Chapter 1, Problem 14RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 14RE

The $x$ intercepts are 0,-1 and 1

The $y$ intercept is 0.

The graph of the equation is symmetric withrespect to the origin.

Explanation of Solution

Given:

  $y=x^3-x$

Calculation:

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

  $\begin{array}{l}0=x^3-x\hfill \\ 0=x\left(x^2-1\right)\hfill \\ x=0\text{ or }{x}^2-1=0\hfill \end{array}$

Solve.

  $\begin{array}{ccccc}x=0& x+1& =0& \text{ or }& x-1=0\\ x& =-1& \text{ or }& x=1& \end{array}$

The $x$ intercepts are 0,-1 and 1

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

  $y=0^3-0=0$

The $y$ intercept is 0

We can test for the symmetry.

  $x-$ Axis:

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

  $-y=x^3-x$

Since $y=x^3-x$ is not equivalent to $-y=x^3-x,$ so the graph of the equation is not symmetricwith respect to the $x$ -axis.

  $y$ -Axis:

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

  $y={(-x)}^3-(-x)=-x^3+x$

Since $y=x^3-x$ is not equivalent to $y=-x^3+x,$ 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 $-y$

  $-y={(-x)}^3-(-x)=-x^3+xy=x^3-x$

Since $y=x^3-x$ is equivalent to $-y={(-x)}^3-x,$ so the graph of the equation is symmetric withrespect to the origin.

Conclusion:

The $x$ intercepts are 0,-1 and 1

The $y$ intercept is 0.

The graph of the equation is symmetric withrespect to the origin.