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

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

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

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

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

$c = \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

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

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{2}{3} |}$

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

$\frac{2 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Cancel the common factor.

$2 π \frac{3}{2}$

Rewrite the expression.

$π \cdot 3$

Move $3$ to the left of $π$ .

$3 π$

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: $\frac{\frac{π}{2}}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $2$ by $2$ .

Phase Shift: $\frac{π \cdot 3}{4}$

Move $3$ to the left of $π$ .

Phase Shift: $\frac{3 π}{4}$

List the properties of the trigonometric function.

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

Period: $3 π$

Phase Shift: $\frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the right)

Vertical Shift: None

Select a few points to graph.

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

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

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

Simplify the result.

Combine $2$ and $\frac{3 π}{4}$ .

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

Multiply $2$ by $3$ .

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

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

Factor $2$ out of $6 π$ .

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Subtract $π$ from $π$ .

$f (\frac{3 π}{4}) = \frac{\cos (\frac{0}{2})}{2}$

Divide $0$ by $2$ .

$f (\frac{3 π}{4}) = \frac{\cos (0)}{2}$

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

$f (\frac{3 π}{4}) = \frac{1}{2}$

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

$\frac{1}{2}$

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

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

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

Simplify the result.

Combine $2$ and $\frac{3 π}{2}$ .

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

Multiply $2$ by $3$ .

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

Reduce the expression by cancelling the common factors.

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

Factor $2$ out of $6 π$ .

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

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3 π$ by $1$ .

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

Simplify the numerator.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Subtract $π$ from $2 π$ .

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

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

$f (\frac{3 π}{2}) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{3 π}{2}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Combine $2$ and $\frac{9 π}{4}$ .

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

Multiply $2$ by $9$ .

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

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

Factor $2$ out of $18 π$ .

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Subtract $π$ from $3 π$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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.

$f (\frac{9 π}{4}) = \frac{- \cos (0)}{2}$

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

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

Multiply $- 1$ by $1$ .

$f (\frac{9 π}{4}) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (\frac{9 π}{4}) = - \frac{1}{2}$

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

$- \frac{1}{2}$

Find the point at $x = 3 π$ .

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

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

Simplify the result.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $2 (π)$ by $1$ .

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

Simplify the numerator.

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Subtract $π$ from $4 π$ .

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

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

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

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

$f (3 π) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (3 π) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{15 π}{4}$ .

Replace the variable $x$ with $\frac{15 π}{4}$ in the expression.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{2 (\frac{15 π}{4})}{3} - \frac{π}{2})}{2}$

Simplify the result.

Combine $2$ and $\frac{15 π}{4}$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{2 (15 π)}{4}}{3} - \frac{π}{2})}{2}$

Multiply $2$ by $15$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{30 π}{4}}{3} - \frac{π}{2})}{2}$

Reduce the expression $\frac{30 π}{4}$ by cancelling the common factors.

Factor $2$ out of $30 π$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{2 (15 π)}{4}}{3} - \frac{π}{2})}{2}$

Factor $2$ out of $4$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{2 (15 π)}{2 (2)}}{3} - \frac{π}{2})}{2}$

Cancel the common factor.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{2 (15 π)}{2 \cdot 2}}{3} - \frac{π}{2})}{2}$

Rewrite the expression.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{\frac{15 π}{2}}{3} - \frac{π}{2})}{2}$

Simplify the numerator.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{15 π}{2} \cdot \frac{1}{3} - \frac{π}{2})}{2}$

Cancel the common factor of $3$ .

Factor $3$ out of $15 π$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{3 (5 π)}{2} \cdot \frac{1}{3} - \frac{π}{2})}{2}$

Cancel the common factor.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{3 (5 π)}{2} \cdot \frac{1}{3} - \frac{π}{2})}{2}$

Rewrite the expression.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{5 π}{2} - \frac{π}{2})}{2}$

Combine the numerators over the common denominator.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{5 π - π}{2})}{2}$

Subtract $π$ from $5 π$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{4 π}{2})}{2}$

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

Factor $2$ out of $4 π$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{2 (2 π)}{2})}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (\frac{15 π}{4}) = \frac{\cos (\frac{2 (2 π)}{2 (1)})}{2}$

Cancel the common factor.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{2 (2 π)}{2 \cdot 1})}{2}$

Rewrite the expression.

$f (\frac{15 π}{4}) = \frac{\cos (\frac{2 π}{1})}{2}$

Divide $2 π$ by $1$ .

$f (\frac{15 π}{4}) = \frac{\cos (2 π)}{2}$

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

$f (\frac{15 π}{4}) = \frac{\cos (0)}{2}$

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

$f (\frac{15 π}{4}) = \frac{1}{2}$

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

$\frac{1}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{3 π}{4} & \frac{1}{2} \\ \frac{3 π}{2} & 0 \\ \frac{9 π}{4} & - \frac{1}{2} \\ 3 π & 0 \\ \frac{15 π}{4} & \frac{1}{2} \end{array}$

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

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

Period: $3 π$

Phase Shift: $\frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{3 π}{4} & \frac{1}{2} \\ \frac{3 π}{2} & 0 \\ \frac{9 π}{4} & - \frac{1}{2} \\ 3 π & 0 \\ \frac{15 π}{4} & \frac{1}{2} \end{array}$

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

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