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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
1장, 문제 1.41c

Identify the symmetry (if any) in the graphs of the following equations.
y2−4x2=4y^2-4x^2=4

검증된 단계별 안내
1
First, recognize that the given equation is in the form of a conic section. Specifically, it resembles the equation of a hyperbola: \( y^2 - 4x^2 = 4 \).
To analyze symmetry, consider the standard forms of symmetry: symmetry about the x-axis, y-axis, and the origin. For hyperbolas, symmetry is typically about the axes or the origin.
Check for symmetry about the x-axis by replacing \( y \) with \( -y \) in the equation. Substitute \( -y \) into the equation: \( (-y)^2 - 4x^2 = 4 \). Simplify to see if the equation remains unchanged.
Check for symmetry about the y-axis by replacing \( x \) with \( -x \) in the equation. Substitute \( -x \) into the equation: \( y^2 - 4(-x)^2 = 4 \). Simplify to see if the equation remains unchanged.
Check for symmetry about the origin by replacing both \( x \) with \( -x \) and \( y \) with \( -y \). Substitute into the equation: \( (-y)^2 - 4(-x)^2 = 4 \). Simplify to see if the equation remains unchanged. Analyze the results to determine the symmetry of the graph.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m

주요 개념

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

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations, such as reflection or rotation. Common types of symmetry include even symmetry (about the y-axis), odd symmetry (about the origin), and symmetry about a line. Identifying symmetry helps in understanding the behavior of functions and their graphs.
추천 영상:
가이드 코스
06:15
Graphing The Derivative

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The equation given, y² - 4x² = 4, represents a hyperbola, which is characterized by its two branches that open away from each other. Understanding the properties of conic sections is essential for analyzing their graphs and symmetries.
추천 영상:
가이드 코스
5:33
Parabolas as Conic Sections

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, replacing y with -y reflects the graph across the x-axis, while replacing x with -x reflects it across the y-axis. These transformations are crucial for determining the symmetry of a graph, as they can reveal how the graph behaves under various operations.
추천 영상:
가이드 코스
5:25
Intro to Transformations
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