How to determine if a graph is symmetric with respect to the origin?

Graphs can be used to predict whether a function is even or odd. This is usually determined by checking the symmetry of a particular function about the y-axis or the origin. If the graph is symmetric about the y-axis, then it is even, whereas, if the graph is symmetric about the origin, then it is odd. Let's see how we can check the symmetry with respect to the origin.

Answer: To determine if a graph is symmetric with respect to the origin, we have to check whether we get the same equation back when we replace y by -y and x by -x.

Let's understand with the help of an example.

Explanation:

For a graph to be odd, we check its symmetry about the origin.

Assume the function y = x³ - x⁵.

Now, replace, x by -x and y by -y.

Therefore, -y = ((-x)³ - (-x)⁵) 

⇒-y = -x³ + x⁵

Now, we multiply both sides by -1. Then we get:

⇒y =  x³ - x⁵

Hence, we get the original equation back. Hence, it is symmetrical to the origin (and is an odd function). 

Similarly, you can prove that f(x) = 2x³ + x² + x + 3, is not symmetrical to the origin.

Hence, to determine if a graph is symmetric with respect to the origin, we have to check whether we get the same equation back when we replace y by -y and x by -x.