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) 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:

- As x → -∞, y → ?
- As x → +∞, y → ?

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

**Q: What is the end behavior of the function y=5x ^{3}+7x^{2}-2x-1**

A: OK. Let’s look at the left end behavior first:

As x approaches -∞, what is the function (y-value) doing?

Imagine x=-1000000 (some super large and super negative number, like the idea of -∞), we have:

y=5(-1000000)^{3}+7(-1000000)^{2}-2(-1000000)-1

Don’t do the actual math. Just think:

Is this number large or small?

Is it positive or negative?

I can look at the x^{3} term and see that it dominates this function. x^{2} and x are small peanuts compared to x^{3}. So, in reaity, in polynomials, I can focus on the term of the largest degree:

y=5(-1000000)^{3}~~+7(-1000000)~~^{2}-2(-1000000)-1

y=5(-1000000)^{3}

This number gives y = negative and super large.

So, I can jump to conclusions here…

As x → -∞, y → -∞

(As x approaches negative infinity, y approaches negative infinity).

Now, let’s look at the right end behavior:

As x approaches +∞, what is the function (y-value) doing?

Imagine x=+1000000 (some super large and super positive number, like the concept of +∞), we have:

y=5(+1000000)^{3}+7(+1000000)^{2}-2(+1000000)-1

And, by the same reasoning, we can focus on the term of largest degree:

y=5(+1000000)^{3}+~~7(+1000000)~~^{2}-2(+1000000)-1

y=5(+1000000)^{3} = super large and super positive

So, as x → +∞, y → +∞

(As x approaches positive infinity, y approaches positive infinity)

*Note: in this example, y behavior mimicked x behavior, this isn’t always the case!*