End Behavior
The end behavior of a polynomial functions describes how the relationship between input and outputs at the far left and far right of the graph. In other words, as x becomes increasingly negative, approaching negative infinity, how do the outputs behave? Or, as x becomes increasingly large, approaching positive infinity, how do the outputs behave?
In this section you’ll learn what end behavior is, how to identify end behavior by looking at the leading coefficient and the sign of the leading coefficient, and how that ties into the number of x – intercepts.
Read through the notes below, watch the video, try the practice problems. Learning new material is always difficult and confusing. In fact, it requires you to develop and then relief confusion, and often frustration. But, the end result is worth it because learning is an activity worth it’s own efforts.
When we discuss “end behavior” of a polynomial function we are talking about what happens to the outputs (y values) when x is really small, or really large. Another way to say this is, what do the far left and far right of the graph look like?
For the graph to the left, we can describe the end behavior on the left as “going up.” We can describe the behavior on the right as “going up,” as well.
In truth, on the left, as you learned in the last section, the graph is decreasing. On the right it is increasing.
In this section you will learn to separate all possible polynomial functions into two basic categories, even and odd. You will learn the possible end behavior possibilities of each based on the sign of the leading coefficient and whether the degree is even or odd. Let’s get to it!
Even v Odd
To determine if a polynomial function is even or odd you examine the degree. If the degree is even, we call the polynomial function even. If the degree is odd, we call the polynomial function odd. That’s it!
So, f(x) = 5x7 is an odd function because the degree is 7.
The reason we separate even and odd functions is because of the sign of a negative base raised to an exponent. If the exponent is even, the product is always positive. If the exponent is odd, the product is negative. To be explicitly clear, look at the exploration below.
This happens because a negative number times another negative number is positive. So every pair of odd numbers has an even product.
(-2)(-2) = 4
But, when you have an odd number of that base, the pairs will all be positive, with one extra negative number left over, making the final product negative.
(-2)(-2)(-2) = -8
Let’s do a simple chart comparing two functions, one a degree of 5, the other a degree of 4. We will only plug in negative numbers.
In the table above we only showed negative inputs. That is because a positive base to the power of any integer is positive. Let’s look at the graph of each now. The first table will compare a rather global view of each graph. But the second will zoom in on the “left” side of the coordinate plane, where the x – values are negative. This will allow us to see how the table above translates to the graph.
Remember, a graph is a picture of all solutions to an equation (in our context). Look at how the graph of each function behaves in quadrants 2 and 3. In quadrants 2 and 3, x is always negative, and x is the input.
For small values of x (small meaning negative numbers and on the “left” of the coordinate plane), the odd function is increasing while the even function is decreasing. For both functions, the odd and even, the “right” side of the coordinate plane showed both graphs to be increasing.
Let’s look at each graph again and discuss some common language here.
There is one remaining piece of the puzzle we need to understand. After that, we’ll look at a chart that will help make all of this clear and easily referenced.
In both of our examples, the leading coefficient was positive. If you recall with quadratic equations, when a is negative, the parabola goes down. One way to think of this is that it is flipped over. The same holds true for other polynomial functions. Let’s see a t-chart to show how this works algebraically. Then, we’ll look at the graphs.
The key here is not to confuse our verbal descriptions with increasing and decreasing. For the graph of y = –x5 we say, “up on the left and down on the right.” However, it is decreasing on both sides. Do you see that odd functions can be described as heading in opposite directions on each side? Even functions either go up both ways, or down both ways. Let’s take a look at that.
Let’s pull it all together now, shall we. In the chart below you can see how the end behavior of even and odd functions looks, for both positive leading coefficients and negative leading coefficients. Remember, odd functions go opposite directions and even functions go the same direction.
End behavior describes what the output (y) or f(x) does as x grows infinitely small (to the left, x → -∞), or as x grows infinitely large (to the right, x → ∞).
As x grows infinitely small, if the outputs are increasing, we say this is “up left.”
As x grows infinitely small, if the outputs are decreasing, we say this is “down left.”
As x grows infinitely large, if the outputs are increasing, we say this is “up right.”
As x grows infinitely large, if the outputs are decreasing, we say this is “down right.”
- Even functions with a positive leading coefficient “go up both ways.”
- Even functions with a negative leading coefficient “go down both ways.”
- Odd functions with a positive leading coefficient “go down left, up right.”
- Odd functions with a negative leading coefficient “go up left, down right.”
To see a review of the assignment and to summarize what we’ve learned to date about polynomial functions, watch the video below.
