Find Amplitude, Period, and Phase Shift y=-1/2cos(2x-2pi)

$y = - \frac{1}{2} \cos (2 x - 2 π)$

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.

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

$b = 2$

$c = 2 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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: $\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

Phase Shift: $\frac{2 π}{2}$

Divide $π$ by $1$ .

Phase Shift: $π$

Step 5

List the properties of the trigonometric function.

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

Period: $π$

Phase Shift: $π$ ( $π$ to the right)

Vertical Shift: None

Step 6