Q. 76 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 69–76, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the x-axis, the y-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to assist in drawing the graph. Verify your results using a graphing utility.

$y=x^3-x$

Step-by-Step Solution

Answer

1. The x-intercepts are -1,0, and 1 and the y-intercept is 0.
2. The equation is symmetric with respect to the origin.
3. The graph is shown below:

Step 1: Given information

The given equation is $y=x^3-x$.

Step 2: Test of symmetry

Perform the test for symmetry with respect to the x-axis. Put $y=-y$

$$\begin{gathered} -y=x^3-x \\ y=x-x^3 \end{gathered}$$

Since after substituting $y=-y$ into the given equation, the resulting equation is not equivalent to the original equation hence the graph of the given equation is not symmetrical about the x-axis.

Perform the test for symmetry with respect to the y-axis. Put $x=-x$

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

Since after substituting $x=-x$ into the given equation, the resulting equation is not equivalent to the original equation hence the graph of the given equation is not symmetrical about the y-axis.

Perform the test for symmetry with respect to the origin. Put $x=-x ; y=-y$

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

Since after substituting $x=-x ; y=-y$ into the given equation, the resulting equation is equivalent to the original equation hence the graph of the given equation is symmetrical about the origin.

Step 3: Determine the x-intercept and y-intercept.

For x-intercept, substitute $0$ for $y$ and then solve for $x$.

$$\begin{gathered} 0=x^3-x \\ x\left(x^2-1\right)=0 \\ x\left(x-1\right)\left(x+1\right)=0 \end{gathered}$$

For y-intercept, substitute $0$ for $x$ and then solve for $y$.

$$\begin{gathered} y=0^3-0 \\ =0 \end{gathered}$$

The graph is shown below:

Step 4: Verify the graph using the graphing utility.

By using the graphing utility, the graph is shown below:

Step 5: Write the conclusion

1. The x-intercept are $\left(0,0\right),\left(1,0\right),\left(-1,0\right)$ and y-intercept is $\left(0,0\right)$.
2. The graph is symmetric with respect to origin.
3. The graph is shown below: