Graph each function over a two-period interval.
y = 1 - cot x

Verified step by step guidance

1. Identify the period of the function. Since the function is \(y = 1 - \cot x\), recall that the cotangent function \(\cot x\) has a period of \(\pi\). Therefore, a two-period interval for \(\cot x\) is \(2\pi\).
2. Determine the interval over which to graph the function. For \(\cot x\), a natural choice is from \(0\) to \(2\pi\) to cover two full periods.
3. Analyze the behavior of \(\cot x\) within one period. \(\cot x = \frac{\cos x}{\sin x}\) has vertical asymptotes where \(\sin x = 0\), which occur at \(x = 0, \pi, 2\pi\). Between these points, \(\cot x\) decreases from \(+\infty\) to \(-\infty\).
4. Apply the transformation to the function: \(y = 1 - \cot x\). This means you take the cotangent graph, reflect it vertically (because of the minus sign), and then shift it upward by 1 unit.
5. Sketch the graph over the interval \([0, 2\pi]\) by plotting key points and asymptotes at \(x = 0, \pi, 2\pi\), noting the vertical asymptotes and the shifted values of the function. This will give you the graph of \(y = 1 - \cot x\) over two periods.

Key Concepts

Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). It has vertical asymptotes where sin(x) = 0, i.e., at integer multiples of π. Understanding its periodicity and behavior near asymptotes is essential for graphing.

Periodicity of Trigonometric Functions

The cotangent function has a fundamental period of π, meaning its values repeat every π units. Graphing over a two-period interval involves plotting the function from 0 to 2π or any interval of length 2π, capturing two full cycles of cot(x).

Vertical Shifts in Trigonometric Graphs

The function y = 1 - cot(x) involves a vertical shift of the cotangent graph by 1 unit upwards. This means every point on the cotangent curve is increased by 1, affecting the position of the graph but not its shape or period.