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 function f (x) = sin 2x + cos 3x.
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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 1.8
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
검증된 단계별 안내1
To determine symmetry about the y-axis, check if the function y = f(x) satisfies f(x) = f(-x). Substitute -x into the function: y = e^(-(-x)^2). Simplify to see if it equals the original function.
To determine symmetry about the origin, check if the function y = f(x) satisfies f(-x) = -f(x). Substitute -x into the function: y = e^(-(-x)^2) and compare it to -e^(-x^2).
Simplify the expression e^(-(-x)^2) to e^(-x^2) and compare it to the original function e^(-x^2) to check for y-axis symmetry.
Compare e^(-x^2) with -e^(-x^2) to check for origin symmetry. If they are not equal, the function is not symmetric about the origin.
Conclude whether the function is symmetric about the y-axis, the origin, or neither based on the comparisons made in the previous steps.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 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 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 equation. 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
Exponential functions
Exponential functions, such as y = e^(-x²), are characterized by a constant base raised to a variable exponent. These functions typically exhibit rapid growth or decay. Understanding their general shape and behavior is crucial for analyzing their symmetry properties, as they often do not possess symmetry about the y-axis or the origin.
추천 영상:
6:13Exponential Functions
관련 실천
교과서 질문
Shifting Graphs
The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.
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교과서 질문
Graph the functions in Exercises 37–56.
y = |x − 2|
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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) = x² + x
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교과서 질문
In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x
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교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x - sin x
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