Even and Odd Functions :]

EVEN Functions

An even function can be defined as f(-x)=f(x) algebraically.

• This means that for every input value of x and its opposite value (x and -x), the output will be the same (y).



• Therefore (x,y) and (-x,y), or (x,-y) and (-x,-y) will be reflections of each other about the y axis if plotted on the same graph.



• Because of this, the graph of an even function will always be symmetrical about about the y axis. (Quadrant 1&2, or Quadrant 3&4).



• Any function with an even power (x^2, x^4, x^6, etc.) or with an absolute value will have these characteristics.

Example: All 3 graphs are symmetrical about the y axis.

In the first graph, x^2, If 1 and -1 (x and -x) are inputted into the equation, the outputs (y) will both be 1.

If 5 and -5 are inputted, the outputs will both be 25.

f(-x)=f(x)

f(-5)=f(5); x= -5, 5

25=25; y=y

ODD Functions

An odd function can be defined as f(-x)= -f(x) algebraically.

• If the point (x,y) or (-x,y) is plotted on a graph, (-x,-y) or (x,-y) must also be a point on the graph, respectively, in order for it to be an odd function.



• The graph of an odd function will always be symmetrical about the origin.



• Symmetry may be on Quadrant 1&3, or Quadrant 2&4 in most, but not all, cases.



• Any function with an odd power (x, x^3, x^5, x^7, etc.) will have these characterisitcs.

Example: All 3 graphs are symmetrical about the origin.

In the first graph, x, if 1 and -1 (x and -x) are inputted, the outputs will be 1 and -1 (y and -y) respectively.

If 5 and -5 are inputted, the outputs will be 5 and -5 respectively.