Find Amplitude, Period, and Phase Shift y=1/4cos((2pi)/3theta)

$y = \frac{1}{4} \cos (\frac{2 π}{3} θ)$

Step 1

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

$a = \frac{1}{4}$

$b = \frac{2 π}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Step 3

Find the period of $\frac{\cos (\frac{2 π θ}{3})}{4}$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{2 π}{3} |}$

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Cancel the common factor of $2 π$ .

Cancel the common factor.

$2 π \frac{3}{2 π}$

Rewrite the expression.

$3$

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: $\frac{0}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{2 π})$

Multiply $0$ by $\frac{3}{2 π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $3$

Phase Shift: None

Vertical Shift: None

Step 6