Write an equation of cotangent with period = 3pi, phase shift = -pi, and vertical shift = -1

cot(x) has period pi

cot(x/3) has period 3pi
Now add in the rest.

To write an equation of cotangent function with the given information, we can use the general form of the cotangent function:

y = A * cot(Bx - C) + D

where A represents the amplitude, B represents the period, C represents the phase shift, and D represents the vertical shift.

Given:
Period = 3π
Phase Shift = -π
Vertical Shift = -1

The period of cotangent function is π, so to get a period of 3π, we must multiply by 3. Therefore, B = 3.

The phase shift is -π, which means the function is shifted to the right by π units. So, C = -π.

The vertical shift is down by 1 unit. So, D = -1.

Since the amplitude is not given, we assume it to be 1 as the standard amplitude for the cotangent function. Therefore, A = 1.

Now we can substitute the values into the general form of the cotangent function:

y = 1 * cot(3x + π) - 1

So, the equation of the cotangent function with a period of 3π, phase shift of -π, and vertical shift of -1 is y = cot(3x + π) - 1.

To write the equation of the cotangent function with a period of 3π, a phase shift of -π, and a vertical shift of -1, we can use the following general form of a cotangent function:

y = A * cot(Bx - C) + D

where A represents the amplitude, B represents the frequency (which is equal to 2π divided by the period), C represents the phase shift, and D represents the vertical shift.

Given that the period is 3π, we can find the frequency as follows:

Frequency (B) = 2π / Period = 2π / (3π) = 2/3

Since the phase shift is -π, we'll add that to the angle inside the cotangent function, which is Bx - C:

Bx - C = (2/3)x - (-π) = (2/3)x + π

Finally, considering the vertical shift of -1, we obtain the equation:

y = A * cot((2/3)x + π) - 1

Note that we haven't been given the amplitude (A) in this case. If the amplitude is not specified, it is typically assumed to be 1. Therefore, the final equation of the cotangent function with the given period, phase shift, and vertical shift is:

y = cot((2/3)x + π) - 1