Questions: Determine whether the given graph is symmetric with respect to the x-axis, the y-axis, and the origin. x^4 = y^3(4+x^2)

Determine whether the given graph is symmetric with respect to the x-axis, the y-axis, and the origin. x^4 = y^3(4+x^2)
Transcript text: Determine whether the given graph is symmetric with respect to the $x$-axis, the $y$-axis, and the origin. \[ x^{4}=y^{3}\left(4+x^{2}\right) \]
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Solution

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Solution Steps

Step 1: Identify the Symmetry with Respect to the x-axis

To determine if the graph is symmetric with respect to the x-axis, we need to check if replacing \( y \) with \( -y \) in the equation results in the same equation.

Step 2: Substitute \( y \) with \( -y \)

Given the equation: \[ x^4 = y^3 (4 + x^2) \]

Replace \( y \) with \( -y \): \[ x^4 = (-y)^3 (4 + x^2) \]

Step 3: Simplify the Equation

Simplify the equation: \[ x^4 = -y^3 (4 + x^2) \]

Step 4: Compare the Equations

Compare the original equation \( x^4 = y^3 (4 + x^2) \) with the new equation \( x^4 = -y^3 (4 + x^2) \). They are not the same because of the negative sign in front of \( y^3 \).

Final Answer

No, the graph is not symmetric with respect to the x-axis.

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