Problem 32
Question
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. $$y^{5}=x^{4}+2$$
Step-by-Step Solution
Verified Answer
The graph of the equation \(y^{5} = x^{4} + 2\) is symmetric with respect to the y-axis only.
1Step 1 Title: Test for symmetry with respect to the y-axis
Replace \(x\) by \(-x\) in the equation to get \((-x)^{4} + 2\). This simplifies to \(x^{4} + 2\), which is the original equation. Hence, \(y^{5} = x^{4} + 2\) is symmetric with respect to the y-axis.
2Step 2 Title: Test for symmetry with respect to the x-axis
Replace \(y\) by \(-y\) in the equation to get \((-y)^{5} = x^{4} + 2\). This simplifies to \(-y^{5} = x^{4} + 2\), which is not the original equation. Hence, \(y^{5} = x^{4} + 2\) is not symmetric with respect to the x-axis.
3Step 3 Title: Test for symmetry with respect to the origin
Replace \(x\) by \(-x\) and \(y\) by \(-y\) in the equation to get \((-y)^{5} = (-x)^{4} + 2\). This simplifies to \(-y^{5} = x^{4} + 2\), which is not the original equation. Hence, \(y^{5} = x^{4} + 2\) is not symmetric with respect to the origin.
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