Find Amplitude, Period, and Phase Shift 2/3cos(4t+3pi)

$\frac{2}{3} \cos (4 t + 3 π)$

Step 1

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

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

$b = 4$

$c = - 3 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

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

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{- 3 π}{4}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{3 π}{4}$

Step 5

List the properties of the trigonometric function.

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

Period: $\frac{π}{2}$

Phase Shift: $- \frac{3 π}{4}$ ( $\frac{3 π}{4}$ to the left)

Vertical Shift: None

Step 6