End behavior of Function

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The end behavior of a function is the behavior of the graph of the function as the input values (x-values) approach positive infinity or negative infinity.

To determine the end behavior of a function, you can examine the leading term of the function (the term with the highest exponent). The leading term determines the overall shape of the graph and the direction in which it is "heading" as the x-values get very large.

Here are some examples of common end behaviors:

  • The end behavior is "up" if the function's leading term is a positive constant. This suggests that the graph of the function will approach the y-axis as x approaches positive infinity and will approach negative infinity as x approaches negative infinity.
  • The end behavior is "down" if the function's leading term is a negative constant. This suggests that the graph of the function will approach the x-axis as x approaches positive infinity and will approach positive infinity as x approaches negative infinity.
  • When the leading term of the function is a positive constant time x^2 (or a higher even-numbered power), the end behavior is "up and down." This suggests that the graph of the function will approach the x-axis as x approaches positive infinity and will approach the x-axis again as x approaches negative infinity.
  • When the leading term of the function is a negative constant time x^2 (or a higher even-numbered power), the end behavior is "down and up." This suggests that the graph of the function will approach the y-axis as x approaches positive infinity and will approach the y-axis again as x approaches negative infinity.

The end behavior of a function can be helpful to understand the overall shape and behavior of the graph of the function. It is also a helpful tool to compare the graphs of different functions.

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