Skip to main content
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 39
5장, 문제 39

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter points are identified by dots.)


<IMAGE>

검증된 단계별 안내
1
Identify the type of trigonometric function represented by the graph. Since midpoints and quarter points are marked, check if the graph resembles a sine or cosine wave by noting where it starts and its shape.
Determine the amplitude (a) by measuring the vertical distance from the midline (equilibrium position) to the maximum or minimum point of the graph. This gives the coefficient 'a' in the function \(y = a \sin(bx)\) or \(y = a \cos(bx)\).
Find the period (T) of the function by measuring the horizontal length of one complete cycle of the wave. Use the relationship between period and frequency: \(b = \frac{2\pi}{T}\), where \(b > 0\) is the frequency multiplier inside the function argument.
Since the problem specifies no phase shifts, ensure the function starts at the appropriate point for sine or cosine without horizontal translation. For example, sine typically starts at the midline going upward, cosine starts at the maximum.
Write the equation in the form \(y = a \sin(bx)\) or \(y = a \cos(bx)\) using the values of amplitude 'a' and frequency 'b' found, ensuring \(b > 0\) and no phase shift term is included.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
도움이 되었나요?

주요 개념

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

Basic Form of Sine and Cosine Functions

The general equations for sine and cosine functions are y = a sin(bx + c) + d and y = a cos(bx + c) + d, where a is amplitude, b affects the period, c is the phase shift, and d is the vertical shift. Understanding these parameters helps in matching the graph to its equation.
추천 영상:
5:53
Graph of Sine and Cosine Function

Period and Frequency of Trigonometric Functions

The period of sine and cosine functions is given by 2π divided by the coefficient b (period = 2π/b). Identifying the period from the graph allows determination of b, which controls how many cycles occur over a given interval.
추천 영상:
5:33
Period of Sine and Cosine Functions

Phase Shift and Midpoint Identification

Phase shift c translates the graph horizontally. Since the question specifies no phase shifts, the function should start at standard positions. Using midpoints and quarter points marked on the graph helps confirm the function's alignment and ensures c = 0.
추천 영상:
6:31
Phase Shifts
관련 실천
교과서 질문

Graph each function over a two-period interval.

y = cos (x - π/2 )

991
views
교과서 질문

Graph each function over a two-period interval.

y = sin (x + π/4)

901
views
교과서 질문

Graph each function over a two-period interval.

y = 1 - 2 cot [2(x + π/2)]

776
views
교과서 질문

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 3 cos [π/2 (x - ½)]

606
views
교과서 질문

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 2 - sin(3x - π/5)

680
views
교과서 질문

Graph each function over a one-period interval.

y = -½ cos (πx - π)

815
views