Chapter 1.8, Problem 75E

To determine

To calculate: The x- and y- intercepts for the equation $y=4-\left|x\right|$ and test the symmetry of the equation. Also construct the table to sketch the graph of the equation.

Answer to Problem 75E

The x-intercepts are $\pm 4$ and y-intercept is $4$ . The equation is symmetric about y-axis.Graph of the equation $y=4-\left|x\right|$ is provided below,

  

Explanation of Solution

Given information:

The equation $y=4-\left|x\right|$ .

Formula used:

The function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

Calculation:

It is provided that the equation is $y=4-\left|x\right|$ . Construct a table to evaluate the value of y for different values of x.

Substitute the point $x=-4$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|-4\right|\\ y=4-4\\ y=0\end{array}$

Substitute the point $x=0$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|0\right|\\ y=4-0\\ y=4\end{array}$

Substitute the point $x=2$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|2\right|\\ y=4-2\\ y=2\end{array}$

Substitute the point $x=4$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|4\right|\\ y=4-4\\ y=0\end{array}$

Construct a table with the values obtained above,

  $\begin{array}{ccc}x& y& \left(x,y\right)\\ -4& 0& \left(-4,0\right)\\ 0& 4& \left(0,4\right)\\ 2& 2& \left(2,2\right)\\ 4& 0& \left(4,0\right)\end{array}$

In the coordinate plane plot the points obtained above and connect them through a line.

The graph of the equation is provided below $y=4-\left|x\right|$ .

  

Recall that the x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

Substitute $y=0$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}0=4-\left|x\right|\\ 4=\left|x\right|\\ x=\pm 4\end{array}$

Therefore, x-intercepts are $\pm 4$ .

Recall that the y-intercepts are the points on x-axis where the graph of the equation intersects the y-axis.

Substitute $x=0$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|0\right|\\ y=4-0\\ y=4\end{array}$

Therefore, y-intercept is $4$ .

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

Replace y by $-y$ in the equation $y=4-\left|x\right|$ ,

  $-y=4-\left|x\right|$

The equation is changed. Therefore, the equation $y=4-\left|x\right|$ is not symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}y=4-\left|-x\right|\\ y=4-\left|x\right|\end{array}$

The equation is unchanged. Therefore, the equation $y=4-\left|x\right|$ is symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ and y by $-y$ in the equation $y=4-\left|x\right|$ ,

  $\begin{array}{c}-y=4-\left|-x\right|\\ -y=4-\left|x\right|\end{array}$

The equation is changed. Therefore, the equation $y=4-\left|x\right|$ is not symmetricabout the origin.

Thus, the x-intercepts are $\pm 4$ and y-intercept is $4$ . The equation is symmetric about y-axis.Graph of the equation $y=4-\left|x\right|$ is provided below,