Chapter 1, Problem 45RE

To determine

To sketch:the graph of the given equation

Answer to Problem 45RE

The Equation is symmetric with respect to the origin and the points to locate the graphis $\left(-3,-27\right)$ , $\left(-2,-8\right)$ , $\left(-1,-1\right)$ , $\left(0,0\right)$ , $\left(1,1\right)$ , $\left(2,8\right)$ , $\left(3,27\right)$ .

  

Explanation of Solution

Given: $y=x^3$

Calculation:

X-axis: To test for symmetry with respect to x-axis, replace y by −y in the equation.

Since, $-y=x^3$ is not equivalent with $y=x^3$ . The graph is not symmetric with respect to the x-axis.

Y-axis: To test for symmetry with respect to y-axis, replace x by −x in the equation,

Since, $y=-x^3$ is not equivalent to $y=x^3$ . The graph is not symmetric with respect to the y-axis.

Origin: To test for symmetry with respect to origin, replace x by −x and y by −y, since, $-y=-x^3$ is equivalent to $y=x^3$ . The graph is symmetric with respect to the origin.

To graph by hand, use the equation to obtain several points on the graph. Because of the symmetry with respect to the origin, need to locate points on the graph for which $x\ge 0$ .

$x$ $y=x^3$ points

$-3$ $-27$ $\left(-3,-27\right)$ $-2$   $-8$ $\left(-2,-8\right)$   $-1$ $-1$ $\left(-1,-1\right)$   $0$ $0$ $\left(0,0\right)$   $1$ $1$ $\left(1,1\right)$   $2$ $8$ $\left(2,8\right)$   $3$ $27$ $\left(3,27\right)$

To graph the equation $y=-x^3$ using graphical utility, plot all the points as listed in table and connect them to obtain the graph of equation.

The graph of $y=-x^3$ is given below;

  

Conclusion:

Therefore, the equation is symmetric with respect to the origin and the points to locate the graph is $\left(-3,-27\right)$ , $\left(-2,-8\right)$ , $\left(-1,-1\right)$ , $\left(0,0\right)$ , $\left(1,1\right)$ , $\left(2,8\right)$ , $\left(3,27\right)$ .