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Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 12th EditionCh. 4 - Graphs of the Circular Functions문제 35
5장, 문제 35

Graph each function over a two-period interval.
y= -1 + (1/2) cot (2x - 3π)

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1
Identify the given function: \(y = -1 + \frac{1}{2} \cot(2x - 3\pi)\).
Recall that the cotangent function \(\cot(\theta)\) has a period of \(\pi\). Since the argument of the cotangent is \(2x - 3\pi\), the period of the function changes according to the coefficient of \(x\). The period \(T\) is given by \(T = \frac{\pi}{|2|} = \frac{\pi}{2}\).
Since the problem asks to graph over a two-period interval, determine the interval length: \(2 \times T = 2 \times \frac{\pi}{2} = \pi\). Choose an interval of length \(\pi\) for \(x\), for example from \(x = a\) to \(x = a + \pi\), where \(a\) can be chosen to include the phase shift.
Analyze the phase shift caused by \(-3\pi\) inside the cotangent argument. Set the inside of the cotangent equal to zero to find the horizontal shift: \(2x - 3\pi = 0 \Rightarrow x = \frac{3\pi}{2}\). This means the cotangent function is shifted to the right by \(\frac{3\pi}{2}\).
Plot key points of the cotangent function within the chosen interval, considering the amplitude scaling by \(\frac{1}{2}\) and the vertical shift down by 1. Remember that \(\cot(\theta)\) has vertical asymptotes where \(\sin(\theta) = 0\), i.e., at multiples of \(\pi\). Use these to sketch the graph accurately.

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주요 개념

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

Period of Trigonometric Functions

The period of a trigonometric function is the length of the interval over which the function completes one full cycle. For cotangent, the basic period is π, but when the function argument is multiplied by a factor (like 2 in cot(2x - 3π)), the period changes to π divided by that factor, affecting the graph's horizontal length.
추천 영상:
5:33
Period of Sine and Cosine Functions

Phase Shift in Trigonometric Functions

Phase shift refers to the horizontal translation of a trigonometric graph caused by adding or subtracting a constant inside the function's argument. In cot(2x - 3π), the term -3π shifts the graph horizontally, changing where the function's key points and asymptotes occur along the x-axis.
추천 영상:
6:31
Phase Shifts

Vertical Transformations of Trigonometric Graphs

Vertical transformations include shifts and stretches/compressions applied to the function's output. In y = -1 + (1/2) cot(2x - 3π), the -1 shifts the graph down by one unit, and the factor 1/2 compresses the cotangent's amplitude vertically, altering the height and position of the graph's features.
추천 영상:
5:25
Introduction to Transformations
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