Graphs of Rational Functions, Day 1
Vertical Asymptotes and Holes
Vertical Aymptotes & Holes Exploration
PART A
Some rational functions cannot be simplified – for example g ( x) =
1. a) Graph g (x) and trace the graph near x=3.
1
.
x −3
b) Describe what happens to the graph near x=3.
2. a) Determine another rational function with domain (-∞,∞), x≠ 2 that
can’t be simplified.
b) Graph your function and describe what happens to the graph at x=2.
3. Determine the equation of a rational function that has two vertical
asymptotes: x=-1 and x=2. Graph your function.
PART B
Some rational functions can be simplified to polynomials.
4. a) Simplify f ( x) =
x2 − 4
x−2
b) Graph the function f (x) prior to simplifying it. (Plug the ORIGINAL into
your calculator and sketch).
c) Trace the graph near x=2. Describe what happened to the graph at
x=2.
5. a) Determine another rational function that simplifies to a polynomial with
domain (-∞,∞), x≠ 1 (i.e. it has a restriction of x ≠ 1
b) Graph your function and describe what happens to the graph at x=1.
SUMMARIZE
6. What determines where a rational function has:
a) a hole?
B) a vertical asymptote?
7. Determine the equation of a rational function that has
both a vertical asymptote and a “hole”. Graph your function.
8. Determine the equation of a rational function without any
“holes” or vertical asymptotes. Graph your function.
Find all holes & vertical asymptotes of the functions below
𝟔𝒙 + 𝟑
𝒈 𝒙 =
𝟏𝟎𝒙 + 𝟓
Holes:
V.A:
Holes:
V.A:
f ( x) =
11
3
5x − 45 x
Holes:
V.A:
Holes:
V.A:
Holes:
V.A:
𝒉 𝒙 =
𝟏𝟕𝒙 − 𝟓𝟏
𝒙−𝟑
Holes:
V.A:
Rational Functions Sort:
Sort the following equations into two categories—equations with vertical asymptotes or equations with
holes. Each equation will have either vertical asymptotes or holes, but not both. Glue to
construction paper and
(a) Find any zeroes of the function
(b) Find the location of any holes or the equation of
any vertical asymptotes
Practice!
Identify any vertical asymptotes or holes for the following functions
5) Write the equation of a function that has a hole at x= - 4
6) Write the equation of a function that has a vertical asymptote at x=9
7) Write the equation of a function that has a vertical asmpytote at x = 3 and a hole at x=-5.
8) Write the equation of function with a vertical asymptote at x= - 1 and a hole at x=21.