Chapter 1.2, Problem 90E

To determine

To describe the type of symmetry for the graph of the circle.

Answer to Problem 90E

There are symmetries about x-axis, y-axis and about origin.

Explanation of Solution

Given information:

The equation of circle is $x^2+y^2=1$ and the graph is,

  

Test the symmetry:

Symmetry about x-axis.

Replace y with -y and get the function same as original.

Symmetry about y-axis.

Replace x with -x and get the function same as original.

Symmetry about origin.

Replace y with -y and x with -x, and get the function same as original.

Calculation:

  $x^2+y^2=1$ .

Symmetry about x-axis.

Replace y with -y.

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

The above function is same as the original function.

So, there is a symmetry about x-axis.

Symmetry about y-axis.

Replace x with -x.

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

The above function is same as the original function.

So, there is a symmetry about y-axis.

Symmetry about origin.

Replace y with -y and x with -x.

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

The above function is same as the original function.

So, there is a symmetry about origin.