Chapter 1.8, Problem 68E

To determine

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

Answer to Problem 68E

The x-intercept is $-4$ and y-intercept is $2$ . 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 $y=\sqrt{x+4}$ .

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 $y=\sqrt{x+4}$ . Construct a table to evaluate the value of y for different values of x.

Substitute the point $x=0$ in the equation $y=\sqrt{x+4}$ ,

  $\begin{array}{c}y=\sqrt{0+4}\\ =2\end{array}$

Substitute the point $x=1$ in the equation $y=\sqrt{x+4}$ ,

  $\begin{array}{c}y=\sqrt{1+4}\\ =\sqrt{5}\end{array}$

Substitute the point $x=2$ in the equation $y=\sqrt{x+4}$ ,

  $\begin{array}{c}y=\sqrt{2+4}\\ =\sqrt{6}\end{array}$

Substitute the point $x=3$ in the equation $y=\sqrt{x+4}$ ,

  $\begin{array}{c}y=\sqrt{3+4}\\ =\sqrt{7}\end{array}$

Construct a table with the values obtained above,

  $\begin{array}{ccc}x& y& \left(x,y\right)\\ 0& 2& \left(0,2\right)\\ 1& \sqrt{5}& \left(1,\sqrt{5}\right)\\ 2& \sqrt{6}& \left(2,\sqrt{6}\right)\\ 3& \sqrt{7}& \left(3,\sqrt{7}\right)\end{array}$

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

The graph of the equation is provided below $y=\sqrt{x+4}$ .

  

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 $y=\sqrt{x+4}$ ,

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

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 $y=\sqrt{x+4}$ ,

  $\begin{array}{c}y=\sqrt{0+4}\\ =2\end{array}$

Therefore, y-intercept is 2.

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 $y=\sqrt{x+4}$ ,

  $-y=\sqrt{x+4}$

The equation is changed. Therefore, the equation $y=\sqrt{x+4}$ 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 $y=\sqrt{x+4}$ ,

  $y=\sqrt{-x+4}$

The equation is changed. Therefore, the equation $y=\sqrt{x+4}$ 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 $y=\sqrt{x+4}$ ,

  $-y=\sqrt{-x+4}$

The equation is changed. Therefore, the equation $y=\sqrt{x+4}$ is not symmetricabout the origin.

Thus, the x-intercept is $-4$ and y-intercept is 2. The equation is neither symmetric about x-axis, nor y-axis nor origin. Graph of the equation $y=\sqrt{x+4}$ is provided below,