교과서 질문
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x
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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 8
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 sin 5x
검증된 단계별 안내1
Identify the general form of the sine function: \(y = A \sin(Bx - C) + D\), where \(A\) is the amplitude, \(\frac{2\pi}{B}\) is the period, \(D\) is the vertical translation, and \(\frac{C}{B}\) is the phase shift.
Compare the given function \(y = 2 \sin 5x\) to the general form. Here, \(A = 2\), \(B = 5\), and both \(C\) and \(D\) are 0 since there is no horizontal shift or vertical translation explicitly shown.
Calculate the amplitude, which is the absolute value of \(A\): \(|2| = 2\). This represents the maximum distance from the midline to the peak of the sine wave.
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{5}\). This is the length of one complete cycle of the sine wave along the x-axis.
Determine the vertical translation and phase shift. Since \(D = 0\), there is no vertical translation, and since \(C = 0\), the phase shift is \(\frac{C}{B} = 0\), meaning the graph starts at the origin without horizontal shift.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Amplitude of a Sine Function
Amplitude is the maximum value the sine function reaches from its midline. It is given by the absolute value of the coefficient in front of the sine function. For y = 2 sin 5x, the amplitude is 2, indicating the wave oscillates 2 units above and below the midline.
추천 영상:
5:05
Amplitude and Reflection of Sine and Cosine
Period of a Sine Function
The period is the length of one complete cycle of the sine wave. It is calculated as 2π divided by the absolute value of the coefficient of x inside the sine function. For y = 2 sin 5x, the period is 2π/5, meaning the wave repeats every 2π/5 units.
추천 영상:
5:33Period of Sine and Cosine Functions
Vertical Translation and Phase Shift
Vertical translation shifts the graph up or down and is determined by any constant added or subtracted outside the sine function. Phase shift moves the graph left or right and depends on horizontal shifts inside the function's argument. In y = 2 sin 5x, there is no vertical translation or phase shift since no constants are added or subtracted.
추천 영상:
6:31Phase Shifts
관련 실천
교과서 질문
Match each function with its graph in choices A–I. (One choice will not be used.)
y = sin (x - π/4)
A. <IMAGE> B. <IMAGE> C. <IMAGE>
D. <IMAGE> E. <IMAGE> F. <IMAGE>
G. <IMAGE> H. <IMAGE> I. <IMAGE>
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교과서 질문
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 1 + 2 sin ¼ x
588
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교과서 질문
Match each function with its graph in choices A–F.
y = tan (x - π )
A. <IMAGE> B. <IMAGE> C. <IMAGE>
D. <IMAGE> E. <IMAGE> F. <IMAGE>
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교과서 질문
Fill in the blank(s) to correctly complete each sentence.
The graph of y = 3 + 5 cos (x + π/5) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)
883
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교과서 질문
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -2 + 3 cos (x - π/6) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)
666
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