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Chapter 1: Problem 30

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$x^{2} y^{2}+5 x y=2$$

Short Answer

Expert verified
The graph of the equation \(x^{2} y^{2}+5 x y=2\) is symmetric with respect to the origin, but not with respect to the x-axis or the y-axis.

Step by step solution

01

Test for symmetry with respect to the y-axis

To test for y-axis symmetry, replace \(x\) with \(-x\). With this, the equation becomes \((-x)^{2} y^{2}+5 (-x) y=2\), which simplifies to \(x^{2} y^{2}-5 x y=2\). This is not the same equation as the original, so the graph is not symmetric with respect to the y-axis.
02

Test for symmetry with respect to the x-axis

To test for x-axis symmetry, replace \(y\) with \(-y\). With this, the equation becomes \(x^{2} (-y)^{2}+5 x (-y)=2\), which simplifies to \(x^{2} y^{2}-5 x y=2\). This is not the same equation as the original, so the graph is not symmetric with respect to the x-axis.
03

Test for symmetry with respect to the origin

To test for origin symmetry, replace \(x\) and \(y\) with \(-x\) and \(-y\) respectively. With this, the equation becomes \((-x)^{2} (-y)^{2}+5 (-x) (-y)=2\), which simplifies to \(x^{2} y^{2}+5 x y=2\). This is the same equation as the original, so the graph is symmetric with respect to the origin.

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Key Concepts

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

Understanding Y-Axis Symmetry in Graphs
Graphs that show symmetry with respect to the y-axis have a special quality: if you fold the graph along the y-axis, both halves will match up perfectly. To determine if this symmetry exists, we replace every instance of the variable \(x\) in the equation with \(-x\).

Once the replacement is made, simplify the equation. You are looking to see if the equation still looks exactly the same as it did before making any substitutions.
  • If the new equation is identical to the original, then the graph is symmetric with respect to the y-axis.
  • If the new equation is different, then it is not symmetric with the y-axis.

For example, in the exercise provided, after replacing \(x\) with \(-x\), the equation became \(x^2 y^2 - 5xy = 2\) instead of \(x^2 y^2 + 5xy = 2\). The change in the expression indicates that there is no y-axis symmetry in this graph.
Exploring X-Axis Symmetry in Graphs
A graph that is symmetric about the x-axis has the same appearance above and below this axis. This is like a mirror image across the horizontal line of the graph.

To check for x-axis symmetry, replace each occurrence of \(y\) in the equation with \(-y\).
  • Simplify the equation and observe if it remains unchanged.
  • If it stays the same, the graph is symmetric with respect to the x-axis.
  • If it looks different, there is no x-axis symmetry.

In our exercise example, after replacing \(y\) with \(-y\), the resulting equation \(x^2 y^2 - 5xy = 2 \) differs from the original. Hence, this graph does not possess symmetry concerning the x-axis.
Examining Origin Symmetry in Graphs
Origin symmetry in graphs is when a graph looks the same when rotated 180 degrees around the center origin point (0,0).

To determine if such symmetry is present, both variables need to undergo sign changes: replace \(x\) with \(-x\) and \(y\) with \(-y\).
  • After transforming the equation, check if it reverts to the original form.
  • If it does, then indeed, the graph has symmetry about the origin.
  • If it doesn't, then origin symmetry is absent.

When applied to the given exercise, substituting both \(x\) and \(y\) resulted in returning back to the original equation \(x^2 y^2 + 5xy = 2\). Therefore, this confirms that the graph of the given equation is symmetric about the origin.

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