Chapter 1, Problem 10RE

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 10RE

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

Explanation of Solution

Given:

  $y=5x$

Calculation:

In order to find the $x$ -intercept, substitute 0 for $y$ in the given equation and evaluate.

  $\begin{array}{l}0=5x\hfill \\ x=0\hfill \end{array}$

The $x$ intercept is 0

Now, substitute 0 for $x$ and evaluate to find the $y$ -intercept.

  $\begin{array}{l}y=5(0)\\ =0\end{array}$

The $y$ intercept is 0

Test for the symmetry.

  $x_{-}Axis.$ :

In order to test for symmetry with respect to the $x$ -axis, replace $y$ by $-y$

  $-y=5x$

Since $y=5x$ is not equivalent to $-y=5x,$ so the graph of the equation is not symmetric withrespect to the $x$ -axis.

  $y-$ Axis:

Replace $x$ by $-x,$ to test for symmetry with respect to the $y$ -axis.

  $y=5(-x)=-5x$

Since $y=5x$ is not equivalent to $y=-5x,$ so the graph of the equation is not symmetric with respect to the $y$ -axis.

Origin:

To test for symmetry with respect to the origin, replace $x$ by $-x,$ and $y$ by $-y$

  $-y=5(-x)$

  $-y=-5x$ Or $y=5x$

Since $y=5x$ is equivalent to $-y=-5x,$ so the graph of the equation is symmetric with respect to the origin.

Conclusion:

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