Transcript Section 1.6 Powers, Polynomials, and Rational Functions
Section 1.6
Powers, Polynomials, and Rational Functions
• Often in this class we will deal with functions of the form
y
kx n
,
k
and
n
are constants
• Functions of this form are called
power functions
– Notice the variable is being raised to an exponent – Contrast this with an exponential function where the variable is in the exponent
• Which of the following are power functions and identify the
k
and the
n y
kx n
1 .
y
2 .
y
3
x
/
x
3 .
y
x
3
x
2 4 .
y
x
3
x
• Power functions can be odd, even or neither – How can we decide?
– What about the following?
1 .
y
0 .
25
x
5 2 .
y
x
1 / 4 • What about the end behavior of a power function versus an exponential – Which grows faster?
• What happens if we add or subtract power functions?
• A
polynomial
is a sum (or difference) of power functions whose exponents are
nonnegative integers
• What determines the
degree
of a polynomial?
• For example
y
3
x
2 10
x
1 • What is the leading term in this polynomial?
• We have the general form of a polynomial which can be written as
a x n n
• Where
n
a n
1
x n
1
a x
1
a
0 is a positive integer called the degree of
p
– Each power function is called a
term
– The constants
a n , a n-
1
,… a
0
,
are called
coefficients
– The term
a
0 is called the
constant term
– The term
a n x n
is called the
leading term
End Behavior The shape of the graph of a polynomial function depends on the degree.
Degree EVEN Degree ODD
a n
>0
a n
<0
a n
>0
a n
<0
• What are the zeros (or roots) of a polynomial?
– Where the graph hits the
x
-axis – The input(s) that make the polynomial equal to 0 • How can we find zeros of a polynomial?
• For example, what are the zeros of
h
(
x
) (
x
3 )(
x
5 ) • Notice this polynomial is in its
factored form
– It is written as a product of its linear factors • A polynomial of degree
n
can have at most
n
real zeros
Behavior of Polynomials
m
(
x
)
x
2 (
x
3 )
n
(
x
)
x
2 (
x
3 ) 3
x x
What behavior do you notice at the zeros of these functions?
• When a polynomial,
p
, has a repeated linear factor, then it has a
multiple root
– If the factor (
x
-
k
) is repeated an
even
number of times, the graph does not cross the
x
-axis at
x
=
k.
It ‘bounces’ off. The higher the (even) exponent, the flatter the graph appears around
x
=
k.
– If the factor (
x
-
k
) is repeated an
odd
number of times, the graph does cross the
x
-axis at
x
=
k.
It appears to flatten out. The higher the (odd) exponent, the flatter it appears around
x
=
k
.
• If
r
can be written as the ratio of polynomial functions
p
(
x
) and
q
(
x
),
r
(
x
)
p
(
x q
(
x
) ) then
r
is called a
rational function
• The long-run behavior is determined by the leading terms of both
p
and
q
– These functions often have
horizontal asymptotes
which define their long run behavior
• We have three cases • The degree of
p
< the degree of
q
– The horizontal asymptote is the line
y
= 0 • The degree of
p
> the degree of
q
– There is no horizontal asymptote • The degree of
p
= the degree of
q
– The horizontal asymptote is the ratio of the coefficients of the leading terms of
p
and
q
• Let’s consider the following functions
f
(
x
)
x x
3 2
g
(
x
) (
x
3
x
2 1 )(
x
3 )
h
(
x
)
x
2
x
1 2
k
(
x
)
x
2 5
x
4 • How do we find their
x
-intercepts?
• What are they?
• What happens if the denominators equal 0?
• What are their horizontal asymptotes?