# 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 = x^{3} - x^{5}.

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

Therefore, -y = ((-x)^{3} - (-x)^{5})

⇒-y = -x^{3} + x^{5}

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

⇒y = x^{3} - x^{5}

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) = 2x^{3} + x^{2} + 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.

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