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Chapter 2: Problem 54

Determine whether the graph is symmetric with respect to the \(x\) -axis, the \(y\) -axis, and the origin. $$\left(x^{2}+y^{2}\right)^{2}=2 x y$$ (GRAPH CAN'T COPY)

Short Answer

Expert verified
The graph is symmetric with respect to the origin only.

Step by step solution

01

- Symmetry with respect to the x-axis

To check if the graph is symmetric with respect to the x-axis, replace every \( y \) with (\( - y \) ) and simplify: \begin{align*} (x^2 + (-y)^2)^2 & = 2x(-y) => (x^2 + y^2)^2 & = -2xy \rightarrow (1) As equation by replacing \( y\) with \( -y \) is changed it is not symmetric with respect to the \( x-axis. \)
02

- Symmetry with respect to the y-axis

To check if the graph is symmetric with respect to the y-axis, replace every x with - x and simplify: \begin{align*} ((-x)^2 + y^2)^2 = 2(-x) y \ \rightarrow (x^2 + y^2)^2 = -2xy => (2) As equation by replacing \(x \) with \( -x \) is same, thus graph is not symmetric with respect to the \( y \) -axis.
03

- Symmetry with respect to the origin

To check if the graph is symmetric with respect to the origin, replace every \(x \) with \( - x \) and every \( y \) with \( -y \) and simplify. \begin{align*} ((-x)^2 + (-y)^2)^2 = 2(-x)(-y) \rightarrow (x^2 + y^2)^2 = 2xy \rightarrow (3) As the original equation is \(\rightarrow (2xy \rightarrow at equation after replacing \) (1) same after graph is symmetric with respect to origin.

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Key Concepts

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

x-axis Symmetry
To determine if a graph is symmetric with respect to the x-axis, you replace all instances of \(y\) in the equation with \(-y\). This transformation checks if flipping the graph over the x-axis leaves the equation unchanged. After substituting \(-y\) for \(y\) in our equation:

\[ (x^2 + (-y)^2)^2 = 2x(-y) \rightarrow (x^2 + y^2)^2 = -2xy \]. We see that the equation changes, so the graph is not symmetric with respect to the x-axis. Remember: if the result matches the original equation, that would mean the graph is symmetric about the x-axis.
y-axis Symmetry
Checking for y-axis symmetry involves replacing every instance of \(x\) with \(-x\) in the equation. This transformation reflects the graph across the y-axis. Let's apply this change to our equation:

\[ ((-x)^2 + y^2)^2 = 2(-x)y \rightarrow (x^2 + y^2)^2 = -2xy \].

As we can observe, the equation does not revert to its original form after this substitution. Therefore, the graph is not symmetric about the y-axis. For a graph to be y-axis symmetric, the equation should stay the same after replacing \(x\) with \(-x\).
Origin Symmetry
Origin symmetry is checked by replacing \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. This transformation is like flipping the graph over both the x-axis and y-axis. Applying this to our equation, we get:

\[ ((-x)^2 + (-y)^2)^2 = 2(-x)(-y) \rightarrow (x^2 + y^2)^2 = 2xy \].

In this case, the equation reverts to its original form. Hence, the graph is symmetric with respect to the origin. Remember: origin symmetry means the equations must remain the same after flipping both variables.

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