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

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

Step 1

Rewrite the expression as $- \frac{2 \cos (x - \frac{π}{2})}{3} + 2$ .

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

Step 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.

$a = - \frac{2}{3}$

$b = 1$

$c = \frac{π}{2}$

$d = 2$

Step 3

Find the amplitude $| a |$ .

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

Step 4

Find the period using the formula $\frac{2 π}{| b |}$ .

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$2 π$

Step 5

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: $\frac{\frac{π}{2}}{1}$

Divide $\frac{π}{2}$ by $1$ .

Phase Shift: $\frac{π}{2}$

Step 6

List the properties of the trigonometric function.

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

Period: $2 π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $2$

Step 7

Select a few points to graph.

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

Simplify each term.

Simplify the numerator.

Combine the numerators over the common denominator.

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

Subtract $π$ from $π$ .

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

Divide $0$ by $2$ .

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

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

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

Multiply $2$ by $1$ .

$f (\frac{π}{2}) = 2 - \frac{2}{3}$

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

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$f (\frac{π}{2}) = \frac{6 - 2}{3}$

Subtract $2$ from $6$ .

$f (\frac{π}{2}) = \frac{4}{3}$

The final answer is $\frac{4}{3}$ .

$\frac{4}{3}$

Find the point at $x = π$ .

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

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

Simplify the result.

Simplify each term.

Simplify the numerator.

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Subtract $π$ from $2 π$ .

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

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

$f (π) = 2 - \frac{2 \cdot 0}{3}$

Multiply $2$ by $0$ .

$f (π) = 2 - \frac{0}{3}$

Divide $0$ by $3$ .

$f (π) = 2 - 0$

Multiply $- 1$ by $0$ .

$f (π) = 2 + 0$

Add $2$ and $0$ .

$f (π) = 2$

The final answer is $2$ .

$2$

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

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

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

Simplify the result.

Simplify each term.

Simplify the numerator.

Combine the numerators over the common denominator.

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

Subtract $π$ from $3 π$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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.

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

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

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

Multiply $- 1$ by $1$ .

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

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

$f (\frac{3 π}{2}) = 2 + \frac{2}{3}$

Multiply $- - \frac{2}{3}$ .

Multiply $- 1$ by $- 1$ .

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

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

$f (\frac{3 π}{2}) = 2 + \frac{2}{3}$

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

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$f (\frac{3 π}{2}) = \frac{6 + 2}{3}$

Add $6$ and $2$ .

$f (\frac{3 π}{2}) = \frac{8}{3}$

The final answer is $\frac{8}{3}$ .

$\frac{8}{3}$

Find the point at $x = 2 π$ .

Replace the variable $x$ with $2 π$ in the expression.

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

Simplify the result.

Simplify each term.

Simplify the numerator.

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

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

Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

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

Subtract $π$ from $4 π$ .

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

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

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

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

$f (2 π) = 2 - \frac{2 \cdot 0}{3}$

Multiply $2$ by $0$ .

$f (2 π) = 2 - \frac{0}{3}$

Divide $0$ by $3$ .

$f (2 π) = 2 - 0$

Multiply $- 1$ by $0$ .

$f (2 π) = 2 + 0$

Add $2$ and $0$ .

$f (2 π) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{5 π}{2}$ .

Replace the variable $x$ with $\frac{5 π}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Simplify the numerator.

Combine the numerators over the common denominator.

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

Subtract $π$ from $5 π$ .

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

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

Factor $2$ out of $4 π$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

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

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

$f (\frac{5 π}{2}) = 2 - \frac{2 \cos (0)}{3}$

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

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

Multiply $2$ by $1$ .

$f (\frac{5 π}{2}) = 2 - \frac{2}{3}$

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

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$f (\frac{5 π}{2}) = \frac{6 - 2}{3}$

Subtract $2$ from $6$ .

$f (\frac{5 π}{2}) = \frac{4}{3}$

The final answer is $\frac{4}{3}$ .

$\frac{4}{3}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{2} & \frac{4}{3} \\ π & 2 \\ \frac{3 π}{2} & \frac{8}{3} \\ 2 π & 2 \\ \frac{5 π}{2} & \frac{4}{3} \end{array}$

Step 8

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

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

Period: $2 π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ \frac{π}{2} & \frac{4}{3} \\ π & 2 \\ \frac{3 π}{2} & \frac{8}{3} \\ 2 π & 2 \\ \frac{5 π}{2} & \frac{4}{3} \end{array}$

Step 9

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

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