Chapter 1, Problem 19RE

To determine

The intercepts, then test for the symmetry and sketch the graph.

Explanation of Solution

Given:

The equation is given as:

  $y=x^3+5$

The intercept can be determined by putting the value of the other variable equal to the zero.

The x intercept is written as (x,0) and the y-intercept is written as (0,y).

Hence, the x -intercept:

  $\begin{array}{l}y=x^3+5\\ 0=x^3+5\\ x=-\sqrt[3]{5}\end{array}$

Similarly, the y-intercept:

  $\begin{array}{l}y=x^3+5\\ y=5\end{array}$

Therefore, the intercepts are $\left(-\sqrt[3]{5},0\right)\text{and }\left(0,5\right)$ .

Now, the test for the symmetry:

Firstly replace the y with the −y and if there is no effect on the equation then the graph will be symmetric with respect to the x-axis.

  $\begin{array}{l}y=x^3+5\mathrm{....}\left(1\right)\\ \left(-y\right)=x^3+5\\ y=-\left(x^3+5\right)\mathrm{......}\left(2\right)\end{array}$

As equation 1 and 2 are different then the system is non-symmetric.

Similarly, replace x with −x and if there is no effect on the equation then the graph is symmetric along the y-axis.

  $\begin{array}{l}y=x^3+5\mathrm{....}\left(1\right)\\ y={\left(-x\right)}^3+5\\ y=-x^3+5\mathrm{....}\left(2\right)\end{array}$

As the equation 1 and 2 are not equal, hence the graph is not symmetric along the y-axis.

Now, replacing the x by −x and y by −y, if there is no effect on the equation then the system will be symmetric along the origin.

  $\begin{array}{l}y=x^3+5\mathrm{....}\left(1\right)\\ -y={\left(-x\right)}^3+5\\ y=x^3-5\mathrm{....}\left(2\right)\end{array}$

As the equation 1 and 2 are not equal then the system is not symmetric along the origin also.

Now the graph is drawn as: