View
250
Download
2
Category
Preview:
Citation preview
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
Recommended