POLYNOMIAL FUNCTIONS AND MODELS

1
SECTION 3.1

• POLYNOMIAL FUNCTIONS AND MODELS

2
POLYNOMIAL FUNCTIONS
A polynomial is a function of the form f(x) a n
x n a n-1 x n-1 . . . a1x a0 where an, a
n-1, . . ., a1, a0 are real numbers and n is a
nonnegative integer. The domain consists of all
real numbers.
3
POLYNOMIAL FUNCTIONS
Which of the following are polynomial functions?
4
POLYNOMIAL FUNCTIONS
SEE TABLE 1
5
POLYNOMIAL FUNCTIONS
The graph of every polynomial function is smooth
and continuous no sharp corners and no gaps or
holes.
6
POLYNOMIAL FUNCTIONS
When a polynomial function is factored
completely, it is easy to solve the equation f(x)
0 and locate the x-intercepts of the
graph. Example f(x) (x - 1)2 (x 3)
0 The zeros are 1 and - 3
7
POLYNOMIAL FUNCTIONS
If f is a polynomial function and r is a
real number for which f (r ) 0, then r is
called a (real) zero of f , or root of f.
If r is a (real) zero of f , then (a) r is an
x-intercept of the graph of f. (b) (x - r) is a
factor of f.
8
POLYNOMIAL FUNCTIONS
If (x - r)m is a factor of a polynomial f and (x
- r)m1 is not a factor of f, then r is called a
zero of multiplicity m of f. Example f(x)
(x - 1)2 (x 3) 0 1 is a zero of
multiplicity 2.
9
POLYNOMIAL FUNCTIONS
For the polynomial f(x) 5(x - 2)(x 3)2(x -
1/2)4 2 is a zero of multiplicity 1 - 3 is a zero
of multiplicity 2 1/2 is a zero of multiplicity 4
10
INVESTIGATING THE ROLE OF MULTIPLICITY
For the polynomial f(x) x2(x - 2) (a) Find the
x- and y-intercepts of the graph. (b) Graph the
polynomial on your calculator. (c) For each
x-intercept, determine whether it is of odd or
even multiplicity. What happens at an x-intercept
of odd multiplicity vs. even multiplicity?
11
EVEN MULTIPLICITY
If r is of even multiplicity The sign of f(x)
does not change from one side to the other side
of r. The graph touches the x-axis at r.
12
ODD MULTIPLICITY
If r is of odd multiplicity The sign of f(x)
changes from one side to the other side of
r. The graph crosses the x-axis at r.
13
TURNING POINTS
When the graph of a polynomial function changes
from a decreasing interval to an increasing
interval (or vice versa), the point at the change
is called a local minima (or local maxima). We
call these points TURNING POINTS.
14
EXAMPLE
Look at the graph of f(x) x3 - 2x2 How many
turning points do you see? Now graph
y x3, y x3 - x, y x3 3x2 4
15
EXAMPLE
Now graph
y x4, y x4 - (4/3)x3, y x4 - 2x2
How many turning points do you see on these
graphs?
16
THEOREM
If f is a polynomial function of degree n, then
f has at most n - 1 turning points. In fact, the
number of turning points is either exactly n -
1or less than this by a multiple of 2.
17
GRAPH

• P(x ) x2 P2(x) x3
• P1(x) x4 P3(x) x5

18

• When n (or the exponent) is even, the graph on
  both ends goes to .
• When n is odd, the graph goes in opposite
  directions on each end, one toward , the
  other toward - .

19
EXAMPLE

• Determine the direction the arms of the graph
  should point. Then, confirm your answer by
  graphing.
• f(x) - 0.01x 7

20
EXAMPLE

• Graph the functions below in the same plane,
  first using - 10,10 by - 1000, 1000, then
  using - 10, 10 by - 10000, 10000

• p(x) x 5 - x 4 - 30x 3 80x 3
• p(x) x 5

21

• The behavior of the graph of a polynomial as x
  gets large is similar to that of the graph of the
  leading term.

22
THEOREM
For large values of x, either positive or
negative, the graph of the polynomial f(x) a n
x n a n-1 x n-1 . . . a1x a0 resembles
the graph of the power function y a n x n
23
EXAMPLE
DO EXAMPLES 9 AND 10
24

• CONCLUSION OF SECTION 3.1