Chapter 9, Problem 5CR

To determine

The symmetry with respect to the $x-$ axis, the $y-$ axis and the origin for the equation $x^2+y^3=2x^4$ .

Answer to Problem 5CR

Solution:

The graph of the equation $x^2+y^3=2x^4$ is not symmetric with respect to both $x$-axis and the origin, but it is symmetric with respect to $y$ -axis .

Explanation of Solution

Given information:

The equation $x^2+y^3=2x^4$ .

To test for symmetry with respect to $x$ -axis, replace $y$ by $-y$ . If it results equivalent to original equation, then the equation is symmetric with respect to $x$ -axis.

Replace $y$ by $-y$ in the given equation,

  $\Rightarrow x^2+{\left(-y\right)}^3=2x^4$

  $\Rightarrow x^2-y^3=2x^4$ .

This is not equivalent to the equation $x^2+y^3=2x^4$ .

Hence, the graph of the equation $x^2+y^3=2x^4$ is not symmetric with respect to $x$ -axis.

To test for symmetry with respect to $y$ -axis, replace $x$ by $-x$ . If it results equivalent to original equation, then the equation is symmetric with respect to $y$ -axis.

Replace $x$ by $-x$ in the given equation,

  $\Rightarrow {\left(-x\right)}^2+y^3=2{\left(-x\right)}^4$ .

  $\Rightarrow x^2+y^3=2x^4$ .

This is equivalent to the equation $x^2+y^3=2x^4$ .

Hence, the graph of the equation $x^2+y^3=2x^4$ is symmetric with respect to $y$ -axis.

To test for symmetry with respect to origin, replace $x$ by − $x$ and $y$ by $-y$ . If it results equivalent to the original equation, then the equation is symmetric with respect to the origin.

Replace $x$ by − $x$ and $y$ by $-y$ in the given equation,

  $\Rightarrow {\left(-x\right)}^2+{\left(-y\right)}^3=2{\left(-x\right)}^4$

  $\Rightarrow x^2-y^3=2x^4$

This is not equivalent to the equation $x^2+y^3=2x^4$ .

Hence, the graph of the equation $x^2+y^3=2x^4$ is not symmetric with respect to the origin.