Chapter 2 Polynomial and Rational Functions

2.1 Quadratic Functions

Definition of a polynomial function

Let n be a nonnegative integer so n={0,1,2,3…}

Let be real numbers with

The function given by

Is called a polynomial function of x with degree n

Example:

This is a 4th degree polynomial

)(xf nn xa

11n

n xa xa1 0a 22xa

0121 ,,,...,, aaaaa nn 0na

123)( 24 xxxxf

Polynomial Functions are classified by degree

For example: In Chapter 1 Polynomial function , with

Example:

This function hasdegree 0, is a horizontal line and is calleda constant function.

0a

y

x–2

2

axf )(

2)( xf

Polynomial Functions are classified by degree

In Chapter 1

A Polynomial function , is a line whose slope is m and y-intercept is (0,b)

Example:

This function has a degree of 1,and is called a linear function.

0m

y

x–2

2

bmxxf )(

32)( xxf

Section 2.1 Quadratic Functions

Definition of a quadratic function

Let a, b, and c be real numbers with . The function given by f(x)=Is called a quadratic function

This is a special U shaped curve called a … ?

0acbxax 2

Parabola !Parabolas are

symmetric to a line called the axis of symmetry.

The point where the axis intersects with the parabola is the vertex.

y

x

–2

2

The simplest type of quadratic is When sketching

Use as a reference.(This is the simplest type of graph)

a>1 the graph of y=af(x)

is a vertical stretch of the

graph y=f(x)

0<a<1 the graph of y=af(x)

is a vertical shrink of the graph y=f(x)

Graph on your calculator

, ,

)(xf 2ax

)(xf 2ax

y 2x

y

x

–2

2

2)( xxf 23)( xxf 2)4

1()( xxf

Standard Form of a quadratic Function

)(xf 0,)( 2 akhxa

The graph of f(x) is a parabola whose axis is the vertical line x=h and whose vertex is the point ( , ).

-shifts the graph right or left -shifts the graph up or down

For a>0 the parabola opens up a<0 the parabola opens down

kh

hk

NOTE!

Example of a Quadratic in Standard Form

Graph :

Where is the Vertex? ( , )

Graph:

Where is the Vertex? ( , )

0,)( 2 akhxa

2)( xxf

4)2()( 2 xxf

y

x

–2

2

)(xf

Identifying the vertex of a quadratic function

One way to find the vertex is to put the quadratic function in standard form by completing the square.

Where is the vertex? ( , )

782)( 2 xxxf y

x–2

2

0,)()( 2 akhxaxf

Identifying the vertex of a quadratic function

Another way to find the vertex is to use

the Vertex Formula

If a>0, f has a minimum x

If a<0, f has a maximum x

a b c

NOTE: the vertex is: ( , )

To use Vertex Formula-

To use completing the square start

with to get

)2(,

2 a

bf

a

b

782)( 2 xxxf

cbxaxxf 2)(

cbxaxxf 2)( f x a x h k( ) ( ) 2

Identifying the vertex of a quadratic function(Example)

Find the vertex of the parabola ( , )

The direction the parabola opens?________

By completing the square? By the Vertex Formula

86)( 2 xxxf

ba2

Identifying the x-Intercepts of a quadratic function

The x-intercepts are found as follows

86)( 2 xxxf

Identifying the x-Intercepts of a quadratic function (continued)

Standard form is:

Shape:_______________

Opens up or down?_____

X-intercepts are

y

x

–2

2

f x x( ) ( ) 3 12

Identifying the x-Intercepts of a Quadratic Function (Practice)

Find the x-intercepts of y

x

–2

2

f x x x( ) 2 6 82

Writing the equation of a Parabola in Standard Form

Vertex is:

The parabola passes through point

*Remember the vertex is

Because the parabola passed through we have:

)2,1()6,3(

0,)( 2 akhxa)(xf),( kh

)6,3(

Writing the equation of a Parabola in Standard Form (Practice)

Vertex is:

The parabola passes through point

Find the Standard Form of the equation.

( , )3 1( , )4 1

f x a x h k( ) ( ) 2

Homeworkp.95-96 #1-8 all, 9-33x3

p. 96 # 36-60x3