The document discusses graphs of rational functions. It provides examples of graphs along vertical and horizontal asymptotes, as well as graphs around roots. The document also contains exercises asking the reader to identify roots, poles, and orders from sign charts and sketch graphs of various rational functions.

1. Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+ + +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA

2. Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch the graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e. x2 – 1 = 0  x = ± 1.
Both have order 1, so
the sign changes at
each of these values.
As x±∞, R(x) resembles
x2/x2 = 1, i.e. it has y = 1
as the HA. Note the
graph stays above the
HA to the far left,
and below to the far right.
++ –
x=2
++

3. Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch the graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e. x = 2.
As x ±∞, the graph of R(x)
resembles the graph of the
quotient of the leading terms
x2/x = x, or y = x.
Hence there is no HA.
x=3
Do the sign-chart. Construct the
middle part of the graph.
+–+–
x= –1

4. Graphs of Rational Functions
+ +
+ +
The graphs along poles.
Even-order PolesOdd-order Poles
Even-order RootsOdd-order Roots
The graphs around a roots. (See 2.9.)
order = 3, 5, 7…

5. Exercise A. Following are the sign charts of
simplified factorable rational function
with their roots, poles, and their orders given.
a. Write down a (any) rational formula in the
simplified factored form of any given sign chart.
b. Sketch its graphs (its “mid-section”).
–1
1.
ord=1 ord=1
1– – – –1
2.
ord=2ord=1
1+ + +
–1
3.
ord=1 ord=1
1 – – – –1
4.
ord=2ord=2
1 + + +
–1
7.
1 3 –1
8.
ord=1 ord=2
1 3
ord=2
+ +– –
= root
= pole
(asymptote)
ord=2 ord=1 ord=1
–1
5.
1 3 –1
6.
ord=1 ord=2
1 3
ord=2
+ +– –
ord=2 ord=1 ord=1
Graphs of Rational Functions

6. B. For each of the following rational functions,
identify its roots, poles and their orders.
a. Make a sign charts (as in A).
b. Determine its horizontal behavior. Sketch its graph.
1. R(x) = 2. x – 3x + 2
2 R(x) = –1
5. R(x) = 6. x – 3x + 2
x – 3 R(x) =
x + 2
3. R(x) = 4. 5 – xx + 1
–2 R(x) = 1
9. (x + 3)2R(x) = x – 2
7. R(x) = 8. 3 – 2xx + 2
–3x R(x) =
x
10. (x + 3)2R(x) = x + 2
11. (x – 3)(x + 3)
R(x) =
x – 2
12. R(x) =
(x – 3)(x + 3)
x – 5
Graphs of Rational Functions

7. B. For each of the following rational functions,
identify its roots, poles and their orders.
a. Make a sign charts (as in A).
b. Determine its horizontal behavior. Sketch its graph.
13. (x – 2)2R(x) = 14.
(x – 4)(x + 5)
R(x) =
15. (x – 3)(x + 3)
R(x) = 16. R(x) =
(x – 3)(x + 3)
(x – 3)(x + 3) (x + 2)2
(x + 3)(x + 5)
(2x – 1)(x + 1)
(x – 4)(x + 5)
17. x(x – 3)(x + 3)
R(x) = 18. R(x) =
(x – 3)2(x + 3)
(2x – 1)(x + 1)
Graphs of Rational Functions

8. (Answers to odd problems) Exercise A.
1. 3.
x – 1
x + 1
(x + 1)(x – 1)
1
–
5. (x + 1)2–
(x – 1)(x – 3) 7.
(x + 1)2(x – 3)
(x – 1)
Graphs of Rational Functions

9. Exercise B
1.
ord=1
–2 + + +– – – 3.
ord=1
–1+ + + – – –
5.
–2
ord=1ord=1
3+ + + +– – –
7. –2
ord=1ord=1
0+ + +– – – –
9.
–3
ord=1ord=2
2 + + +– – – – –
Graphs of Rational Functions

10. 11. –3
ord=1ord=1
2+ +– – – – – 3
ord=1
+ +
13. –3
ord=1ord=1
2+ + – – – 3
ord=1
+ +– – –
15. 17.
–5
ord=1ord=1
-3+ + – – –
ord=1
+ ++ +
ord=1
3 – – – 4 –5
ord=1ord=1
-3– –
ord=1
+ ++ +
ord=1
3– – 4– –
ord=1
0
Graphs of Rational Functions