Chapter 1.8, Problem 57E

To determine

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

Answer to Problem 57E

The x-intercept is 4 and y-intercept is 4. The equation is neither symmetric about x-axis, nor y-axis nor origin. Graph of the equation is provided below,

  

Explanation of Solution

Given information:

The equation $y=-x+4$ .

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 $y=-x+4$ . Construct a table to evaluate the value of y for different values of x.

Substitute the point $x=0$ in the equation $y=-x+4$ ,

  $\begin{array}{c}y=0+4\\ =4\end{array}$

Substitute the point $x=1$ in the equation $y=-x+4$ ,

  $\begin{array}{c}y=-1+4\\ =3\end{array}$

Substitute the point $x=2$ in the equation $y=-x+4$ ,

  $\begin{array}{c}y=-2+4\\ =2\end{array}$

Substitute the point $x=3$ in the equation $y=-x+4$ ,

  $\begin{array}{c}y=-3+4\\ =1\end{array}$

Construct a table with the values obtained above,

  $\begin{array}{ccc}x& y& \left(x,y\right)\\ 0& 4& \left(0,4\right)\\ 1& 3& \left(1,3\right)\\ 2& 2& \left(2,2\right)\\ 3& 1& \left(3,1\right)\end{array}$

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

The graph of the equation is provided below $y=-x+4$ .

  

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 $y=-x+4$ ,

  $\begin{array}{c}0=-x+4\\ x=4\end{array}$

Therefore, x-intercept is 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 $y=-x+4$ ,

  $\begin{array}{c}y=-0+4\\ y=4\end{array}$

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 yby$-y$ in the equation $y=-x+4$ ,

  $\begin{array}{c}-y=-x+4\\ x-y=4\end{array}$

The equation is changed.Therefore, the equation $y=-x+4$ 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 $y=-x+4$ ,

  $\begin{array}{c}y=-\left(-x\right)+4\\ y=x+4\end{array}$

The equation is changed. Therefore, the equation $y=-x+4$ is not 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 $y=-x+4$ ,

  $\begin{array}{c}-y=-\left(-x\right)+4\\ -y=x+4\end{array}$

The equation is changed. Therefore, the equation $y=-x+4$ is not symmetricabout the origin.

Thus, the x-intercept is 4 and y-intercept is 4. The equation is neither symmetric about x-axis, nor y-axis nor origin. Graph of the equation $y=-x+4$ is provided below,