In this article, I will show you how you can tell if a graph is even or odd just by looking at it. Knowing if a given Function is Even or Odd is a big help and great benefit as it helps you to graph the function by telling you, which half of the points you need to graph.
To tell it simply, whenever a function is even the graph will be shown as asymmetrical on the y-axis and If the said function is odd then the graph will become symmetrical on the origin i.e (0,0)
Note: There is actually a third possibility of what a function might be, Neither. That’s true a function can be Even, Odd, or Neither too.
Check If the function satisfies f(x)=−f(−x) f ( x ) = − f ( − x ) and If it does, it will be known as odd. If the function in question does not satisfy either of the rules, it will be determined as neither the even nor the odd.
What is the Even function?
According to the mathematical definition, an even function is
f(–x) = f(x) for any value of x.
A quick example of it is f(x) = x3 because f(x)=f(-x) for all of the x.
For example, f(4) = 10, and f(–4) = 10.
In short, you can say, the opposite input yields the same output every time.
As you can see in the picture above, the graph looks like a mirror image on both sides.
What is Odd Function?
According to the mathematical definition, an odd function is f(–x) = –f(x) for any value of x. So, the opposite inputs always give the opposite outputs. As you can see, these graphs have a 180-degree symmetry on their origin. Even If you were to turn this graph upside down, it will still look the same as before.
As shown above, f(x) = x4, it is an odd function because f(-x)=-f(x) for all x. For example, f(4) = 28 and f(–4) = –28.
- Make a graph of the function
You can use a graphing calculator or a physical graph paper to draw the graph of the function. Choose any value for ‘x’ and insert it in the function, now, we can calculate the resulting value of ‘y’.
After plotting several of these points onto the graph, we can connect them to see the function graph.
- Testing a function across the y-axis for symmetry
Symmetry means a mirror image of the function you are looking at. When you see a part of the graph matching on both the positive and negative sides of the y-axis then it means that the graph is symmetrical across the y-axis. What it means for the graph is that the function is symmetrical across the y-axis which in turn means the function is even.
You can also check if the symmetry is even by selecting any of the individual’s points, if the y-value of any selected x is the same as the y-value of the -x then the function is even.
- Testing a Function for its origin symmetry
The Origin is referred to the starting point of the graph that is (0,0). What it entails is that any positive result for chosen x-value will directly correspond to the negative result for -x and the same can be said for the Vice Versa. Also, odd functions will display origin symmetry.
When selecting sample values for x and also their opposite corresponding -x values, you will always get the opposite results.
- Looking at a function for no Symmetry
This is the final example, looking from side to side of a function to see if it has symmetry. While looking at the graph you will see it is not a mirror image across the y-axis or around the origin.
I hope this little article was helpful and gave you enough insight on how to tell if a graph is even or odd.