Find Amplitude, Period, and Phase Shift y=-3cos(3/2x)

$y = - 3 \cos (\frac{3}{2} x)$

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.

$a = - 3$

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

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $- 3 \cos (\frac{3 x}{2})$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{3}{2} |}$

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Multiply $2 π \frac{2}{3}$ .

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

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: $\frac{0}{\frac{3}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{3})$

Multiply $0$ by $\frac{2}{3}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{4 π}{3}$

Phase Shift: None

Vertical Shift: None

Step 6