Graphs of Rational Functions
Vertical Asymptotes and Holes
Warm Up
1.
f ( x) =
1
x
VA
HA
Vertical Asymptote
State x= or “none”
Horizontal Asymptote
State y= or “none”
Domain
and
Range
Domain:
___________
Range:
___________
2.
f (x) =
Domain:
1
x+2
___________
Range:
___________
3.
f ( x) =
Domain:
1
+5
x−4
___________
Range:
___________
4.
−1
f ( x) =
−2
x−6
Domain:
___________
Range:
___________
Graph
(draw asymptotes
as dashed lines)
Vertical Aymptotes & Holes Exploration
PART A
Some rational functions cannot be simplified – for example g ( x) =
1. a) Graph g (x) and insert the picture of the graph here.
1
.
x −3
b) Describe what happens to the graph near x=3.
2. a) Create another rational function with domain (-∞,∞), x≠ 2 that can’t be simplified.
Function:
b) Graph your function and insert the picture of the graph here.
c) 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 and insert the
picture of the graph here..
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).
Insert the picture of the graph here.
c) 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
Function:
b) Graph your functionand insert the picture here.
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.
PRACTICE TIME
Find all holes & vertical asymptotes of the functions below
𝟔𝒙 + 𝟑
𝒈 𝒙 =
Holes:
𝟏𝟎𝒙 + 𝟓
V.A:
Holes:
V.A:
Holes:
V.A:
Holes:
V.A:
𝒉 𝒙 =
𝟏𝟕𝒙 − 𝟓𝟏
𝒙−𝟑
Holes:
V.A:
HOMEWORK
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.