How to Tell If a Graph is Even or Odd

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Understanding the symmetry of a graph can be a useful tool in mathematics, especially in calculus. Two types of symmetry that are commonly used in graphing are even and odd symmetry. An even function is symmetric with respect to the y-axis, while an odd function is symmetric with respect to the origin.

In this article, we will discuss how to determine if a graph is even or odd, and provide examples of functions that exhibit each type of symmetry.

What is Even Symmetry?

A function is said to have even symmetry if its graph is symmetric with respect to the y-axis. This means that if you reflect the graph across the y-axis, the graph will look exactly the same as the original. Mathematically, this can be expressed as:

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f(-x) = f(x)

where f(x) is the function and f(-x) is the reflection of the function across the y-axis.

For example, the graph of y = x^2 is an example of an even function. If we reflect this graph across the y-axis, we get the same graph:

Graph of y = x^2

This function can be expressed as f(x) = x^2. To check if this function is even, we substitute -x for x:

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f(-x) = (-x)^2 = x^2 = f(x)

Since f(-x) = f(x), the function is even.

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What is Odd Symmetry?

A function is said to have odd symmetry if its graph is symmetric with respect to the origin. This means that if you reflect the graph across the origin, the graph will look exactly the same as the original. Mathematically, this can be expressed as:

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f(-x) = -f(x)

where f(x) is the function and f(-x) is the reflection of the function across the y-axis.

For example, the graph of y = x^3 is an example of an odd function. If we reflect this graph across the origin, we get the same graph:

Graph of y = x^3

This function can be expressed as f(x) = x^3. To check if this function is odd, we substitute -x for x:

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f(-x) = (-x)^3 = -x^3 = -f(x)

Since f(-x) = -f(x), the function is odd.

How to Determine if a Graph is Even or Odd

To determine if a graph is even or odd, we can use the following steps:

  1. Check for symmetry with respect to the y-axis. If the graph is symmetric with respect to the y-axis, it is even.
  2. Check for symmetry with respect to the origin. If the graph is symmetric with respect to the origin, it is odd.
  3. If the graph is not symmetric with respect to either the y-axis or the origin, it is neither even nor odd.
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Let’s use the function f(x) = x^4 - 3x^2 + 2 as an example. To check if this function is even or odd, we follow the steps above:

  1. We check for symmetry with respect to the y-axis. To do this, we substitute -x for x in the function:
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    f(-x) = (-x)^4 - 3(-x)^2 + 2
    = x^4

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