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

검증된 단계별 안내

Identify the given function: \(y = \frac{1}{2} \cot(4x)\).

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 \(T = \frac{\pi}{4}\).

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 \(\frac{\pi}{4}\).

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.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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