what are the period, phase shift, and vertical shift of y = csc3(x + 4) + 6?\no period: $\frac{pi}{3}$…

Explanation:

Step1: Recall csc - function period formula

The general form of a cosecant - function is $y = A\csc(B(x - C))+D$. The period of $y = \csc(x)$ is $2\pi$. For the function $y=\csc(B(x - C))+D$, the period $T$ is given by $T=\frac{2\pi}{|B|}$. Here, $B = 3$, so $T=\frac{2\pi}{3}$.

Step2: Determine the phase - shift

The phase - shift of the function $y = A\csc(B(x - C))+D$ is given by $C$. In the function $y=\csc(3(x + 4))+6=\csc(3(x-(-4)))+6$, $C=-4$. A negative value of $C$ indicates a shift to the left. So, the phase - shift is 4 units to the left.

Step3: Determine the vertical - shift

The vertical - shift of the function $y = A\csc(B(x - C))+D$ is given by $D$. In the function $y=\csc(3(x + 4))+6$, $D = 6$. A positive value of $D$ indicates a shift up. So, the vertical - shift is 6 units up.

Answer:

period: $\frac{2\pi}{3}$; phase shift: 4 units left; vertical shift: 6 units up