Even and Odd Functions

The function y=f(x) can be known as an:

• EVEN function f(-x)=f(x)

• ODD function f(-x)=-f(x)

Think of y as an even power of x such as y=x^2 or y=x^6. This is an even function because (-x)^2=x^2(any input of x gives the result of the opposite) and (-x)^6=x^6. Meaning that if your x is 3 and you input it into the even function as f(-3) then it will equal its opposite which is 3. When graphing an even function then the function is symmetric at the y-axis. For example any graph in quadrant one will reflect onto quadrant two. An graph in quadrant 3 will reflect onto quadrant four. Picture a circle along the y-axis. The y-axis is the division of that circle, meaning that the pictures have to look similar. If you were to take that circle and fold it (the y-axis would be the fold) then the picture would seem like its folded into one
.

Think of y as an odd function of x such as (-x)^1=-x or (-x)^5=-x^5.
Think of f(-x)=-f(x) as a diagonal. Lets say on the graph you have point (a,b) so this means that point(-a,-b) is on the graph as well. Another example is if you have (-a,b) then you have (a,-b). the odd function is symmetric about the origin. Basically, this is also taking the opposite of each point. Point in quadrant 1 will reflect into quadrant 3 such as points in quadrant 2 will reflect onto quadrant 4. How do these points look on a graph? Well remembering the example using the circle, besides having the division line as the y-axis we will also have the x-axis dividing that circle. So, the circle is picture on both the y-axis and the x-axis where the interception of both lines is considered to be the middle of the circle. Okay so now that you have your circle centered, fold it diagonal. The picture will fold as one, just like the even function, but instead of folding side to side this folds diagonally. TAKE NOTICE: not every point and graph you come across is a circle-ish form so better yet as the book says" A graph is symmetric about the origin if a rotation of 180 degrees about the origin leaves the graph unchanged." This means that whatever points you take, flip them 180 degrees on the graph and you should have the same graph you started with HENCE in its diagonal form.