Chapter 1, Problem 10RE

In Problems 10-14, test each equation for symmetry with respect to the $x-axis$, the $y-axis$, and the origin.

$2x=3y^2$

To determine

To test: The equation $2x=3y²$ for symmetry, with respect to the $x\text{-axis}$, $y\text{-axis}$ and the origin

Answer to Problem 10RE

Solution:

$\text{The graph of the given equation is symmetric, with respect to }x\text{-axis}$.

$\text{The graph of the given equation is not symmetric, with respect to }y\text{-axis}$.

$\text{The graph of the given equation is not symmetric, with respect to the origin}$.

Given:

The equation $2x=3y²$

Explanation of Solution

Replace $y$ by $-y$ in the equation $2x=3y²$.

$2x=3\left(-y\right)²$

$2x=3y²$, which is equivalent to the given equation.

Hence the graph is symmetric, with respect to the $x\text{-axis}$.

Replace $x$ by $-x$.

$2\left(-x\right)=3y²$

$-2x = 3y²$, which is not equivalent to the given equation.

Hence the graph is not symmetric, with respect to the $y\text{-axis}$.

Replace $x$ by $-x$ and $y$ by $-y$.

$2\left(-x\right)=3\left(-y\right)²$

$-2x = 3y²$, which is not equivalent to the given equation.

Hence the graph is not symmetric, with respect to the origin.