교과서 질문
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph two periods of the function f (x) = 3 cot (x/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 1.6
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = x²/⁵
검증된 단계별 안내1
To determine symmetry about the y-axis, check if replacing x with -x in the function yields the same function. For the given function y = x^(2/5), replace x with -x to get y = (-x)^(2/5).
Simplify the expression (-x)^(2/5). Since the exponent 2/5 is a positive rational number, (-x)^(2/5) simplifies to x^(2/5) because the square of a negative number is positive.
Since y = (-x)^(2/5) simplifies to y = x^(2/5), the function is symmetric about the y-axis.
To check for symmetry about the origin, replace both x with -x and y with -y in the original equation. This gives -y = (-x)^(2/5).
Since -y = x^(2/5) does not simplify to the original equation y = x^(2/5), the function is not symmetric about the origin. Therefore, the graph of the function is symmetric about the y-axis only.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Symmetry about the y-axis
A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph of the function is a mirror image across the y-axis.
추천 영상:
06:21
Properties of Functions
Symmetry about the origin
A function is symmetric about the origin if replacing x with -x and y with -y results in the same function. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.
추천 영상:
06:21Properties of Functions
Analyzing function symmetry
To determine the symmetry of a function, one can evaluate the function at both x and -x. By comparing the results, one can conclude whether the function is symmetric about the y-axis, the origin, or neither. This analysis is crucial for understanding the behavior of the graph and its visual representation.
추천 영상:
06:21Properties of Functions
관련 실천
교과서 질문
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x⁻⁵
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교과서 질문
Shifting Graphs
Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.
y = −√x Right 3
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교과서 질문
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the upper branch of the hyperbola y² − 16x² = 1.
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교과서 질문
In Exercises 35 and 36, find the (a) domain and (b) range.
𝔂 = { -x - 2, -2 ≤ x ≤ - 1
{ x, -1 < x ≤ 1
{ -x + 2, 1 < x ≤ 2
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교과서 질문
Theory and Examples
In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.
a. y = x⁴
b. y = x⁷
c. y = x¹⁰
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