what are the period, phase shift, and vertical shift of ( y = csc3(x + 4) + 6 )?\nperiod: ( \frac{pi}{3} )…

Explanation:

Step1: Recall the general form of (y = A\csc(B(x - C))+D)

The period of (y = A\csc(B(x - C))+D) is (\frac{2\pi}{|B|}), the phase - shift is (C) (right if (C>0), left if (C < 0)), and the vertical shift is (D) (up if (D>0), down if (D < 0)). For the function (y=\csc[3(x + 4)]+6=\csc(3(x-(- 4)))+6), we have (A = 1), (B = 3), (C=-4), (D = 6).

Step2: Calculate the period

Using the formula for the period (\text{Period}=\frac{2\pi}{|B|}), with (B = 3), we get (\text{Period}=\frac{2\pi}{3}).

Step3: Determine the phase - shift

Using the formula for the phase - shift (C), with (C=-4), the phase - shift is (4) units to the left.

Step4: Determine the vertical shift

Using the formula for the vertical shift (D), with (D = 6), the vertical shift is (6) units up.

Answer:

period: (\frac{2\pi}{3}); phase shift: (4) units left; vertical shift: (6) units up (the third option)