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

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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.

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.

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 \(B = \frac{1}{2}\).

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.

Identify the vertical translation \(D\): Since there is no constant added or subtracted outside the cotangent function, the vertical translation is \(0\).

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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.

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Introduction to Trigonometric Functions

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.

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Introduction to Cotangent Graph

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.

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