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Determining the End Behavior of a Function

How do you determine the end behavior of a function?  And, what does this mean?

When looking at a graph, the “end behavior” is referring to what is happening all the way to the far left of the graph and all the way to the far right of the graph.  Your goal is to analyze the y-value (height or function value) of the function when x is really large and negative, and then again when x is really large and positive.  What is the pattern on each end?  What is the “end behavior”?

Notationally, we are thinking:

  1. As x → -∞, y → ?
  2. As x → +∞, y → ?

OK, so let’s try this on a polynomial example:

Q:  What is the end behavior of the function y=5x3+7x2-2x-1

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End Behavior

1.  Find the end behavior of the following functions:

(a)  Consider:  y = (x²-2x)/(-x³-5x²+4)
as x goes to ∞, y goes to ?
as x goes to – ∞, y goes ?

(b) Consider y = (2x+1)/(-4x+1)
as x goes to ∞, y goes ?
as x goes to – ∞, y goes ?

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