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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 1 - Functions문제 1.3.70
1장, 문제 1.3.70

General Sine Curves


For


f(x) = A sin ((2π/B)(x – C) +D


identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.


y = L/2π sin (2πt/L), L > 0

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1
Identify the general form of the sine function: \( f(x) = A \sin\left(\frac{2\pi}{B}(x - C)\right) + D \). Here, \( A \) is the amplitude, \( B \) determines the period, \( C \) is the horizontal shift, and \( D \) is the vertical shift.
Compare the given function \( y = \frac{L}{2\pi} \sin\left(\frac{2\pi t}{L}\right) \) with the general form. Notice that \( A = \frac{L}{2\pi} \), \( B = L \), \( C = 0 \), and \( D = 0 \).
Determine the amplitude: The amplitude is the absolute value of \( A \), which is \( |A| = \left|\frac{L}{2\pi}\right| \). This represents the maximum vertical displacement of the sine wave from its midline.
Determine the period: The period of the sine function is given by \( B \), which is \( L \). This represents the length of one complete cycle of the sine wave.
Sketch the graph: Plot the sine wave with the identified parameters. The wave has an amplitude of \( \frac{L}{2\pi} \), a period of \( L \), no horizontal shift (\( C = 0 \)), and no vertical shift (\( D = 0 \)). The graph oscillates symmetrically about the x-axis.

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주요 개념

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

Amplitude (A)

The amplitude of a sine function, represented by A, determines the height of the wave from its midline to its peak. It indicates how far the function's values stretch above and below the midline (y=0). In the context of the given function, a larger A results in a taller wave, while a smaller A compresses the wave vertically.
추천 영상:
3:40
Introduction to Cotangent Graph Example 1

Period (B)

The period of a sine function, denoted by B, is the distance along the x-axis required for the function to complete one full cycle. It is calculated using the formula Period = 2π/B. In the provided function, adjusting B alters the frequency of the wave, with a smaller B leading to more cycles within a given interval.
추천 영상:
5:43
Introduction to Tangent Graph

Phase Shift (C) and Vertical Shift (D)

The phase shift, represented by C, indicates how much the graph of the sine function is shifted horizontally, while the vertical shift, denoted by D, moves the graph up or down. Specifically, C affects the starting point of the wave, and D adjusts the midline of the wave. Together, they help position the sine curve in the coordinate plane.
추천 영상:
5:25
Intro to Transformations
관련 실천
교과서 질문

Functions


In Exercises 1–6, find the domain and range of each function.


f(x) = 1 + x²

484
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교과서 질문

In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x² + 1

228
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교과서 질문

Functions and Graphs


Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

573
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교과서 질문

In Exercises 39–42, express the given quantity in terms of sin x and cos x.


sin (2π − x)

225
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교과서 질문

In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x⁴ + 1

x³ - 2x

227
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교과서 질문

Graph the functions in Exercises 37–56.


y = (x + 2)³/² + 1

213
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