교과서 질문
Graph the functions in Exercises 37–56.
y = (x + 2)³/² + 1
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.3.18
Graph the functions in Exercises 13–22. What is the period of each function?
−cos 2πx
검증된 단계별 안내1
Identify the function to be graphed: \(-\cos(2\pi x)\). This is a cosine function with a negative sign, which means it is reflected over the x-axis.
Determine the amplitude of the function. The amplitude is the absolute value of the coefficient in front of the cosine function. Here, the amplitude is \(1\) because the coefficient is \(-1\).
Find the period of the function. The period of a cosine function \(\cos(bx)\) is given by \(\frac{2\pi}{b}\). In this case, \(b = 2\pi\), so the period is \(\frac{2\pi}{2\pi} = 1\).
Sketch the graph of the function. Start by plotting key points: the maximum, minimum, and intercepts within one period. Since the function is \(-\cos(2\pi x)\), it starts at a minimum point at \(x = 0\), reaches a maximum at \(x = 0.5\), and returns to a minimum at \(x = 1\).
Label the x-axis with the period and the y-axis with the amplitude. The graph should show one complete cycle from \(x = 0\) to \(x = 1\), with the curve starting and ending at the minimum value of \(-1\), peaking at \(1\) in the middle.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Period of a Function
The period of a function is the length of the interval over which the function repeats itself. For trigonometric functions, the period can be determined from the coefficient of the variable inside the function. For example, the function f(x) = -cos(2πx) has a period of 1, as it completes one full cycle as x varies from 0 to 1.
추천 영상:
6:22
Graphs of Secant and Cosecant Functions
Cosine Function
The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic, with a standard period of 2π. The function -cos(θ) reflects the cosine graph across the x-axis, affecting its amplitude but not its period.
추천 영상:
5:53Graph of Sine and Cosine Function
Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting their values over a specified interval. Key features to consider include amplitude, period, phase shift, and vertical shift. For the function -cos(2πx), the graph will oscillate between -1 and 1, repeating every 1 unit along the x-axis.
추천 영상:
6:04Introduction to Trigonometric Functions
관련 실천
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In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 1
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교과서 질문
Graph the functions in Exercises 37–56.
y = (1/x²) − 1
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교과서 질문
Graphing
In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.
y = (1/2x) − 1
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교과서 질문
Evaluate sin (5π/12).
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