Chapter 1.8, Problem 78E

To determine

To calculate:Whether the equation $x=y^4-y^2$ is symmetric about x-axis or y-axis or origin.

Answer to Problem 78E

The equation $x=y^4-y^2$ is symmetric only with respect to x-axis.

Explanation of Solution

Given information:

The equation $x=y^4-y^2$ .

Formula used:

The function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Calculation:

It is provided that the equation is $x=y^4-y^2$ . Construct a table to evaluate the value of y for different values of x.

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

Replace y by $-y$ in the equation $x=y^4-y^2$ ,

  $\begin{array}{c}x={\left(-y\right)}^4-{\left(-y\right)}^2\\ x=y^4-y^2\end{array}$

The equation is unchanged. Therefore, the equation $x=y^4-y^2$ is symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ in the equation $x=y^4-y^2$ ,

  $-x=y^4-y^2$

The equation is changed. Therefore, the equation $x=y^4-y^2$ is not symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ and y by $-y$ in the equation $x=y^4-y^2$ ,

  $\begin{array}{c}-x={\left(-y\right)}^4-{\left(-y\right)}^2\\ -x=y^4-y^2\end{array}$

The equation is changed. Therefore, the equation $x=y^4-y^2$ is not symmetricabout the origin.

Thus, the equation is symmetric only about x-axis.