Name
November 6 (F), 7(E), 10(A), 2014
Math 4 notes and problems
page 1
Rational functions with holes and asymptotes
Objective: Graph rational functions with holes and asymptotes.
In past lessons, we’ve observed two ways that a rational function graph can have two kinds of
discontinuities: removable discontinuities (holes) and infinite discontinuities (vertical asymptotes).
These are the only two kinds of discontinuities a rational function can have. Many rational
functions have both kinds of discontinuities in the same graph. The problem #9 in the last
assignment gave us a first look at a function that has both, and most of the graphs you’ll be
graphing today will have both holes and asymptotes.
Here are some general strategies that help with drawing these kinds of graphs:
•
Begin by finding the zeros of the denominator. This tells you the domain of the rational
function (the domain is always all real numbers except for the zeros of the denominator).
Find these zeros by factoring if you can. Otherwise you may need to graph the denominator
on a calculator to find its zeros.
•
The graph will have a discontinuity at each x-value excluded from the domain. Determine
whether each discontinuity is removable (hole) or infinite (vertical asymptote). You can
find the removable discontinuities by factoring both the numerator and the denominator,
and reducing the fraction as much as possible. If the reduced fraction no longer has the
discontinuity, that means the discontinuity is removable (a hole). If a discontinuity cannot
be eliminated by reducing the fraction, then it is infinite (a vertical asymptote).
•
First try graphing the reduced fraction, because usually it will be simpler to graph than the
original function. For example, it might just be a polynomial or a transformed 1x .
divisor
Sometimes transformed 1x ’s can be revealed by dividing to get the form quotient + remainder
(see the example below). In more complicated cases, you might need the calculator to help
with the graphing.
•
Now modify the graph by putting holes at any x-values you previously identified as having
removable discontinuities. Check your graph against the domain that you originally
identified; it should have either a hole or a vertical asymptote at each omitted x-value.
Example: Graph the function
5 x 2 + 26 x + 21
.
x 2 + 5x + 4
•
The denominator factors as (x + 1)(x + 4), so the domain is all real numbers except x = –1
and x = –4.
•
The function factors as
•
Dividing shows that
•
Finally, there should be a hole at x = –1. Calculate the y-coordinate of the hole using the
5(−1) + 21 16
reduced function:
= .
(−1) + 4
3
•
See the top of page 2 for the final graph.
( x + 1)(5 x + 21)
5 x + 21
which reduces to
. So the discontinuity at x
( x + 1)( x + 4)
x+4
= –1 is removable (a hole) but the one at x = –4 is infinite (a vertical asymptote).
5 x + 21
1
=
+ 5 . So the graph has the
x+4
x+4
the left by 4 and up by 5.
1
x
shape, but translated to
Name
November 6 (F), 7(E), 10(A), 2014
Graph for the preceding example
Math 4 notes and problems
page 2
5 x 2 + 26 x + 21
:
x 2 + 5x + 4
Exercises
Directions for 1 through 4: For each given rational function:
Identify the domain, and whether each discontinuity is removable (hole) or infinite (vertical
asymptote).
• Sketch a graph of the function. For these functions it should be possible to make the graphs
without a calculator by using the strategies from page 1.
• Write limit statements describing the function near each discontinuity.
− x+5
6 x − 11
1. f ( x) = 2
2. f ( x) =
x−2
x − 8 x + 15
•
3.
2 x 2 − 3x − 2
f ( x) =
x 2 − 2x
4.
5 x 2 − 80
f ( x) = 2
x − 16
Also do these exercises from the textbook: section 2.6 exercises 5, 7, and 11-18.