 6.3.2.1.3

Subtract $π$ from $- π$ .

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

Step 6.3.2.1.4

Cancel the common factor of $- 2$ and $2$ .

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Step 6.3.2.1.4.1

Factor $2$ out of $- 2 π$ .

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

Step 6.3.2.1.4.2

Cancel the common factors.

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Step 6.3.2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.3.2.1.4.2.2

Cancel the common factor.

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

Step 6.3.2.1.4.2.3

Rewrite the expression.

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

Step 6.3.2.1.4.2.4

Divide $- π$ by $1$ .

$f (\frac{π}{6}) = 2 \sin (- π) + 2$

Step 6.3.2.1.5

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

$f (\frac{π}{6}) = 2 \sin (0) + 2$

Step 6.3.2.1.6

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

$f (\frac{π}{6}) = 2 \cdot 0 + 2$

Step 6.3.2.1.7

Multiply $2$ by $0$ .

$f (\frac{π}{6}) = 0 + 2$

Step 6.3.2.2

Add $0$ and $2$ .

$f (\frac{π}{6}) = 2$

Step 6.3.2.3

The final answer is $2$ .

$2$

Step 6.4

Find the point at $x = \frac{π}{3}$ .

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Step 6.4.1

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

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

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

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.1.1

Factor $3$ out of $- 3$ .

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

Step 6.4.2.1.1.1.2

Cancel the common factor.

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

Step 6.4.2.1.1.1.3

Rewrite the expression.

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

Step 6.4.2.1.1.2

Rewrite $- 1 π$ as $- π$ .

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

Step 6.4.2.1.2

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

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

Step 6.4.2.1.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 6.4.2.1.4

Combine the numerators over the common denominator.

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

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.1.5.2

Subtract $π$ from $- 2 π$ .

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

Step 6.4.2.1.6

Move the negative in front of the fraction.

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

Step 6.4.2.1.7

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

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

Step 6.4.2.1.8

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

$f (\frac{π}{3}) = 2 \cdot 1 + 2$

Step 6.4.2.1.9

Multiply $2$ by $1$ .

$f (\frac{π}{3}) = 2 + 2$

Step 6.4.2.2

Add $2$ and $2$ .

$f (\frac{π}{3}) = 4$

Step 6.4.2.3

The final answer is $4$ .

$4$

Step 6.5

Find the point at $x = \frac{π}{2}$ .

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Step 6.5.1

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

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

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

Simplify each term.

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Step 6.5.2.1.1.1

Combine $- 3$ and $\frac{π}{2}$ .

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

Step 6.5.2.1.1.2

Move the negative in front of the fraction.

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

Step 6.5.2.1.2

Combine the numerators over the common denominator.

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

Step 6.5.2.1.3

Subtract $π$ from $- 3 π$ .

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

Step 6.5.2.1.4

Cancel the common factor of $- 4$ and $2$ .

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Step 6.5.2.1.4.1

Factor $2$ out of $- 4 π$ .

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

Step 6.5.2.1.4.2

Cancel the common factors.

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Step 6.5.2.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.5.2.1.4.2.2

Cancel the common factor.

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

Step 6.5.2.1.4.2.3

Rewrite the expression.

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

Step 6.5.2.1.4.2.4

Divide $- 2 π$ by $1$ .

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

Step 6.5.2.1.5

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

$f (\frac{π}{2}) = 2 \sin (0) + 2$

Step 6.5.2.1.6

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

$f (\frac{π}{2}) = 2 \cdot 0 + 2$

Step 6.5.2.1.7

Multiply $2$ by $0$ .

$f (\frac{π}{2}) = 0 + 2$

Step 6.5.2.2

Add $0$ and $2$ .

$f (\frac{π}{2}) = 2$

Step 6.5.2.3

The final answer is $2$ .

$2$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: $2$

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

Step 8