(2)Algebraic Functions-Polynomial for Uploading

8/12/2019 (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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Any polynomial function has

domain .

If , the polynomial functionP

is said to be of degree n.

xP

RPD

0na


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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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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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The constant function i.e. y = k, k is any constant.

Examples:

1. f(x) = 5

2. g(x) = -7


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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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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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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:



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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

(2)Algebraic Functions-Polynomial for Uploading

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