교과서 질문
Graph each function over a two-period interval.
y = 1 - cot x
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
5장, 문제 33
Graph each function over a two-period interval.
y = -1 + 2 tan x
검증된 단계별 안내1
Identify the base function and its characteristics: The given function is \(y = -1 + 2 \tan x\). The base function here is \(\tan x\), which has a period of \(\pi\).
Determine the interval for graphing: Since the problem asks for a two-period interval, and the period of \(\tan x\) is \(\pi\), the interval will be \([0, 2\pi]\) or any other interval of length \(2\pi\).
Analyze the transformations applied to the base function: The function has a vertical stretch by a factor of 2 (due to the coefficient 2 in front of \(\tan x\)) and a vertical shift downward by 1 unit (due to the \(-1\)).
Identify the vertical asymptotes of the function: The vertical asymptotes of \(\tan x\) occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. These asymptotes will remain the same for the transformed function.
Sketch the graph by plotting key points and asymptotes over the two-period interval, applying the vertical stretch and shift to the \(\tan x\) values, and marking the asymptotes at the appropriate \(x\)-values.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Period of the Tangent Function
The tangent function has a fundamental period of π, meaning its values repeat every π units along the x-axis. When graphing over a two-period interval, you plot the function from an initial point to two times π, capturing two full cycles of the tangent curve.
추천 영상:
5:43
Introduction to Tangent Graph
Vertical Asymptotes of Tangent
Tangent has vertical asymptotes where the function is undefined, occurring at x = (2k+1)π/2 for all integers k. These asymptotes divide the graph into sections and indicate where the function approaches infinity or negative infinity, crucial for accurate graphing.
추천 영상:
4:42Asymptotes
Transformations of Trigonometric Functions
The function y = -1 + 2 tan x involves vertical scaling by 2 and a vertical shift downward by 1. Scaling affects the steepness of the graph, while shifting moves the entire curve down, altering the midline and range but not the period or asymptotes.
추천 영상:
4:22Domain and Range of Function Transformations
관련 실천
교과서 질문
Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.
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교과서 질문
Graph each function over a two-period interval.
y= -1 + (1/2) cot (2x - 3π)
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교과서 질문
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin (x + π)
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교과서 질문
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos [π/2 (x - ½)]
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교과서 질문
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -¼ cos (½ x + π/2)
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