Rational functions: Horizontal and oblique asymptotes NAME:
This worksheet is designed to help us understand why and where horizontal (and oblique) asymptotes occur on the graphs of rational functions.
We can think of horizontal and oblique asymptotes as the end behavior of rational functions. In other words, horizontal and oblique asymptotes will be determined by what
the y values approach at the (left and right) ends of the graph.
Recall a rational function is a function that can be written as a fraction where the top and bottom are both polynomial functions. (Recall that the degree of a polynomial function is the highest exponent of the x’s.)
We have seen that the end behavior of a polynomial can be found by looking at the sign (positive or negative) and degree of the leading term. We will extend this to say that the end behavior (horizontal or oblique asymptotes) of a rational function can be determined by looking at the leading term on top and the leading term on bottom.
We will see three cases: 1. degree on top equal to degree on bottom, 2. degree on top less than degree on bottom, and 3. degree on top greater than degree on bottom.
Case 1: Degree on top equal to degree on bottom
1. Graph 4 3 2 2 2 + + = x x x
y in the standard window. Quickly copy the graph onto your paper.
3. Now let’s consider this function 4 3 2 2 2 + + = x x x
y . Recall the top polynomial x
x y=2 2+3
has the same end behavior as 2 2x
y = . Likewise, the bottom polynomial has the same end behavior as 2
x
y = . So if we want to know what happens at the end of the graph of 4 3 2 2 2 + + = x x x
y , we could think of the function 2 2 2
x x
y= . What does this simplify to? This is the ho rizontal asymptote of
4 3 2 2 2 + + = x x x
y . Notice you should get the same answer that you got in #2.
Case 2: Degree on top is less than degree on bottom
4. Let’s start off by looking at the function
x
5. You should have gotten that the graph approaches zero at the ends. Complete the table below to see how the y values approach zero as we approach the ends of the graph. Use the Value (TI83 or TI82) or EVAL (TI86 or TI85) function to make this easier. Set your xmin and x max to [-5000, 5000]. You will not be able to see the graph clearly but that’s okay. Remember 2E-4 is interpreted as .0002 and -2E-4 is interpreted as -.0002.
x x y= 1 -5000 -1000 -100 -50 -10 0 The y value is undefined when x is 0. 10 50 100 1000 5000
Let’s look at a specific rational function whose degree on top is less than the degree on bottom.
As we approach the
ends of the graph,
the y values get
closer and closer to
zero. This is because
we’re dividing 1 by
bigger and bigger
numbers.
Rational functions
whose degree on top
is less than the
degree on bottom
will follow this
pattern.
As we approach the
ends of the graph,
the y values get
closer and closer to
zero. This is because
we’re dividing 1 by
bigger and bigger
numbers.
Rational functions
whose degree on top
is less than the
degree on bottom
6. Consider the function 3 7 5 6 3 4 3 2 − + + − = x x x x
y . Graph it on the standard window and copy it below.
Now, the end behavior can be determined by looking at the function 3 2 5 4 x x y= . If we simplify this, we get
x y
5 4
= . (Do you see why?) Just like how
x
y = 1 went towards zero at the ends of the graph,
x y
5 4
= will go towards zero at the ends. To see this, set your xmin and x max to [-1000, 1000] and trace out to the ends of the graph of
3 7 5 6 3 4 3 2 − + + − = x x x x
y . You’ll see that the y values approach zero as x gets really big or really small. This means that the horizontal asymptote of
3 7 5 6 3 4 3 2 − + + − = x x x x y is y = 0.
Case 3: Degree on top is greater than degree on bottom
7. We’ll use the function
7 5 8 4 5 2 3 − − + = x x x x
8. Notice how the ends of the graph do not approach a single value but rather go off at an angle. This indicates an oblique asymptote. You can trace out to the ends and you’ll notice the y values just getting bigger and bigger. There is no horizontal asymptote because the y values do not approach a single value. When the degree on top is greater
than the degree on bottom, the rational function does not have a horizontal asymptote. It
is possible to find the equation of the oblique asymptote. However, we do not need to know how to do this. In these cases, just say there is no horizontal asymptote.
9. Without graphing tell the horizontal asymptotes of the following functions. Write your answers in the form “y = ??”. If there is no horizontal asymptote, just write “none”.
