Graph y=-1/2cos(2x+2/3pi)

$y = - \frac{1}{2} \cos (2 x + \frac{2}{3} π)$

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.

$a = - \frac{1}{2}$

$b = 2$

$c = - \frac{2 π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Step 3

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Step 4

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

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: $\frac{- \frac{2 π}{3}}{2}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $- 2 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{- π}{3}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $π$

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

Vertical Shift: None

Step 6

Select a few points to graph.

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

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

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

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply $2 (- \frac{π}{3})$ .

Multiply $- 1$ by $2$ .

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

Combine $- 2$ and $\frac{π}{3}$ .

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

Move the negative in front of the fraction.

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

Combine the numerators over the common denominator.

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

Add $- 2 π$ and $2 π$ .

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

Divide $0$ by $3$ .

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

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

$f (- \frac{π}{3}) = - \frac{1}{2}$

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Find the point at $x = - \frac{π}{12}$ .

Replace the variable $x$ with $- \frac{π}{12}$ in the expression.

$f (- \frac{π}{12}) = - \frac{\cos (2 (- \frac{π}{12}) + \frac{2 π}{3})}{2}$

Simplify the result.

Simplify the numerator.

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{π}{12}$ into the numerator.

$f (- \frac{π}{12}) = - \frac{\cos (2 (\frac{- π}{12}) + \frac{2 π}{3})}{2}$

Factor $2$ out of $12$ .

$f (- \frac{π}{12}) = - \frac{\cos (2 (\frac{- π}{2 (6)}) + \frac{2 π}{3})}{2}$

Cancel the common factor.

$f (- \frac{π}{12}) = - \frac{\cos (2 (\frac{- π}{2 \cdot 6}) + \frac{2 π}{3})}{2}$

Rewrite the expression.

$f (- \frac{π}{12}) = - \frac{\cos (\frac{- π}{6} + \frac{2 π}{3})}{2}$

Move the negative in front of the fraction.

$f (- \frac{π}{12}) = - \frac{\cos (- \frac{π}{6} + \frac{2 π}{3})}{2}$

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{π}{12}) = - \frac{\cos (- \frac{π}{6} + \frac{2 π}{3} \cdot \frac{2}{2})}{2}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$f (- \frac{π}{12}) = - \frac{\cos (- \frac{π}{6} + \frac{2 π \cdot 2}{3 \cdot 2})}{2}$

Multiply $3$ by $2$ .

$f (- \frac{π}{12}) = - \frac{\cos (- \frac{π}{6} + \frac{2 π \cdot 2}{6})}{2}$

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = - \frac{\cos (\frac{- π + 2 π \cdot 2}{6})}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$f (- \frac{π}{12}) = - \frac{\cos (\frac{- π + 4 π}{6})}{2}$

Add $- π$ and $4 π$ .

$f (- \frac{π}{12}) = - \frac{\cos (\frac{3 π}{6})}{2}$

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 π$ .

$f (- \frac{π}{12}) = - \frac{\cos (\frac{3 (π)}{6})}{2}$

Cancel the common factors.

Factor $3$ out of $6$ .

$f (- \frac{π}{12}) = - \frac{\cos (\frac{3 π}{3 \cdot 2})}{2}$

Cancel the common factor.

$f (- \frac{π}{12}) = - \frac{\cos (\frac{3 π}{3 \cdot 2})}{2}$

Rewrite the expression.

$f (- \frac{π}{12}) = - \frac{\cos (\frac{π}{2})}{2}$

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

$f (- \frac{π}{12}) = - \frac{0}{2}$

Simplify the expression.

Divide $0$ by $2$ .

$f (- \frac{π}{12}) = - 0$

Multiply $- 1$ by $0$ .

$f (- \frac{π}{12}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{π}{6}$ .

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

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

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Add $π$ and $2 π$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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.

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

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

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

Multiply $- 1$ by $1$ .

$f (\frac{π}{6}) = - \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (\frac{π}{6}) = \frac{1}{2}$

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Find the point at $x = \frac{5 π}{12}$ .

Replace the variable $x$ with $\frac{5 π}{12}$ in the expression.

$f (\frac{5 π}{12}) = - \frac{\cos (2 (\frac{5 π}{12}) + \frac{2 π}{3})}{2}$

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$f (\frac{5 π}{12}) = - \frac{\cos (2 (\frac{5 π}{2 (6)}) + \frac{2 π}{3})}{2}$

Cancel the common factor.

$f (\frac{5 π}{12}) = - \frac{\cos (2 (\frac{5 π}{2 \cdot 6}) + \frac{2 π}{3})}{2}$

Rewrite the expression.

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π}{6} + \frac{2 π}{3})}{2}$

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π}{6} + \frac{2 π}{3} \cdot \frac{2}{2})}{2}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π}{6} + \frac{2 π \cdot 2}{3 \cdot 2})}{2}$

Multiply $3$ by $2$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π}{6} + \frac{2 π \cdot 2}{6})}{2}$

Combine the numerators over the common denominator.

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π + 2 π \cdot 2}{6})}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{5 π + 4 π}{6})}{2}$

Add $5 π$ and $4 π$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{9 π}{6})}{2}$

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9 π$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{3 (3 π)}{6})}{2}$

Cancel the common factors.

Factor $3$ out of $6$ .

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{3 (3 π)}{3 (2)})}{2}$

Cancel the common factor.

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{3 (3 π)}{3 \cdot 2})}{2}$

Rewrite the expression.

$f (\frac{5 π}{12}) = - \frac{\cos (\frac{3 π}{2})}{2}$

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

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

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

$f (\frac{5 π}{12}) = - \frac{0}{2}$

Simplify the expression.

Divide $0$ by $2$ .

$f (\frac{5 π}{12}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{5 π}{12}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Simplify the numerator.

Multiply $2 \frac{2 π}{3}$ .

Combine $2$ and $\frac{2 π}{3}$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Add $4 π$ and $2 π$ .

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

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 π$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

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

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

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

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

$f (\frac{2 π}{3}) = - \frac{1}{2}$

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{3} & - \frac{1}{2} \\ - \frac{π}{12} & 0 \\ \frac{π}{6} & \frac{1}{2} \\ \frac{5 π}{12} & 0 \\ \frac{2 π}{3} & - \frac{1}{2} \end{array}$

Step 7

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

Amplitude: $\frac{1}{2}$

Period: $π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{3} & - \frac{1}{2} \\ - \frac{π}{12} & 0 \\ \frac{π}{6} & \frac{1}{2} \\ \frac{5 π}{12} & 0 \\ \frac{2 π}{3} & - \frac{1}{2} \end{array}$

Step 8

Graph y=-1/2cos(2x+2/3pi)

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