Graph each function over a one-period interval.
y = ½ cot (4x)

Verified step by step guidance

1. Identify the given function: \(y = \frac{1}{2} \cot(4x)\).
2. Recall that the cotangent function \(\cot(\theta)\) has a period of \(\pi\). For \(\cot(4x)\), the period is adjusted by the coefficient inside the argument, so the period \(T\) is given by \(T = \frac{\pi}{4}\).
3. Determine the one-period interval for \(x\). Since the period is \(\frac{\pi}{4}\), you can choose an interval of length \(\frac{\pi}{4}\), for example, \(\left(0, \frac{\pi}{4}\right)\) or any other interval of length \(\frac{\pi}{4}\).
4. Find the key points within the chosen interval where the cotangent function has important features: vertical asymptotes occur where \(\sin(4x) = 0\), and zeros occur where \(\cos(4x) = 0\). These points help in sketching the graph.
5. Plot the function \(y = \frac{1}{2} \cot(4x)\) over the chosen interval by marking the vertical asymptotes, zeros, and the general shape of the cotangent curve scaled by \(\frac{1}{2}\) in amplitude.

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, and its graph repeats every π units. Understanding its shape and behavior is essential for graphing transformations.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle before it repeats. For cot(x), the period is π. When the function is modified as cot(bx), the period changes to π/|b|. Identifying the correct period is crucial for determining the interval to graph.

Amplitude and Vertical Scaling

Amplitude refers to the vertical stretch or compression of a function. For cotangent, which has no maximum or minimum, the coefficient affects the steepness of the curve. In y = ½ cot(4x), the factor ½ compresses the graph vertically, making the curve less steep.