Problem 75 Test the equation for symmetry. ... [FREE SOLUTION]

91影视

Precalculus: Mathematical for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 75

Test the equation for symmetry. $$x^{4} y^{4}+x^{2} y^{2}=1$$

Short Answer

Expert verified

The equation is symmetric about the x-axis, y-axis, and the origin.

Step by step solution

Test for Symmetry About the X-axis

To test for symmetry about the x-axis, we replace $y$ with $-y$ in the equation and see if it remains unchanged. Substitute in the equation: \[ x^{4} (-y)^{4} + x^{2} (-y)^{2} = 1 \]This simplifies to:\[ x^{4} y^{4} + x^{2} y^{2} = 1 \]Since the equation remains unchanged, the equation is symmetric about the x-axis.

Test for Symmetry About the Y-axis

To test for symmetry about the y-axis, replace $x$ with $-x$ in the equation: \[ (-x)^{4} y^{4} + (-x)^{2} y^{2} = 1 \]This simplifies to:\[ x^{4} y^{4} + x^{2} y^{2} = 1 \]The equation remains unchanged, indicating the equation is symmetric about the y-axis.

Test for Symmetry About the Origin

To test for symmetry about the origin, replace $x$ with $-x$ and $y$ with $-y$:\[ (-x)^{4} (-y)^{4} + (-x)^{2} (-y)^{2} = 1 \]This simplifies to:\[ x^{4} y^{4} + x^{2} y^{2} = 1 \]Since the equation remains unchanged, it is also symmetric about the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetry About the X-axis

Understanding symmetry about the x-axis is quite straightforward. Imagine drawing a graph and flipping it vertically over the x-axis. If the graph appears unchanged, it exhibits symmetry about the x-axis. Mathematically, we test this by substituting $ y $ with $ -y $ in an equation.

If the equation remains unchanged after this substitution, then it indicates the graph's symmetry about the x-axis.
This means every point $ (x, y )$ on the graph has a corresponding point $ (x, -y) $.
A typical example would be the equation $ x^2 = y^2 $, which is symmetric with respect to the x-axis.

For our original equation $ x^{4} y^{4} + x^{2} y^{2} = 1 $, substituting $ y $ with $ -y $ led to the same equation, confirming x-axis symmetry.

Symmetry About the Y-axis

Finding symmetry about the y-axis might remind you of looking in a mirror placed along the y-axis. If the graph looks exactly the same, it is symmetric about the y-axis. To confirm this with an equation, substitute $ x $ with $ -x $.

If the equation looks the same after this swap, it shows that the graph is symmetric with respect to the y-axis.
This implies that for each point $ (x, y) $, there exists a corresponding point $ (-x, y) $.
Common examples include equations like $ y^2 = x^2 $, which are symmetric about the y-axis.

In our given equation $ x^{4} y^{4} + x^{2} y^{2} = 1 $, replacing $ x $ with $ -x $ resulted in the same equation, confirming its symmetry about the y-axis.

Symmetry About the Origin

Symmetry about the origin is slightly different. It's like rotating the graph around the origin (0,0) by 180 degrees. For an equation to be symmetric about the origin, replacing $ x $ with $ -x $ and $ y $ with $ -y $ should yield an unchanged equation.

After making these substitutions, if the equation remains identical, the graph is symmetric about the origin.
This suggests every point $ (x, y) $ is mirrored as $ (-x, -y) $.
An example of origin symmetry is the equation $ y = x^3 $. If reflected, it aligns perfectly with itself.

Our original equation $ x^{4} y^{4} + x^{2} y^{2} = 1 $ also shows symmetry about the origin, which we confirmed because replacing with $ -x $ and $ -y $ maintains the same equation.

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91影视

Test the equation for symmetry. $$x^{4} y^{4}+x^{2} y^{2}=1$$

Short Answer

Step by step solution

Test for Symmetry About the X-axis

Test for Symmetry About the Y-axis

Test for Symmetry About the Origin

Key Concepts

Symmetry About the X-axis

Symmetry About the Y-axis

Symmetry About the Origin

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