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PMC 0045
PreCalculus for Engineering
Foundation in Engineering
ONLINE NOTES
Topic 5
Polynomial and Rational Functions
FOSEE , MULTIMEDIA UNIVERSITY
(436821T)
MELAKA CAMPUS, JALAN AYER KEROH LAMA, 75450 MELAKA, MALAYSIA.
Tel 606 252 3594 Fax 606 231 8799
URL: http://fosee.mmu.edu.my/~asd/
Mathematics Department
(MD)
Centre for Foundation Studies and Extension Education
(FOSEE)
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PreCalculus for Engineering
Topic 5
TOPIC 5: POLYNOMIAL AND RATIONAL FUNCTIONS
References
:
1.
Sullivan, M. (2002),
, 6
th
Ed., Prentice Hall.
2.
Blitzer, R., Goh Wei Wei, et.al. (2005),
,
Prentice Hall.
Objectives:
1.
Identify the properties of polynomial functions.
2.
Graph the polynomial functions using the graphing techniques.
3.
Know how to graph rational functions by using method of shifting and reflections.
4.
Know and understand the Remainder and Factor Theorems also the Synthetic
Division.
Contents:
5.1
Quadratic Functions
5.2
Polynomial Functions and Their Graphs
5.3
Dividing Polynomials; Remainder and Factor Theorems
5.4
Rational Functions and Their Graphs
5.1
QUADRATIC FUNCTIONS
A quadratic function is a function of the form:
f
(
x
) =
ax
2
+
bx
+
c
,
where
a
,
b
and
c
are real numbers and
0
≠
a
Graphing a Quadratic Function
.
Characteristics of the graph of a Quadratic Function
c
bx
ax
x
f
+
+
=
2
)
(
.
1.
Vertex =


a
b
f
a
b
2
,
2
2.
Axis of symmetry: The line
a
b
x
2

=
3.
Parabola opens up if
a
> 0,
f
min =

a
b
f
2
4.
Parabola opens down if
a
< 0,
f
max =

a
b
f
2
The
x
intercepts of a Quadratic Function.
b
2
 4
ac
No.of distinct
x
intercepts
> 0
2
= 0
1
(at its vertex)
< 0
0
(did not cross / touch
x
axis)
____________________________________________________________________________________
Mathematics Dept
2/10
PreCalculus for Engineering
Topic 5
The
y
intercept of a Quadratic Functions
 the value of
f
(
x
) when
x
= 0
Example:
Graph
f
(
x
) = 3
x
2
+ 6
x
+1
Solution:
We have
a
= 3,
b
= 6,
c
= 1.
Since
a
= 3 < 0, the parabola opens down.
For vertex,
x
coordinate =
a
b
2

=
6
6


= 1 ;
y
coordinate =
f
(1) = 4
So, the vertex is
(1,4) and
f
has a maximum value of 4.
Axis of symmetry:
x
= 1
x
intercept:
b
2
 4
ac
= 6
2
 4(3)(1) = 36 + 12 = 48 >0
( 2
x
intercepts )
x
=
a
ac
b
b
2
4
2

±

x
= 0.15 and
x
= 2.15
y
intercept :
f
(0) = 1
ie. The intercepts are (0.15,0), (2.15,0), (0,1)
Note:
1.
By using the axis of symmetry and
y
intercept, we can locate point (2,1).
2.
We must plot at least 3 points on the graph.


5.2
POLYNOMIAL FUNCTIONS AND THEIR GRAPHS
Definition:
A polynomial function is a function of the form:
0
1
1
1
......
)
(
a
x
a
x
a
x
a
x
f
n
n
n
n
+
+
+
+
=


where
a
n
,
a
n1
,….
.,
a
1
,
a
0
are real numbers and
n
is nonnegative integers.
Note:
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This note was uploaded on 10/30/2010 for the course FOSEE CVL1040 taught by Professor None during the Spring '09 term at Multimedia University, Cyberjaya.
 Spring '09
 none

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