Find Amplitude, Period, and Phase Shift 2sin(-3x+2pi)-12

$2 \sin (- 3 x + 2 π) - 12$

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

$a = 2$

$b = - 3$

$c = - 2 π$

$d = - 12$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period using the formula $\frac{2 π}{| b |}$ .

Find the period of $2 \sin (- 3 x + 2 π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$\frac{2 π}{3}$

Find the period of $- 12$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$\frac{2 π}{3}$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{2 π}{3}$

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: $\frac{- 2 π}{- 3}$

Dividing two negative values results in a positive value.

Phase Shift: $\frac{2 π}{3}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{2 π}{3}$ ( $\frac{2 π}{3}$ to the right)

Vertical Shift: $- 12$