2.3   Exploring Symmetry

Suppose  is a point on the Cartesian coordinate system.

(i)                  If we change its abscissa from  to , we get its reflection with respect to the y-axis (see depiction below):

(ii)                If we change its ordinate from  to , we get its reflection with respect to the x-axis (see depiction below):

(iii)               If we change both its abscissa from  to  and its ordinate from  to , we get its reflection with respect to the origin (see depiction below):

Now, given an equation in two variables x and y, we know that its graph consists of all the points (ordered pairs) (x,y) that satisfy the equation.

(i)                  If we change x to –x in the equation, every point on its graph is reflected about the y-axis. The resulting graph of the transformed equation is a reflection of the original equation’s graph about the y-axis.

(ii)                If we change y to –y in the equation, every point on its graph is reflected about the x-axis. The resulting graph of the transformed equation is a reflection of the original equation’s graph about the x-axis.

(iii)               If we change x to –x and y to –y simultaneously in the equation, every point on its graph is reflected about the origin. The resulting graph of the transformed equation is a reflection of the original equation’s graph about the origin.

Symmetry

Definition 1

The above definition is graphical in nature in the sense that we need to examine the graph to determine if exhibits any symmetry. In view of the discussion above, if we know an equation in two variables x and y, we may determine its symmetry (if any) by examining the equation as follows:

Definition 2

Example 1

Consider the following equation: .

Determine if the graph of this equation is symmetric with respect to the y-axis, the x-axis, or the origin.

Solution

(i)                  The substitution of –x for x results in:

, which is the same as the original equation. Therefore, the graph of this equation is symmetric with respect to the y-axis.

(ii)                The substitution of –y for y results in:

, which is the same as the original equation. Therefore, the graph of this equation is symmetric with respect to the x-axis.

(iii)               The substitutions of –x for x and –y for y result in:

, which is the same as the original equation. Therefore, the graph of this equation is symmetric with respect to the origin.

The graph of the original equation is depicted below (it does indeed exhibit all the three symmetries we verified algebraically):

Example 2

Consider the following equation: .

Determine if the graph of this equation symmetric with respect to the y-axis, the x-axis, or the origin.

Solution

(i)                  The substitution of –x for x results in:

.

The resulting equation is not the same as the original. Therefore, the graph of this equation is not symmetric with respect to the y-axis.

(ii)                The substitution of –y for y results in:

The resulting equation is not the same as the original. Therefore, the graph of this equation is not symmetric with respect to the x-axis.

(iii)               The substitutions of –x for x and –y for y result in:

.

The resulting equation is the same as the original. Therefore, the graph of this equation is symmetric with respect to the origin.

The graph of the original equation is as follows: (it does indicate symmetry with respect to the origin)

Exploration

This interactive exploration is designed to help us develop insight into symmetry both graphically and algebraically.

Step 1. Enter an equation in the two variables x and y where indicated.

Step 2. Push the button “graph equation”. This will graph the equation (you may have to adjust the viewing window in some cases to have a complete view of the graph).

Step 3. Choose any of the transformations and push its associated button: “reflection w/r to the x-axis”, “reflection w/r to the y-axis”, or “reflection w/r to the origin”. Note that the transformed equation is displayed as well as its graph.

Activity

1.      Use the interactive exploration page to check symmetry for each of the following equations:

(i)                 

(ii)               

(iii)              

(iv)