Find Amplitude, Period, and Phase Shift sin(3x+5)

$\sin (3 x + 5)$

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

$a = 1$

$b = 3$

$c = - 5$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\sin (3 x + 5)$ .

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 $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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: $\frac{- 5}{3}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{5}{3}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2 π}{3}$

Phase Shift: $- \frac{5}{3}$ ( $\frac{5}{3}$ to the left)

Vertical Shift: None