Find Amplitude, Period, and Phase Shift y=csc(3(x+4))+6

$y = \csc (3 (x + 4)) + 6$

Step 1

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

$a = 1$

$b = 3$

$c = - 12$

$d = 6$

Step 2

Since the graph of the function $c s c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 3

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

Find the period of $\csc (3 x + 12)$ .

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

$\frac{2 π}{3}$

Find the period of $6$ .

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 $3$ .

$\frac{2 π}{3}$

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

$\frac{2 π}{3}$

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: $\frac{- 12}{3}$

Divide $- 12$ by $3$ .

Phase Shift: $- 4$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{2 π}{3}$

Phase Shift: $- 4$ ( $4$ to the left)

Vertical Shift: $6$

Step 6