1. Symmetries of Graphs of Equations in x and y

2. Symmetries of Graphs of Equations in x and y
Terminology
Graphical Interpretation
Test for symmetry
The graph is symmetric with respect to y-axis
(1) Substitution of –x for x leads to the same equation
The graph is symmetric with respect to x-axis
(2) Substitution of –y for y leads to the same equation
The graph is symmetric with respect to origin
(3) Substitution of –x for x and Substitution of –y for y
leads to the same equation
(x,y)
(-x,y)
(x,y)
(x,-y)
(x,y)
(-x,-y)

3. Example 3 Complete the graph of the following if
a) Symmetric w.r.t y-axis
b) Symmetric w.r.t origin
c) Symmetry with respect to y-axis
d) Symmetry with respect to origin
e) Symmetric w.r.t x-axis

4. (-y) = -2 (-x)5 + 4( -x )3 +7(-x)
Example 4
Determine whether an equation is symmetric w.r.t y-axis, x-axis ,origin or none
a) y = 3x4 + 5x2 – b) y = -2x5 +4x3 +7x c) y = x3 +x2
Substitute x by –x
y = ( -x )5 + ( -x )2
= - x5 + x2
Different equation
Solution:
Substitute x by –x and y by - y
(-y) = -2 (-x)5 + 4( -x )3 +7(-x)
= 2x5 – 4x3 – 7x
= - (-2x5 +4x3 +7 )
Same equation
Substitute x by –x
y = 3( -x )4 + 5 ( -x )2 - 4
= 3x4 + 5x2 – 4
Same equation
Symmetry w.r.t y-axis
Even if we substitute –y for y, we get different equations
Symmetry w.r.t origin

5. The End