Transcript Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions

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 an , an1 ,..., a2 , a1 , a0 be real numbers with an  0
The function given by
f (x)  a n x n  an1 x n1      a2 x 2  a1 x  a0
Is called a polynomial function of x with degree n
Example: f ( x)  3x 4  2 x 2  x  1
This is a 4th degree polynomial
Polynomial Functions are classified by
degree
For example:
In Chapter 1
Polynomial function
f ( x)  a , with a  0
y
2
f ( x)  2
Example:
This function has
degree 0, is a
horizontal line and is called
a constant function.
x
–2
Polynomial Functions are classified by
degree
In Chapter 1
y
A Polynomial function
f ( x)  mx  b , m  0
is a line whose slope is m
and y-intercept is (0,b)
Example: f ( x)  2 x  3
This function has a degree
of 1,and is called a
linear function.
2
x
–2
Section 2.1 Quadratic Functions
Definition of a quadratic function
Let a, b, and c be real numbers with a  0.
The function given by f(x)= ax 2  bx  c
Is called a quadratic function
This is a special U shaped curve called a … ?
Parabola !
y
Parabolas are
symmetric to a
line called the
axis of
symmetry.
The point where
the axis
intersects with
the parabola is
the vertex.
2
x
–2
The simplest type of quadratic is f (x)  ax
2
ax
f
(x
)

When sketching
2
y

x
Use
as a reference.
(This is the simplest type of graph)
y
2
a>1 the graph of y=af(x)
is a vertical stretch of the
graph y=f(x)
2
x
–2
0<a<1 the graph of y=af(x)
is a vertical shrink of the graph y=f(x)
Graph on your calculator
1 2
2
2
f ( x)  x , f ( x )  3 x , f ( x )  ( ) x
4
Standard Form of a quadratic Function
f (x)  a( x  h)  k , a  0
2
The graph of f(x) is a parabola whose axis
is the vertical line x=h and whose vertex is
the point (h, k).
NOTE!
h -shifts the graph right or left
k -shifts the graph up or down
For a>0 the parabola opens up
a<0 the parabola opens down
Example of a Quadratic in Standard Form
f (x)  a( x  h) 2  k , a  0
Graph : f ( x)  x
Where is the Vertex? ( , )
2
y
2
Graph:
f ( x)  ( x  2)  4
2
x
–2
Where is the Vertex? ( , )
Identifying the vertex of a quadratic function
Another way to find the vertex is to use   b , f (  b ) 
2a 
 2a
the Vertex Formula
If a>0, f has a minimum x
If a<0, f has a maximum x
f ( x)  2 x  8 x  7
2
a
b
c
NOTE:
the vertex is: ( , )
2
To use Vertex Formula- f ( x)  ax  bx  c
To use completing the square start
2
2
with f ( x)  ax  bx  c to get f ( x )  a ( x  h)  k
Identifying the vertex of a quadratic function
(Example)
f ( x)   x  6 x  8
2
Find the vertex of the parabola ( , )
b
The direction the parabola opens?________
By completing the square? By the Vertex Formula 2a
Identifying the x-Intercepts of a
quadratic function
The x-intercepts are
found as follows
f ( x)   x  6 x  8
2
 x  6x  8  0
2
 ( x  6 x  8)  0
 ( x  2)( x  4)  0
x2  0
x2
x4
x4  0
2
( 2,0)
( 4,0)
Identifying the x-Intercepts of a
quadratic function (continued)
Standard form is: f ( x )  ( x  3) 2  1
y
Shape:_______________
Opens up or down?_____
2
x
X-intercepts are
( 2,0) ( 4,0)
–2
Identifying the x-Intercepts of a
Quadratic Function (Practice)
Find the x-intercepts of f ( x)  2 x  6 x  8
2
y
2
x
–2
Writing the equation of a Parabola
in Standard Form
(1,2)
Vertex is:
The parabola passes through point
(3,6)
f (x)  a( x  h)  k , a  0
2
*Remember the vertex is ( h, k )
Because the parabola passed through (3,6) we have:
 6  a(3  1)  2
 6  4a  2
2
f
(x
)

2  a
 2( x  1)  2
2
Writing the equation of a Parabola
in Standard Form (Practice)
Vertex is: (3,1)
The parabola passes through point (4,1)
Find the Standard Form of the equation.
f ( x )  a ( x  h)  k
2
Minimum and Maximum Values of
Quadratic Functions
• 1. If a>0, f has a minimum value at
b
x
2a
• 2. If a<0, f has a maximum value at
b
x
2a
Baseball
• A baseball is hit at a point 3
feet above the ground at a
velocity of 100 feet per second
and at an angle of 45 degrees
with respect to the ground.
The path of the baseball is
given by the function
f(x)=-0.0032x2 + x + 3, where
f(x) is the height of the
baseball (in feet) and x is the
horizontal distance from home
plate (in feet). What is the
maximum height reached by
the baseball?
Cost
• A soft drink manufacturer
has daily production costs
of C ( x)  70,000  120 x  0.055x 2
where C is the total cost
(in dollars) and x is the
number of units
produced. Estimate
numerically the number of
units that should be
produced each day to
yield a minimum cost.
Grants
• The numbers g of grants
awarded from the National
Endowment for the
Humanities fund from 1999 to
2003 can be approximated by
the model
g (t )  99.14t 2  2, 201.1t  10,896
9≤t≤13 where t represents
the year, with t=9
corresponding to 1999.
Using this model, determine
the year in which the number
of grants awarded was
greatest.
Homework
• Page 99-102
1-4 all, 6, 8-20 even, 27,28,29-33 odd, 40-44
even, 55,57,61