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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 2 - Functions and Graphs문제 23
3장, 문제 23

Determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of those, or none of these. x^2 + y^2 =17

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To determine symmetry with respect to the y-axis, replace x with -x in the equation. Substitute -x for x in the equation x^2 + y^2 = 17, resulting in (-x)^2 + y^2 = 17. Simplify the expression to check if it matches the original equation.
To determine symmetry with respect to the x-axis, replace y with -y in the equation. Substitute -y for y in the equation x^2 + y^2 = 17, resulting in x^2 + (-y)^2 = 17. Simplify the expression to check if it matches the original equation.
To determine symmetry with respect to the origin, replace both x with -x and y with -y in the equation. Substitute -x for x and -y for y in the equation x^2 + y^2 = 17, resulting in (-x)^2 + (-y)^2 = 17. Simplify the expression to check if it matches the original equation.
After performing the substitutions and simplifications for each type of symmetry, compare the resulting equations to the original equation x^2 + y^2 = 17. If the resulting equation matches the original, the graph is symmetric with respect to that axis or the origin.
Based on the results of the comparisons, conclude whether the graph is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.

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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. For example, a graph is symmetric with respect to the y-axis if replacing x with -x in the equation yields the same equation. Similarly, it is symmetric with respect to the x-axis if replacing y with -y results in the same equation, and it is symmetric with respect to the origin if replacing both x and y with their negatives preserves the equation.
추천 영상:
가이드 코스
02:16
Graphs and Coordinates - Example

Circle Equation

The equation x^2 + y^2 = r^2 represents a circle centered at the origin with radius r. In this case, the equation x^2 + y^2 = 17 describes a circle with a radius of √17. Understanding the standard form of a circle's equation is crucial for analyzing its symmetry properties, as circles inherently exhibit symmetry about both axes and the origin.
추천 영상:
5:18
Circles in Standard Form

Testing for Symmetry

To determine the symmetry of a graph, specific tests can be applied. For y-axis symmetry, substitute -x into the equation; for x-axis symmetry, substitute -y; and for origin symmetry, substitute both -x and -y. If the resulting equation remains unchanged for any of these substitutions, the graph exhibits that type of symmetry. This methodical approach is essential for analyzing the given equation.
추천 영상:
07:42
Properties of Parabolas
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