Transcript f(x)

October 5th
copyright2009merrydavidson
Warm up
Prove inverses by
composition
f(x)= -3x + 8
g(x)= 8 – x
3
Happy
Birthday to:
Emily
Wiegmann
2.1 Polynomial Functions
Examples of polynomial
functions
f(x) = 3x + 5
f(x) = 4
f(x) = x2
f(x) = 5x3
Linear
Constant
Quadratic
Cubic
2.1 Polynomial Functions
What do they all have in
common?
Non-negative
integer exponents
of the variables
f(x) = 3x + 5
f(x) = 4
Leading
coefficient
2
f(x) = x
not = 0
f(x) = 5x3
Smooth & continuous
graphs
Linear
Constant
Quadratic
Cubic
Polynomial Functions are
classified by degree.
Constant Functions; degree 0
Linear Functions; degree 1
Quadratic Functions;degree 2
Cubic Functions; degree 3
What is a Polynomial?
Vocabulary:
Degree of a term
f ( x )  2 x y  3 x y  5 xy  y
5
3 2
6
5
The sum of the powers on the variables in one term
Degree of a polynomial
The term with the highest degree
6
Leading Coefficient
The coefficient of the highest degree term.
Not always the first term.
-2
3
4
4
4
Examples: Find the leading coefficient and degree of each
polynomial function.
Polynomial Function
Leading Coefficient
Degree
f ( x )  2 x 5  3 x 3  5 x  1
–2
5
f ( x)  x3  6 x 2  x  7
1
3
14
0
f ( x )  14
Practice 1: Given the following equations determine the following:
1. Determine if the function is a polynomial in 1 variable. Why or why not?
2. If the function is a polynomial; what is the degree of each term,
and of the polynomial.
A.
f  x   3x 2  x  1
Yes, notice powers on the x are
positive integers and coefficients
are real numbers. 2,1, 0 2
C.
f  x 
B.
No, notice power on x
is -1
3
 3x 1
x
f  x  3 x
No, notice power on the x is the fraction 1/2
xx
E.
f  x   3x 2
1
2
No, there are 2
different
 2 y  1 variables.
3
 4x
x
D.
3x 2
f  x 
1
5
Yes, notice powers on the x are
positive integers and coefficients
are real numbers. 3/5,1 2
Where the function
touches or crosses
the x-axis.
What are zero’s
of a function?
y = -x^3+4x
(-2,0),(0,0),(2,0)
Zero’s
are
listed
as
ordered
pairs.
What are the zero’s of the given
function.
y = x^4-5x^2+4
(-2,0),(-1,0),(1,0),(2,0)
Zero’s
are
listed
as
ordered
pairs.
What are the zero’s of the given
function.
y = -2x^4+2x^2
When the
graph is
“sitting” on
the x-axis, it
is a “double”
root.
Multiplicity of 2
(-1,0),(0,0),(1,0)
(-1,0)(0,0)MP2,(1,0)
This is an even degree function so we need an even
number (4) of roots
Summary:
The degree of the polynomial tells
the number of zero’s (x-intercepts).
There is one less “turning point”
than the degree.
y = -x^3+4x
Degree 3
2 turning points
VOCABULARY
Roots/Solutions refer to the
algebraic answer of the
polynomial equation and is
expressed as x = answer.
Ex: What is the solution to
f(x)=x2 – 9?
You would put: x = 3, -3
VOCABULARY
Zero’s/x-intercept’s refer to the
graph of the polynomial
equation and are expressed as
ordered pairs.
Ex: What are the zero’s of
f(x)=x2 – 9?
You would put: (-3,0), (3,0)
2.2 Polynomial Functions of
higher degree
y = x5
y = x^5
Opposite end behavior
Down
f(x)


as x

f(x)

as x
UP
2.2 Polynomial Functions of
higher degree
y = -x^5
y = -x5
as x

f(x)

as x
Opposite end behavior
Up
Down
f(x)


2.2 Polynomial Functions of
higher degree The power
tells your
left arm
what to do.
Summary:
The
coefficient
tells your
right arm
what to do.
Pos xodd
down/up
y = 3x^7
ex: 3x7
Neg xodd
up/down
y = -4x^9
ex: - 4x9
END in the direction of the
coefficient.
2.2 Polynomial Functions of
higher degree
y = x6
as x

f(x)

as x

f(x)

y = x^6
Same end behavior
UP
UP
2.2 Polynomial Functions of
higher degree
y = -x6
y = -x^6
as x

f(x)

as x
Same end behavior
DOWN
DOWN
f(x)


2.2 Polynomial Functions of
higher degree The power
Summary:
The
coefficient
tells your
right arm
what to do.
Pos xeven
tells your
left arm
what to do.
up/up
y = 3x^8
ex: 3x8
Neg xeven
down/down
y = -4x^8
ex: - 4x8
END in the direction of the
coefficient.
Closure:
Positive
coefficient
even power
odd power
Negative
coefficient
LIMIT NOTATION again 
lim f ( x) 
x 
lim f ( x) 
x 
Practice
What is the right hand and left hand
end behavior for the following polynomial functions?
a) -3x3 + 5x2 – 2x + 3
up/down
b) 5x6 + 4x4 + 7x – 10 up/up
c) -2x8 + 3x7 + 4x – 2 down/down
d) 5x7 + 3x2 - 3
down/up
e) (x+2)2(2x-1)
down/up
Now write them in limit notation….
Homework:
WS 3-3