Find Amplitude, Period, and Phase Shift y=1/2sin(x/3-pi)

$y = \frac{1}{2} \sin (\frac{x}{3} - π)$

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

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

$b = \frac{1}{3}$

$c = π$

$d = 0$

Find the amplitude $| a |$ .

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

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: $\frac{π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π \cdot 3$

Move $3$ to the left of $π$ .

Phase Shift: $3 π$

List the properties of the trigonometric function.

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

Period: $6 π$

Phase Shift: $3 π$ ( $3 π$ to the right)

Vertical Shift: None