Precalculus Examples

$y = - 2 \cos (\frac{2}{3} x)$

Step 1

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

$a = - 2$

$b = \frac{2}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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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 $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

Step 3.3

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{3}{2}$

Step 3.5.3

Rewrite the expression.

$π \cdot 3$

Step 3.6

Move $3$ to the left of $π$ .

$3 π$

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: $\frac{0}{\frac{2}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{2})$

Step 4.4

Multiply $0$ by $\frac{3}{2}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $3 π$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Cancel the common factor of $0$ and $3$ .

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Step 6.1.2.1.1

Factor $3$ out of $2 (0)$ .

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

Step 6.1.2.1.2

Cancel the common factors.

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Step 6.1.2.1.2.1

Factor $3$ out of $3$ .

$f (0) = - 2 \cos (\frac{3 (2 \cdot (0))}{3 (1)})$

Step 6.1.2.1.2.2

Cancel the common factor.

$f (0) = - 2 \cos (\frac{3 (2 \cdot (0))}{3 \cdot 1})$

Step 6.1.2.1.2.3

Rewrite the expression.

$f (0) = - 2 \cos (\frac{2 \cdot (0)}{1})$

Step 6.1.2.1.2.4

Divide $2 \cdot (0)$ by $1$ .

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

Step 6.1.2.2

Multiply $2$ by $0$ .

$f (0) = - 2 \cos (0)$

Step 6.1.2.3

The exact value of $\cos (0)$ is $1$ .

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

Step 6.1.2.4

Multiply $- 2$ by $1$ .

$f (0) = - 2$

Step 6.1.2.5

The final answer is $- 2$ .

$- 2$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Combine $2$ and $\frac{3 π}{4}$ .

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

Step 6.2.2.2

Multiply $2$ by $3$ .

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

Step 6.2.2.3

Reduce the expression $\frac{6 π}{4}$ by cancelling the common factors.

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Step 6.2.2.3.1

Factor $2$ out of $6 π$ .

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

Step 6.2.2.3.2

Factor $2$ out of $4$ .

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

Step 6.2.2.3.3

Cancel the common factor.

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

Step 6.2.2.3.4

Rewrite the expression.

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

Step 6.2.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2.2.5

Cancel the common factor of $3$ .

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Step 6.2.2.5.1

Factor $3$ out of $3 π$ .

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

Step 6.2.2.5.2

Cancel the common factor.

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

Step 6.2.2.5.3

Rewrite the expression.

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

Step 6.2.2.6

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{3 π}{4}) = - 2 \cdot 0$

Step 6.2.2.7

Multiply $- 2$ by $0$ .

$f (\frac{3 π}{4}) = 0$

Step 6.2.2.8

The final answer is $0$ .

$0$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Combine $2$ and $\frac{3 π}{2}$ .

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

Step 6.3.2.2

Multiply $2$ by $3$ .

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

Step 6.3.2.3

Reduce the expression by cancelling the common factors.

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Step 6.3.2.3.1

Reduce the expression $\frac{6 π}{2}$ by cancelling the common factors.

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Step 6.3.2.3.1.1

Factor $2$ out of $6 π$ .

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

Step 6.3.2.3.1.2

Factor $2$ out of $2$ .

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

Step 6.3.2.3.1.3

Cancel the common factor.

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

Step 6.3.2.3.1.4

Rewrite the expression.

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

Step 6.3.2.3.2

Divide $3 π$ by $1$ .

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

Step 6.3.2.4

Cancel the common factor of $3$ .

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Step 6.3.2.4.1

Cancel the common factor.

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

Step 6.3.2.4.2

Divide $π$ by $1$ .

$f (\frac{3 π}{2}) = - 2 \cos (π)$

Step 6.3.2.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 6.3.2.6

The exact value of $\cos (0)$ is $1$ .

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

Step 6.3.2.7

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

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Step 6.3.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{2}) = - 2 \cdot - 1$

Step 6.3.2.7.2

Multiply $- 2$ by $- 1$ .

$f (\frac{3 π}{2}) = 2$

Step 6.3.2.8

The final answer is $2$ .

$2$

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Combine $2$ and $\frac{9 π}{4}$ .

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

Step 6.4.2.2

Multiply $2$ by $9$ .

$f (\frac{9 π}{4}) = - 2 \cos (\frac{\frac{18 π}{4}}{3})$

Step 6.4.2.3

Reduce the expression $\frac{18 π}{4}$ by cancelling the common factors.

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Step 6.4.2.3.1

Factor $2$ out of $18 π$ .

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

Step 6.4.2.3.2

Factor $2$ out of $4$ .

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

Step 6.4.2.3.3

Cancel the common factor.

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

Step 6.4.2.3.4

Rewrite the expression.

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

Step 6.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 π}{4}) = - 2 \cos (\frac{9 π}{2} \cdot \frac{1}{3})$

Step 6.4.2.5

Cancel the common factor of $3$ .

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Step 6.4.2.5.1

Factor $3$ out of $9 π$ .

$f (\frac{9 π}{4}) = - 2 \cos (\frac{3 (3 π)}{2} \cdot \frac{1}{3})$

Step 6.4.2.5.2

Cancel the common factor.

$f (\frac{9 π}{4}) = - 2 \cos (\frac{3 (3 π)}{2} \cdot \frac{1}{3})$

Step 6.4.2.5.3

Rewrite the expression.

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

Step 6.4.2.6

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

$f (\frac{9 π}{4}) = - 2 \cos (\frac{π}{2})$

Step 6.4.2.7

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{9 π}{4}) = - 2 \cdot 0$

Step 6.4.2.8

Multiply $- 2$ by $0$ .

$f (\frac{9 π}{4}) = 0$

Step 6.4.2.9

The final answer is $0$ .

$0$

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.

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $2 (π)$ by $1$ .

$f (3 π) = - 2 \cos (2 (π))$

Step 6.5.2.2

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

$f (3 π) = - 2 \cos (0)$

Step 6.5.2.3

The exact value of $\cos (0)$ is $1$ .

$f (3 π) = - 2 \cdot 1$

Step 6.5.2.4

Multiply $- 2$ by $1$ .

$f (3 π) = - 2$

Step 6.5.2.5

The final answer is $- 2$ .

$- 2$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $2$

Period: $3 π$

Phase Shift: None

Vertical Shift: None

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

Step 8