Even / Odd Symmetry I. Function Symmetry. A) Functions can have symmetry that is even or odd. 1) Even symmetry means it is symmetrical to the y-axis. (The left side looks like the right side). a) It looks the same on both sides of the y-axis. 2) Odd symmetry is also called point symmetry, it is a rotational symmetry. (If you rotate the function about the origin 180°, you will get the same shape again.)

Even / Odd Symmetry II. Finding Even/Odd symmetry mathematically. A) A function y = f(x) is even if f(-x) = f(x). 1) If you replace all the x’s with –x’s and simplify, you get the original function. Example: f(x) = 3x2 – 5  3(-x)2 – 5  3x2 – 5 which is f(x). B) A function y = f(x) is odd if f(-x) = –f(x). 1) If you replace all the x’s with –x’s and simplify, you get the original function with all the signs changed. If you then factor out a –1, you get the original function. In other words –1f(x) or –f(x). Example: f(x) = 4x3 – 7x  4(-x)3 – 7(-x)  – 4x3 + 7x Now factor out a –1 which gives us –1(4x3 – 7x) which is –f(x).

Change the value of x to –x. (Change the sign of x). Even / Odd Symmetry III. Even / Odd Symmetry based on ordered pairs. Change the value of x to –x. (Change the sign of x). EVEN symmetry ODD symmetry (2 , 5)  (-2 , 5) (2 , 5)  (-2 , -5) (3 , -7)  (-3 , -7) (3 , -7)  (-3 , 7) (-4 , 9)  (4 , 9) (-4 , 9)  (4 , -9) (-6 , -1)  (6 , -1) (-6 , -1)  (6 , 1) Note the x changed signs, Note the x changed signs, but the y stayed the same. and the y changed signs too. f(-x) = f(x) f(-x) = –f(x)

Even / Odd Symmetry IV. Non-mathematical way to determine even/odd. A) This is NOT an approved method for determining even & odd symmetry. This is only for checking your answers. 1) Even: All the exponents are even (or have no variable). Example: y = -5x6 + 9x4 – 8 is even. proof: -5(-x)6 + 9(-x)4 – 8  -5x6 + 9x4 – 8 which is the same as the original function, so it is even. 2) Odd: All the exponents are odd (or have no exponent). Example: y = -5x7 + 9x3 – 8x is odd. proof: -5(-x)7 + 9(-x)3 – 8(-x)  5x7 – 9x3 + 8x Note: all the signs are switched compared to the original f(x). Factor out a -1 and all the signs are the same, -1(-5x7 + 9x3 – 8x) so f(-x) = -1f(x) which is the definition of odd symmetry.