Graph each function over a two-period interval.
y = 1 - 2 cot [2(x + π/2)]

Verified step by step guidance

1. Identify the given function: \(y = 1 - 2 \cot \left[ 2 \left( x + \frac{\pi}{2} \right) \right]\). Notice that the function involves a cotangent with a horizontal transformation and a vertical shift and stretch.
2. Determine the period of the cotangent function inside the argument. The standard period of \(\cot x\) is \(\pi\). Since the argument is multiplied by 2, the period becomes \(\frac{\pi}{2}\) because period \(= \frac{\pi}{|b|}\) where \(b=2\).
3. Since the problem asks for a two-period interval, calculate the length of the interval to graph: \(2 \times \frac{\pi}{2} = \pi\). So, you will graph the function over an interval of length \(\pi\) in terms of \(x\).
4. Account for the horizontal shift inside the argument: \(x + \frac{\pi}{2}\). This means the graph is shifted to the left by \(\frac{\pi}{2}\). So, choose your \(x\)-interval accordingly, for example from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) or any interval of length \(\pi\) that reflects this shift.
5. Plot key points by evaluating the cotangent function at values where the argument \(2(x + \frac{\pi}{2})\) equals multiples of \(\frac{\pi}{2}\) (where cotangent has zeros and asymptotes). Then apply the vertical transformations: multiply by \(-2\) and add 1 to get the final \(y\) values. Sketch the graph using these points and asymptotes over the chosen two-period interval.

Key Concepts

Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function and has vertical asymptotes where sine is zero. It is periodic with period π, and understanding its shape and behavior is essential for graphing transformations involving cotangent.

Function Transformations (Shifts and Scaling)

Transformations such as horizontal shifts, vertical shifts, and scaling affect the graph of a trigonometric function. In y = 1 - 2 cot[2(x + π/2)], the inside argument involves a horizontal shift and horizontal compression, while the coefficients outside affect vertical stretch and translation.

Period of a Trigonometric Function

The period of cotangent is π, but when the function argument is multiplied by a constant (like 2), the period changes to π divided by that constant. Identifying the correct period is crucial to graphing the function over the specified interval, here a two-period interval.