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

$y = \frac{1}{2} \cos (2 x - π)$

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.

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

$b = 2$

$c = π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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: $\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

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

Period: $π$

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

Vertical Shift: None

Step 6

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.

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

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $π$ from $π$ .

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

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

$f (\frac{π}{2}) = \frac{1}{2}$

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

$\frac{1}{2}$

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

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

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

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine $- π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $- 1$ .

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

Subtract $2 π$ from $3 π$ .

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

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

$f (\frac{3 π}{4}) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{3 π}{4}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π$ .

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

$f (π) = \frac{\cos (2 (π) - π)}{2}$

Simplify the result.

Simplify the numerator.

Subtract $π$ from $2 π$ .

$f (π) = \frac{\cos (π)}{2}$

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.

$f (π) = \frac{- \cos (0)}{2}$

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

$f (π) = \frac{- 1 \cdot 1}{2}$

Multiply $- 1$ by $1$ .

$f (π) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (π) = - \frac{1}{2}$

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

$- \frac{1}{2}$

Find the point at $x = \frac{5 π}{4}$ .

Replace the variable $x$ with $\frac{5 π}{4}$ in the expression.

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

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine $- π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $- 1$ .

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

Subtract $2 π$ from $5 π$ .

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

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

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

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

$f (\frac{5 π}{4}) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{5 π}{4}) = 0$

The final answer is $0$ .

$0$

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

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

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

Simplify the result.

Simplify the numerator.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $π$ from $3 π$ .

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

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

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

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

$f (\frac{3 π}{2}) = \frac{1}{2}$

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

$\frac{1}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{2} & \frac{1}{2} \\ \frac{3 π}{4} & 0 \\ π & - \frac{1}{2} \\ \frac{5 π}{4} & 0 \\ \frac{3 π}{2} & \frac{1}{2} \end{array}$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{π}{2} & \frac{1}{2} \\ \frac{3 π}{4} & 0 \\ π & - \frac{1}{2} \\ \frac{5 π}{4} & 0 \\ \frac{3 π}{2} & \frac{1}{2} \end{array}$

Step 8

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

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