2.1 Symmetry

There are 3 symmetries important in graphing When a figure has line symmetry, it can be folded on its line of symmetry and the two have to coincide exactly There are 3 symmetries important in graphing Respect to x-axis y-axis origin Definition If (a, b) is on graph, so is (a, –b) so is (–a, b) so is (–a, –b) Illustration Test Substituting –y for y results in equivalent equation Substituting –x for x results in equivalent equation Substituting –x for x AND –y for y results in equivalent equation (a, b) (–a, b) (a, b) (a, b) (a, –b) (–a, –b)

Typically can do two types of problems Ex 1) Given the point , determine coordinates of the point that satisfies specified symmetry. origin b) y-axis c) x-axis (a, b)  (–a, –b) (a, b)  (–a, b) (a, b)  (a, –b)

Ex 2) Determine symmetry. Graph using the properties of symmetry. x-axis –y = x2 – 4 y = –x2 + 4 not equivalent y-axis y = (–x)2 – 4 y = x2 – 4 equivalent origin –y = (–x)2 – 4 –y = x2 – 4 not equivalent x y 1 2 –4 –3 x y –1 –2 –3 Choose only positive x’s & negative x’s will match

Ex 3) Use graphing calculator to find symmetry.

We can also have graphs symmetric to line y = x Test: switch x & y and get equivalent eqtn (x, y)  (y, x) Even Function If f (–x) = f (x) symmetric to y-axis Odd Function If f (–x) = – f (x) symmetric to origin

Ex 4) Determine if function is even, odd or neither a) odd b) neither c) even

Homework #201 Pg 61 #1–31 odd, 39, 41–46 all