© 2025 Perplex Learning Incorporated. All rights reserved.
All content on this website has been developed independently from and is not endorsed by the International Baccalaureate Organization. International Baccalaureate and IB are registered trademarks owned by the International Baccalaureate Organization.
When we divide one polynomial, an expression whose terms all include x raised to whole numbers (or 0), f(x) by another polynomial g(x), we call the resulting function R(x)=g(x)f(x) a rational function, just as a rational number is the ratio of two integers.
These functions' graphs are notable for gaps where the function is undefined when the denominator equals 0.
At these "gaps" we often have vertical asymptotes since the value of the function gets infinitely large (positive or negative) as the denominator nears 0.
Discussion
The reciprocal function f(x), which returns the reciprocal of x, is plotted below.
Powered by Desmos
Describe any key visual features of the graph of f.
The graph of f(x)=x1 has the following key features:
• Vertical asymptote at
since as x approaches 0 from positive values, f(x) increases rapidly toward ∞ and as x approaches 0 from negative values, f(x) decreases rapidly toward −∞.
• Horizontal asymptote at
since as ∣x∣ becomes very large, f(x)=x1 becomes very small.
• Domain and range
• Symmetry
– f(−x)=−f(x), so the graph is symmetric under a 180∘ rotation about the origin.
– Since f−1(x)=f(x), it is also symmetric across the line y=x.
The reciprocal function 1/x
The reciprocal function is defined by f(x)=x1.
Notice that f(x) is not defined for x=0. In fact, since x1 gets very large as x approaches 0, f(x) has a vertical asymptote at x=0.
And since for very large x, x1 approaches zero, there is also a horizontal asymptote y=0.
Notice also that x11=x, so f(x)=x1 is self-inverse.
Powered by Desmos
Discussion
Describe the asymptotes of the graph of f.
Powered by Desmos
The graph of f(x)=x−12x+1 has two asymptotes, which can be described as follows:
Vertical asymptote: A vertical asymptote occurs where the denominator is zero and the function is undefined. Setting x−1=0 gives x=1. Graphically, we see that as x approaches 1 from higher values, f increases rapidly toward infinity. Similarly, as x approaches 1 from lower values, f decreases rapidly toward negative infinity. Hence, we still see that f is undefined at x=1. Therefore, the line
is a vertical asymptote.
Horizontal asymptote: To find the horizontal asymptote, consider the behavior as x→±∞. For large values of x, the highest degree terms dominate, so:
Thus, as x→±∞, f(x) approaches 2 from below (on the left) and from above (on the right). Likewise, visually far to the left, the curve rises very gradually toward y=2, and far to the rightm the curve falls very gradually toward y=2. Therefore, the line
is a horizontal asymptote.
Graphs of linear rational functions
A linear rational function has the form
When the denominator is zero the graph will have a vertical asymptote:
And as x gets very large, the +b and +d can be ignored:
So there is a horizontal asymptote at y=ca.
Powered by Desmos
Checkpoint
Find the asymptotes of
Select the correct option
Discussion
The graph of f restricted to the domain x<−2 or x>2 is shown below.
Powered by Desmos
What kind of function does f look like?
The full graph of f is shown below.
Powered by Desmos
How is the tail behavior of f similar to other rational functions we have seen?
Part (a)
The two branches both rise at a constant slope, and the curve appears straight except for the gap in the middle of its domain. Therefore, it is reasonable to conclude that f looks like a line.
Part (b)
As you go far out to the left or right, the graph of f “straightens out” and looks very similar to a line, as we discussed in part (a). This is because, as we can see in the graph, the two branches of the function cling very tightly to the dashed line as ∣x∣ becomes large.
This is exactly the same kind of tail behavior we saw when simpler rational functions approach a horizontal line—only now the line is tilted instead of flat. In both cases the curve hugs that line more and more closely without ever actually reaching it.
Thus, this rational function has revealed a new type of asymptote, which we defined as a line that a function continuously gets closer to but never reaches. We have only seen horizontal and vertical asymptotes, but this function seems to have a sloped asymptote!
Rationals with oblique asymptotes
When a rational function is of the form
there is a vertical asymptote at
By performing polynomial division, we can find the oblique asymptote of f(x), which has the equation
for some constant c determined during the polynomial division.
Powered by Desmos
Example
Find the asymptotes of f(x)=3x−15x2−3x+2.
There is a vertical asymptote at x=31. To find the oblique asymptote, we performing the polynomial division:
So the oblique asymptote has equation y=35x−94.
Alternatively, we know that the division will give
Then,
so
Then, as x becomes large,
so
for large x, which is precisely to say that the line y=35x−94 is an asymptote of f.
Checkpoint
Find the oblique asymptote of 5x10x2+3x+6.
Select the correct option
Discussion
Describe any key features you would expect to see on the graph of y=x2−x−6x−1.
The function can be rewritten by factoring the denominator:
Domain: The denominator is zero when x=3 or x=−2, so the function is defined for all real x except these values:
Vertical asymptotes: Vertical asymptotes occur where the denominator is zero, so there are vertical asymptotes at:
Horizontal asymptote: The degree of the denominator (2) is greater than the degree of the numerator (1). For rational functions, if the degree of the denominator is greater, the horizontal asymptote is y=0. This is because as ∣x∣ becomes large, the denominator grows much faster than the numerator, so the function approaches zero.
Intercepts:
x-intercept: Set the numerator to zero:
So the graph crosses the x-axis at (1,0).
y-intercept: Set x=0:
So the graph crosses the y-axis at (0,61).
Sign of y: To determine where the function is positive or negative, consider the sign of the numerator and denominator in each region divided by the vertical asymptotes and the x-intercept:
For x<−2:
x−1<0 and (x−3)(x+2)>0 so y<0
For −2<x<1:
x−1<0 and (x−3)(x+2)<0 so y>0
For 1<x<3:
x−1>0 and (x−3)(x+2)<0 so y<0
For x>3:
x−1>0 and (x−3)(x+2)>0 so y>0
Summary:
y<0 when x<−2 or 1<x<3
y>0 when −2<x<1 or x>3
Rationals with quadratic denominator
When a rational function is of the form
There will be vertical asymptotes when the quadratic cx2+dx+e=0.
The horizontal asymptote will simply be y=0 since cx2 dominates ax when x is very large.
Additionally, there will be an x-intercept at x=−ab, when the numerator changes sign.
Powered by Desmos
Note that if the numerator and denominator share a root (ie x=−ab is a root of the denominator), then there will only be one asymptote and a "hole" on the x-axis. This has never shown up on exams.
Example
Sketch the graph of y=x2−x−6x−1.
The denominator is (x−3)(x+2), which is zero when x=3 and x=−2.
The numerator is 0 when x=1 (negative to the left and positive to the right). When x<−2, the function is negative (plug in −3)
So we plot the asymptotes x=3 and x=−2, and start tracing from the negative side of the x-axis. At each vertical asymptote the sign will change, and we have an x-intercept at 1.
Powered by Desmos
Exercise
Consider f(x)=x2+x−20x−3.
Find the asymptotes of f.
Find the x-intercept of f.
Select the correct option
0 / 7
Chat
0 / 7