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Chapter 1: Problem 39

Use the algebraic tests to check for symmetry with respect to both axes and the origin. \(x y^{2}+10=0\)

Short Answer

Expert verified
The graph of the equation \(x y^{2}+10=0\) is symmetric with respect to the x-axis and the origin, but not the y-axis.

Step by step solution

01

Check for symmetry with respect to y-axis

To test for symmetry about the y-axis, replace \(x\) with \(-x\) in the equation. This gives us \(-x y^2 + 10\). Because \(-xy^{2}+10\neq xy^{2}+10\), the graph is not symmetric with respect to the y-axis.
02

Check for symmetry with respect to x-axis

To test for symmetry about the x-axis, replace \(y\) with \(-y\) in the equation. This gives us \(x(-y)^{2}+10 = xy^{2}+10\). Since the equation remains the same, the graph is symmetric with respect to the x-axis.
03

Check for symmetry with respect to the origin

To test for symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. This gives us \(-x(-y)^2 + 10 = xy^{2}+10\). Since the equation remains the same, 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.

Y-Axis Symmetry
Symmetry with regard to the y-axis means that if you fold the graph along the y-axis, both halves would match perfectly. It implies that the left side of the graph is a mirror image of the right side around the y-axis.

To test for y-axis symmetry mathematically:
  • Replace the variable \(x\) with \(-x\) in the given equation.
  • Simplify the equation and compare it to the original.
  • If the new equation matches the original one, the graph is symmetric with the y-axis.
For the equation \(x y^2 + 10 = 0\), after replacing \(x\) with \(-x\), we obtain \(-x y^2 + 10\). Since these two expressions are not equivalent, the graph does not exhibit y-axis symmetry. This indicates that the equation's graph is not mirrored when flipped along the vertical y-axis.
X-Axis Symmetry
When a graph is symmetric with respect to the x-axis, it means the graph can be folded over the x-axis resulting in matching halves. In simpler terms, the top half reflects perfectly over to the bottom half along the x-axis.

Here's how we test for x-axis symmetry:
  • Replace \(y\) with \(-y\) in the equation.
  • Simplify the resulting equation.
  • If it matches the original equation, the graph is symmetric about the x-axis.
For the equation \(x y^2 + 10 = 0\), we replace \(y\) with \(-y\), which results in \(x(-y)^2 + 10 = x y^2 + 10\). The equation stays the same, confirming the graph is symmetric with respect to the x-axis. This means anything going up on the graph will identically go down across the x-axis.
Origin Symmetry
Origin symmetry implies that the graph reflects through the origin point \((0,0)\), meaning it looks the same after a 180-degree rotation around the origin. If you imagine placing a pin at the origin and rotating the graph, the two positions would coincide.

To check for origin symmetry with an equation:
  • Replace \(x\) with \(-x\) and \(y\) with \(-y\).
  • Simplify and see if the equation remains unchanged.
  • If it is unchanged, the graph possesses symmetry about the origin.
For the given equation \(x y^2 + 10 = 0\), substituting \(-x\) and \(-y\) gives us \(-x(-y)^2 + 10 = xy^2 + 10\), leading to the same equation. Therefore, this graph is symmetric about the origin, meaning you would see the same graph if you spun it around the origin.

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