For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = cot (x/2 + 3π/4)

Verified step by step guidance

1. Identify the general form of the cotangent function: \(y = \cot(Bx + C) + D\), where \(B\) affects the period, \(C\) affects the phase shift, and \(D\) is the vertical translation.
2. Determine the amplitude: Since cotangent functions do not have a maximum or minimum value and extend infinitely, the amplitude is undefined or does not exist.
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B\) is the coefficient of \(x\) inside the function. In \(y = \cot\left(\frac{x}{2} + \frac{3\pi}{4}\right)\), rewrite \(\frac{x}{2}\) as \(\frac{1}{2}x\), so \(B = \frac{1}{2}\).
4. Find the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). Here, \(C = \frac{3\pi}{4}\) and \(B = \frac{1}{2}\). Substitute these values to express the phase shift.
5. Identify the vertical translation \(D\): Since there is no constant added or subtracted outside the cotangent function, the vertical translation is \(0\).

Key Concepts

Amplitude of Trigonometric Functions

Amplitude refers to the maximum absolute value of a trigonometric function's output. For cotangent and other reciprocal trig functions, amplitude is undefined because their values range from negative to positive infinity, unlike sine and cosine which have fixed amplitudes.

Period of Cotangent Function

The period of a cotangent function y = cot(bx) is given by π divided by the absolute value of b. It represents the length of one complete cycle of the function. For y = cot(x/2 + 3π/4), the coefficient of x is 1/2, so the period is 2π.

Phase Shift and Vertical Translation

Phase shift is the horizontal shift of the graph caused by adding or subtracting a constant inside the function's argument. It is calculated by solving (bx + c) = 0 for x. Vertical translation shifts the graph up or down, but since there is no added constant outside the cotangent function here, vertical translation is zero.