Chapter 1, Problem 118RE

To determine

To graph: The equation $y=-\sqrt{1-x^2}$ and check the equation for symmetry.

Answer to Problem 118RE

The equation $y=-\sqrt{1-x^2}$ is symmetric about $y$ -axis but not about $x$ -axis and the graph is as follows:

  

Explanation of Solution

Given information:

The equation of line $y=-\sqrt{1-x^2}$

Formula used:

The equation is said to have symmetry about the $x$ -axis when the equation remains unchanged on replacing $y$ by $-y$

Graphically it can be said that the graph remains unchanged when reflected in the $x$ -axis.

The equation is said to have symmetry about the $y$ -axis when the equation remains unchanged on replacing $x$ by $-x$

Graphically it can be said that the graph remains unchanged when reflected in the $y$ -axis.

The equation is said to have symmetry about the origin if it is symmetric about $x$ -axis and $y$ -axis that is the equation remains unchanged on replacing $x$ by $-x$ or $y$ by $-y$

Graphically it can be said that the graph remains unchanged when rotated with an angle of ${180}^{\circ }$ about the origin.

Calculation:

It is provided that the equation of line is $y=-\sqrt{1-x^2}$ $\left(1\right)$

Firstly replace $y$ by $-y$ in $\left(1\right)$

  $\begin{array}{l}-y=-\sqrt{1-x^2}\\ y=\sqrt{1-x^2}\end{array}$

The equation obtained is not the same as the original one.

Hence, $y=-\sqrt{1-x^2}$ is not symmetric about $x$ -axis

Next, replace $x$ by $-x$ in $\left(1\right)$

  $\begin{array}{l}y=-\sqrt{1-{\left(-x\right)}^2}\\ y=-\sqrt{1-x^2}\end{array}$

The equation obtained is the same as the original one.

Hence, $y=-\sqrt{1-x^2}$ is symmetric about $y$ -axis.

For plotting:

$x$	$y=-\sqrt{1-x^2}$	$\left(x.y\right)$
$\begin{array}{l}0\\ 1\\ -1\end{array}$	$\begin{array}{l}-1\\ 0\\ 0\end{array}$	$\begin{array}{l}\left(0,-1\right)\\ \left(1,0\right)\\ \left(-1,0\right)\end{array}$

We plot these points to get the graph as follows:

  

Thus, $y=-\sqrt{1-x^2}$ is symmetric about $y$ -axis but not about $x$ -axis.