Symmetries of Graphs of Equations in x and y

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)

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

(-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 –4 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

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