Chapter 1, Problem 9RE

To determine

To list: the intercepts and test for symmetry with respect to the $x$ -axis, the $y$ -axis, and the origin.

Answer to Problem 9RE

The intercepts is the origin (0,0) and the graph is symmetric with respect to the $x$ -axis.

Explanation of Solution

Given:

  $2x=3y^2$

Calculation:

In order to find the $x$ -intercept, let $y=0$ and solve for $x$

  $2x=0$

Divide both the sides by 2

  $x=0$

The $x$ -intercept is (0, 0)

For finding the $y$ -intercept, let $x=0$ and solve for y

  $3y^2=0$

Divide both the sides by 3

  $\begin{array}{l}{y}^2=0\\ y=0\end{array}$

The $y$ -intercepts is (0, 0)

Test for symmetry with respect to the $x$ -axis, $y$ -axis and the origin.

  $x$ -Axis:

For testing symmetry with respect to the $x$ -axis, replace $y$ by $-y$ .

  $\begin{array}{l}2x=3{(-y)}^2\hfill \\ 2x=3y^2\hfill \end{array}$

Since this equation is equivalent to the original equation, the graph of the equation $2x=3y^2$ is symmetric with respect to the $x$ -axis.

  $y-$ Axis:

For testing symmetry with respect to the $y$ -axis, replace $x$ by $-x$ .

  $\begin{array}{l}2(-x)=3y^2\hfill \\ -2x=3y^2\hfill \end{array}$

Since this equation is not equivalent to the original equation, the graph of the equation $2x=3y^2$ is not symmetric with respect to the $y$ -axis.

Origin:

For testing symmetry with respect to the origin, replace $x$ by $-x$ and $y$ by $-y$

  $\begin{array}{l}2(-x)=3{(-y)}^2\hfill \\ -2x=3y^2\hfill \end{array}$

Since this equation is not equivalent to the original equation, the graph of the equation $2x=3y^2$ is not symmetric with respect to the origin.

Therefore, the intercepts is the origin (0,0) and the graph is symmetric with respect to the

  $x$ -axis

Conclusion:

Therefore, the intercepts is the origin (0,0) and the graph is symmetric with respect to the

  $x$ -axis