(2)Algebraic Functions-Polynomial for Uploading
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Algebraic Functions
Two main types of functions:
Algebraic functionsthose functions that can be
obtained by a finite combination of constants andvariables together with the four basic operations
, exponentiation, or root extractions.
Transcendental functionsthose that are not
algebraic.
,,,
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Algebraic Functions
Polynomial Functions
Form:
where nis a non-negative integer and
0122333311 axaxaxaxaxaxaxP nnnn
a0,a1,a2,...,anR
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Polynomial Functions
Any polynomial function has
domain .
If , the polynomial functionP
is said to be of degree n.
xP
RPD
0na
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Polynomial Functions
For particular values of n, special types of the
polynomial function are defined as follows:
If n= 0, the polynomial function is a constantfunction(a polynomial function of degree zero).
If n= 1, the polynomial function is a linear
function(a polynomial function of the first
degree).
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Polynomial Functions
If n= 2, the polynomial function is a
quadratic function(a polynomialfunction of the second degree).
If n= 3, the polynomial function is a
cubic function(a polynomial function
of the third degree).
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Polynomial FunctionsGraphs of Special Polynomial Functions
a.L inear Functions
Standard form: , where m, b
are any real numbers. The domainof the
function is . The rangeis
, if .
f x mx b
R x|xDf
R
f y |y R 0m
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Polynomial FunctionsExample 1: The following are linear functions:
a. b. c.
Note:The domain and range of the linear function are
both set of real numbers.
How to Graph:
1. Find the y-intercept: set x = 0 and solve for y.
2. Find the x-intercept: set y = 0 and solve for x.
h(x) 2x 5
4
g(x) 57x
f(x) 3x 2
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Polynomial FunctionsThe graph of the linear functionis a straight l ine.
Note: Two points determine a line.
The x and y intercepts are points, enough to
draw a line.
Example 2:Sketch the graph of the functions in
Example 1.
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Polynomial FunctionsNote:
There are two special linear functions:
For the linear function , if m = 0,
we have , where b is any real
number. A function of the form is called
constant function.
f x mx b
bxf
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Polynomial Functions
The constant function i.e. y = k, k is any constant.
Examples:
1. f(x) = 5
2. g(x) = -7
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Polynomial Functions
In general:
The graph of a constant function is a
horizontal line at a directed distance of b
units from thex-axis.
R
x|xDf
Rf y y b
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Polynomial Functions
Given the general form of a linear function ,
, if m= 1 and b= 0, the
resulting function is called
the identi ty function.
For any ordered pair that belongs tothe identity function, the values of thex
andy are identical.
y f x mx b
xxf
y,x
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Polynomial FunctionsThe domain of the identity
function is and
the range is
The graph is the line bisecting
the first and the third
quadrants as shown in the
figure.
R x|xDf R y|yR f xxf
x
y
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Polynomial FunctionsGraphs of Special Polynomial Functions
b.Quadratic Functions arepolynomial functions
of degree n= 2 (alternatively referred to as
second degree polynomial).
General form:
where anda, b,and care constants.
cbxaxxfy 2
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Polynomial FunctionsQuadratic Functions
Df = R
The range is denoted by
or
.
24| , if 0
4f
ac bR y y a
a
2
4| , if 0
4f
ac bR y y a
a
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Polynomial FunctionsThe graph of a quadratic function described by an equation of the form
where a, b, and cR and is a parabola.
cbxaxy 2
0a
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Polynomial Functions
An important point on the parabola is the vertex.
The coordinates of the vertex V is given by
Note: is known as the axis of symmetry.
a
bac
a
bV
4
24
,2
xb
2a
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Polynomial FunctionsIf a> 0, the parabola
opensupward
(or the vertex is the
lowest point);
for a> 0.
cbxaxy 2
a
bac
a
b
4
4,
2
2
Figure1.2.6
y
x
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Polynomial FunctionsIf a< 0, the parabola
opens downward(or
the vertex is thehighest point);
for a< 0.
cbxaxy 2
x
y
a
bac
a
b
4
4,
2
2
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Polynomial FunctionsExample 1: Find the vertex and range of thegiven functions.
a. f(x) = x2-2
Solution:
Vf (0,4(1)(2)0
4(1))
0,8
4
0,2
Rf y /y 2
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Vg (22(1), 4(1)(3) (2)
2
4(1))
1,16
4
1,4
Rg
y /y 4
b. g(x) = x2-2x-3
Solution:
Polynomial Functions
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Polynomial Functions
c. F (x) = 4 - x2
Solution: VF (0,4)
RF
y /y 4
, since a < 0 .
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Polynomial FunctionsHOW TO GRAPH QUADRATIC FUNCTION:
1. Find the vertex
Vertex is either the minimum or maximum point
of the graph.
2. Find the intercepts if theres any.
3. Assign arbitrary points and plot.
a
bac
a
bV
4
24
,
2