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Chapter 1: Problem 36

Use the algebraic tests to check for symmetry with respect to both axes and the origin. $$y=x^{4}-x^{2}+3$$

Short Answer

Expert verified
The graph of the function \(y=x^{4}-x^{2}+3\) is symmetric with respect to the y-axis only.

Step by step solution

01

Test for y-axis symmetry

To test for y-axis symmetry, replace \(x\) with \(-x\). The equation becomes \(y=(-x)^{4}-(-x)^{2}+3 = x^{4} - x^{2} + 3\). The equation is the same as the original, indicating that it is symmetric about the y-axis.
02

Test for x-axis symmetry

To check for x-axis symmetry, replace \(y\) with \(-y\). The equation becomes \(-y=x^{4}-x^{2}+3\). Although the right side of the equation doesn't change, this results in \(-y\), not \(y\). This shows that the function is not symmetric about the x-axis.
03

Test for origin symmetry

To verify symmetry with respect to the origin, replace \(x\) with \(-x\), and \(y\) with \(-y\). This yields \(-y=(-x)^{4}-(-x)^{2}+3 = x^{4} - x^{2} + 3\). While this matches the result from the y-axis test, it doesn't correspond to \(-y\), hence the function isn't symmetric with respect to the origin. Hence, the graph is only symmetric with respect to the y-axis.

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