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Chapter 2: Problem 43

Check for symmetry with respect to both axes and the origin. \(x y=2\)

Short Answer

Expert verified
The given equation \(x y=2\) is not symmetric with respect to either the x-axis or the y-axis,but it is symmetric with respect to the origin.

Step by step solution

01

Check for Symmetry about the x-axis

Replace \(y\) with \(-y\) in the equation and see if the resulting equation is equivalent to the original equation. If \(x (-y) = 2\), then this simplifies to \(-x y = 2\), which is not equivalent to the original equation \(x y = 2\). Therefore, the equation is not symmetric about the x-axis.
02

Check for Symmetry about the y-axis

Replace \(x\) with \(-x\) in the equation and see if the resulting equation is equivalent to the original equation. If \((-x) y = 2\), then this simplifies to \(-x y = 2\), which is not equivalent to the original equation \(x y = 2\). Therefore, the equation is not symmetric about the y-axis.
03

Check for Symmetry about the Origin

Replace both \(x\) and \(y\) with \(-x\) and \(-y\), respectively, in the equation and see if the resulting equation is equivalent to the original equation. If \((-x)(-y) = 2\), then this simplifies to \(x y = 2\), which is equivalent to the original equation. Therefore, the equation is symmetric about the origin.

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Key Concepts

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

Understanding X-Axis Symmetry
To determine if a graph displays x-axis symmetry, replace the variable \( y \) with \( -y \) in the equation. This transformation checks whether flipping the graph over the x-axis results in the same graph.
  • If the resulting equation matches the original, the graph is symmetric with respect to the x-axis.
  • If it does not, as in the case of \( xy = 2 \) which becomes \(-xy = 2\), then the graph does not possess x-axis symmetry.
This kind of symmetry means that for every point \( (x, y) \), there is a corresponding point \( (x, -y) \) on the graph.
Exploring Y-Axis Symmetry
Y-axis symmetry involves replacing the \( x \) variable with \( -x \) in your equation. By doing this, we are testing if reflecting the graph across the y-axis retains the original equation.
  • If the transformed equation equals the original, then y-axis symmetry exists.
  • For the equation \( xy = 2 \), replacing \( x \) with \( -x \) results in \(-xy = 2\), which is not the same as the original equation.
Thus, the graph of \( xy = 2 \) is not symmetric about the y-axis. Y-axis symmetry implies that for each point \( (x, y) \), there is an equivalent point \( (-x, y) \) on the graph.
Demystifying Origin Symmetry
Origin symmetry can be a bit trickier to spot than other types. This involves replacing both \( x \) with \( -x \) and \( y \) with \( -y \). Essentially, we are checking if a 180-degree rotation about the origin brings us back to our original graph.
  • Should the modified equation be equivalent to the original, then the equation has origin symmetry.
  • For example, transforming \( xy = 2 \) to \((-x)(-y) = 2\) simplifies back to \(xy = 2\).
This confirms that the graph for \( xy = 2 \) is symmetric about the origin. Origin symmetry suggests that for every point \( (x, y)\), there is a corresponding point \( (-x, -y) \). This type of symmetry indicates that the graph can be effectively rotated around the origin without changing its appearance.

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Most popular questions from this chapter

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=-\frac{x}{4}\)

The point \((8,2)\) on the graph of \(f(x)=\sqrt[3]{x}\) has been shifted to the point \((5,0)\) after a rigid transformation. Identify the shift and write the new function \(h\) in terms of \(\bar{f}\)

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x+7\)

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In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x-5\)
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