Chapter 1, Problem 22RE

To determine

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

Explanation of Solution

Given:

The equation is given as:

  $y=\left|x\right|+9$

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:

  $\begin{array}{l}y=\left|x\right|+9\\ 0=\left|x\right|+9\\ \left|x\right|=-9\end{array}$

Since, the above value is not possible, therefore the x-intercept will not exist.

Similarly, the y-intercept:

  $\begin{array}{l}y=\left|x\right|+9\\ y=9\end{array}$

Therefore, the intercepts is $\left(0,9\right)$ .

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.

  $\begin{array}{l}y=\left|x\right|+9\mathrm{....}\left(1\right)\\ \left(-y\right)=\left|x\right|+9\\ y=-\left(\left|x\right|+9\right)\mathrm{......}\left(2\right)\end{array}$

As equation 1 and 2 are different then the 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.

  $\begin{array}{l}y=\left|x\right|+9\mathrm{....}\left(1\right)\\ y=\left|-x\right|+9\\ y=\left|x\right|+9\mathrm{....}\left(2\right)\end{array}$

As the equation 1 and 2 are equal, hence the graph is 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.

  $\begin{array}{l}y=\left|x\right|+\mathrm{9....}\left(1\right)\\ -y=\left|-x\right|+9\\ y=-\left(\left|x\right|+9\right)\mathrm{....}\left(2\right)\end{array}$

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: