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

$y = \frac{1}{2} \cos (x + π)$

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.

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

$b = 1$

$c = - π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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 $1$ .

$2 π$

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: $\frac{- π}{1}$

Divide $- π$ by $1$ .

Phase Shift: $- π$

Step 5

List the properties of the trigonometric function.

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

Period: $2 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Find the point at $x = - π$ .

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

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

Simplify the result.

Simplify the numerator.

Add $- π$ and $π$ .

$f (- π) = \frac{\cos (0)}{2}$

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

$f (- π) = \frac{1}{2}$

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

$\frac{1}{2}$

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

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

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

Simplify the result.

Simplify the numerator.

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $- π$ and $2 π$ .

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

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

$f (- \frac{π}{2}) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (- \frac{π}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 0$ .

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

$f (0) = \frac{\cos ((0) + π)}{2}$

Simplify the result.

Simplify the numerator.

Add $0$ and $π$ .

$f (0) = \frac{\cos (π)}{2}$

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.

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

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

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

Multiply $- 1$ by $1$ .

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

Move the negative in front of the fraction.

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

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

$- \frac{1}{2}$

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

Simplify the numerator.

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $π$ and $2 π$ .

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

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

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

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

$f (\frac{π}{2}) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (\frac{π}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π$ .

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

$f (π) = \frac{\cos ((π) + π)}{2}$

Simplify the result.

Simplify the numerator.

Add $π$ and $π$ .

$f (π) = \frac{\cos (2 π)}{2}$

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

$f (π) = \frac{\cos (0)}{2}$

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

$f (π) = \frac{1}{2}$

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

$\frac{1}{2}$

List the points in a table.

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

Step 7

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

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

Period: $2 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: None

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

Step 8

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

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