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

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

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.

$a = - \frac{1}{3}$

$b = \frac{1}{2}$

$c = - \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{3}$

Step 3

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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: $\frac{- \frac{π}{3}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{3} \cdot 2$

Multiply $- \frac{π}{3} \cdot 2$ .

Multiply $2$ by $- 1$ .

Phase Shift: $- 2 \frac{π}{3}$

Combine $- 2$ and $\frac{π}{3}$ .

Phase Shift: $\frac{- 2 π}{3}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{2 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{3}$

Period: $4 π$

Phase Shift: $- \frac{2 π}{3}$ ( $\frac{2 π}{3}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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

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

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

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $- 2 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Combine the numerators over the common denominator.

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

Add $- π$ and $π$ .

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

Divide $0$ by $3$ .

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

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

$f (- \frac{2 π}{3}) = - \frac{1}{3}$

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Find the point at $x = \frac{π}{3}$ .

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

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

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $3$ by $2$ .

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $π$ and $2 π$ .

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

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

Factor $3$ out of $3 π$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$f (\frac{π}{3}) = - \frac{0}{3}$

Simplify the expression.

Divide $0$ by $3$ .

$f (\frac{π}{3}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{π}{3}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Add $2 π$ and $π$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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.

$f (\frac{4 π}{3}) = - \frac{- \cos (0)}{3}$

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

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

Multiply $- 1$ by $1$ .

$f (\frac{4 π}{3}) = - \frac{- 1}{3}$

Move the negative in front of the fraction.

$f (\frac{4 π}{3}) = \frac{1}{3}$

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Find the point at $x = \frac{7 π}{3}$ .

Replace the variable $x$ with $\frac{7 π}{3}$ in the expression.

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{\frac{7 π}{3}}{2} + \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{3} \cdot \frac{1}{2} + \frac{π}{3})}{3}$

Multiply $\frac{7 π}{3} \cdot \frac{1}{2}$ .

Multiply $\frac{7 π}{3}$ by $\frac{1}{2}$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{3 \cdot 2} + \frac{π}{3})}{3}$

Multiply $3$ by $2$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{6} + \frac{π}{3})}{3}$

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

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2})}{3}$

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

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

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2})}{3}$

Multiply $3$ by $2$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π}{6} + \frac{π \cdot 2}{6})}{3}$

Combine the numerators over the common denominator.

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π + π \cdot 2}{6})}{3}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{7 π + 2 \cdot π}{6})}{3}$

Add $7 π$ and $2 π$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{9 π}{6})}{3}$

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

Factor $3$ out of $9 π$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{3 (3 π)}{6})}{3}$

Cancel the common factors.

Factor $3$ out of $6$ .

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{3 (3 π)}{3 (2)})}{3}$

Cancel the common factor.

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{3 (3 π)}{3 \cdot 2})}{3}$

Rewrite the expression.

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{3 π}{2})}{3}$

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

$f (\frac{7 π}{3}) = - \frac{\cos (\frac{π}{2})}{3}$

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

$f (\frac{7 π}{3}) = - \frac{0}{3}$

Simplify the expression.

Divide $0$ by $3$ .

$f (\frac{7 π}{3}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{7 π}{3}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{10 π}{3}$ .

Replace the variable $x$ with $\frac{10 π}{3}$ in the expression.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{\frac{10 π}{3}}{2} + \frac{π}{3})}{3}$

Simplify the result.

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{10 π}{3} \cdot \frac{1}{2} + \frac{π}{3})}{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $10 π$ .

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{2 (5 π)}{3} \cdot \frac{1}{2} + \frac{π}{3})}{3}$

Cancel the common factor.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{2 (5 π)}{3} \cdot \frac{1}{2} + \frac{π}{3})}{3}$

Rewrite the expression.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{5 π}{3} + \frac{π}{3})}{3}$

Combine the numerators over the common denominator.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{5 π + π}{3})}{3}$

Add $5 π$ and $π$ .

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{6 π}{3})}{3}$

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

Factor $3$ out of $6 π$ .

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{3 (2 π)}{3})}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{3 (2 π)}{3 (1)})}{3}$

Cancel the common factor.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{3 (2 π)}{3 \cdot 1})}{3}$

Rewrite the expression.

$f (\frac{10 π}{3}) = - \frac{\cos (\frac{2 π}{1})}{3}$

Divide $2 π$ by $1$ .

$f (\frac{10 π}{3}) = - \frac{\cos (2 π)}{3}$

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

$f (\frac{10 π}{3}) = - \frac{\cos (0)}{3}$

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

$f (\frac{10 π}{3}) = - \frac{1}{3}$

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{2 π}{3} & - \frac{1}{3} \\ \frac{π}{3} & 0 \\ \frac{4 π}{3} & \frac{1}{3} \\ \frac{7 π}{3} & 0 \\ \frac{10 π}{3} & - \frac{1}{3} \end{array}$

Step 7

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

Amplitude: $\frac{1}{3}$

Period: $4 π$

Phase Shift: $- \frac{2 π}{3}$ ( $\frac{2 π}{3}$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{2 π}{3} & - \frac{1}{3} \\ \frac{π}{3} & 0 \\ \frac{4 π}{3} & \frac{1}{3} \\ \frac{7 π}{3} & 0 \\ \frac{10 π}{3} & - \frac{1}{3} \end{array}$

Step 8

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

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