Chapter 4.5, Problem 7AYU

To determine

To find: The graph of the equation $2x^3-x{y}^2=4$ is symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, the origin, or none of these.

Answer to Problem 7AYU

$x\text{-axis}$ symmetry.

Explanation of Solution

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

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

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

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

Here $\left( x, y\right)$ is on the graph and so is $\left( x,-y\right)$, then the graph is symmetric, with respect to the $x\text{-axis}$.

Here $\left(x, y\right)$ is on the graph and so is not $\left( -x,y\right)$, then the graph is not symmetric, with respect to the $y\text{-axis}$.

If $\left(x, y\right)$ is is on the graph and so is not $\left( -x,-y\right)$, then the graph is not symmetric, with respect to the origin.