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) 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

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:


Don’t do the actual math.  Just think:

Is this number large or small?

Is it positive or negative?

I can look at the x3 term and see that it dominates this function. x2 and x are small peanuts compared to x3. So, in reaity, in polynomials, I can focus on the term of the largest degree:



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:


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


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!

Graph the system of equations

Q:  Graph the system of linear equations, determine the number of solutions, and find the solution:

y = 4x + 2 and y =- 2x – 3

A: Okay, for the sake of talking about the lines I will name them:

Line 1:  y = 4x + 2

Line 2:  y =- 2x – 3

A “solution” to a system of linear equations is the point where the lines intersect.  So, we need to graph both of these lines on the same grid and see if and where they intersect.

Step 1:  Graph Line 1 and Line 2 on the same grid:

Tricks for graphing line 1:  Since line 1 is in the form y = mx + b, we can use the method of finding the slope and the y-intercept to graph the line.

The slope of line 1 is 4 and the y-intercept of line 1 is 2.

On your x-y graph, go up 2 on the y-axis and put a point [this is the y-intercept, or the “b” value]

Now, starting from the y-intercept point 2:  go up 4 and right 1 (this is the slope 4/1).  Create another point.  Connect the dots to make a line.  You should have:


Now, we need the same strategy to graph line 2 on the same grid.

Line 2:  y =- 2x – 3

The y-intercept is -3 and the slope is -2.  Plot the y-intercept of -3 on the y-axis.  From that point:  go down 2, right 1 to create another point.  Connect the dots to make a line.  Your graph should now look like:

So, the question now is:  How many solutions are there and what is the solution.  Recall, the solution is (x, y) point of intersection.
Clearly, there is 1 solution.  The lines do cross at 1 point.

What is that point?  Well, you have to estimate that point from the graph.  This is one problem with solving a system by graphing — it can be inaccurate.  Depending on the “neatness” of your graph, your answer will vary.

To me, the answer appears to be around the point (-.8, -1.3)

*To get the accurate solution, we solve these types of problems algebraically.