College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 1, Equations and Graphs - Section 1.2 - Graphs of Equations in Two Variables; Circles - 1.2 Exercises - Page 103: 99

Answer

$\underline{\text{The equation is:}}$ Symmetric with respect to the $X-$axis. Symmetric with respect to the $Y-$axis. Symmetric with respect to the origin.

Work Step by Step

$\color{red}{\text{If $(x,y)$ exists on the graph,}}$ $\color{red}{\text{then the graph is symmetric about the:}}$ 1. $X-$Axis if $(x,−y)$ exists on the graph, 2. $Y-$Axis if $(−x,y)$ exists on the graph, 3. Origin if $(−x,−y)$ exists on the graph $\color{blue}{\text{Test for symmetry about the $X-$axis by plugging in $−y$ for $y$.}}$ $x^4(-y)^4+x^2(-y)^2=1$ Remove parentheses. $x^4y^4+x^2y^2=1$ Since the equation is identical to the original equation, it is symmetric to the $X-$axis. Symmetric with respect to the $X-$axis. $\color{blue}{\text{Test for symmetry about the $Y-$axis by plugging in $−x$ for $x$.}}$ $(-x)^4y^4+(-x)^2y^2=1$ Remove parentheses. $x^4y^4+x^2y^2=1$ Since the equation is identical to the original equation, it is symmetric with respect to the $Y-$axis. Symmetric with respect to the $Y-$axis. $\color{blue}{\text{Test for symmetry about the origin axis by plugging in $−x$ for $x$ and $-y$ for $y$.}}$ $(-x)^4(-y)^4+(-x)^2(-y)^2=1$ Remove parentheses. $x^4y^4+x^2y^2=1$ Since the equation is identical to the original equation, it is symmetric with respect to the origin. Symmetric with respect to the origin.
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