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

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

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.

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

$b = \frac{π}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

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: $\frac{0}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{π})$

Multiply $0$ by $\frac{3}{π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

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

Period: $6$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

Find the point at $x = 0$ .

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

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

Simplify the result.

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

Factor $3$ out of $π (0)$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (0)$ by $1$ .

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

Simplify the numerator.

Multiply $π$ by $0$ .

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

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

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

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

$- \frac{1}{2}$

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

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

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

Simplify the result.

Combine $π$ and $\frac{3}{2}$ .

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

Move $3$ to the left of $π$ .

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

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the expression.

Divide $0$ by $2$ .

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

Multiply $- 1$ by $0$ .

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

The final answer is $0$ .

$0$

Find the point at $x = 3$ .

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

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

Simplify the result.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (3) = - \frac{\cos (π)}{2}$

Simplify the numerator.

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.

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

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

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

Multiply $- 1$ by $1$ .

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

Move the negative in front of the fraction.

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

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

$\frac{1}{2}$

Find the point at $x = \frac{9}{2}$ .

Replace the variable $x$ with $\frac{9}{2}$ in the expression.

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

Simplify the result.

Combine $π$ and $\frac{9}{2}$ .

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

Move $9$ to the left of $π$ .

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

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

$f (\frac{9}{2}) = - \frac{0}{2}$

Simplify the expression.

Divide $0$ by $2$ .

$f (\frac{9}{2}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{9}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 6$ .

Replace the variable $x$ with $6$ in the expression.

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

Simplify the result.

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

Factor $3$ out of $π (6)$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (2)$ by $1$ .

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

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

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

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

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

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

$- \frac{1}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & - \frac{1}{2} \\ \frac{3}{2} & 0 \\ 3 & \frac{1}{2} \\ \frac{9}{2} & 0 \\ 6 & - \frac{1}{2} \end{array}$

Step 7

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

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

Period: $6$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & - \frac{1}{2} \\ \frac{3}{2} & 0 \\ 3 & \frac{1}{2} \\ \frac{9}{2} & 0 \\ 6 & - \frac{1}{2} \end{array}$

Step 8

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

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