Question

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

Short answer

Graph of Equation $x^2+y^3=2x^4$is symmetric with to only y-axis.

Step-by-step solution

Step 1. Given data 

The given equation is

$x^2+y^3=2x^4$

Step 2. Symmetry with x-axis

Substitute$-y$for role="math" localid="1646853294064" $y$in the equation to test the symmetric with respect to the x-axis.

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

The equation did not match with the original equation

So the graph of the equation is not symmetric with respect to the x-axis.

Step 3. Symmetry with y-axis

Substitute$-x$for$x$ in the equation to test the symmetric with respect to the y-axis.

role="math" localid="1646853182518" $$\begin{gathered} {\left(-x\right)}^2+y^3=2{\left(-x\right)}^4 \\ {x}^2+y^3=2x^4 \end{gathered}$$

The equation matches the original equation

So the graph of the equation is symmetric with respect to the y-axis.

Step 4. Symmetry with origin

Substitute$-x$for$x$and$-y$for$y$in the equation to test the symmetric with respect to the origin.

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

The equation did not match with the original equation

So the graph of the equation is not symmetric with respect to the origin.