Transcript QUADRATIC FUNCTIONS

SECTION 3.1
POLYNOMIAL FUNCTIONS AND
MODELS
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.
POLYNOMIAL FUNCTIONS
Which of the following are polynomial
functions?
4
f(x) 2 - 3x
2
x - 2
h(x) 3
x - 1
g(x) x
F(x) 0
G(x) 8
POLYNOMIAL FUNCTIONS
SEE TABLE 1
POLYNOMIAL FUNCTIONS
The graph of every polynomial function
is smooth and continuous: no sharp
corners and no gaps or holes.
POLYNOMIAL FUNCTIONS
When a polynomial function is factored
completely, it is easy to solve the
equation f(x) = 0 and locate the xintercepts of the graph.
Example:
f(x) = (x - 1)2 (x + 3) = 0
The zeros are 1 and - 3
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.
POLYNOMIAL FUNCTIONS
If (x - r)m is a factor of a polynomial f and
(x - r)m+1 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.
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
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?
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.
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.
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.
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
EXAMPLE
Now graph: y = x4, y = x4 - (4/3)x3,
y = x4 - 2x2
How many turning points do you see
on these graphs?
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.
GRAPH:
P(x ) = x2
P2(x) = x3
4
x
5
x
P1(x) =
P3(x) =
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 - .
EXAMPLE:
Determine the direction the
arms of the graph should
point. Then, confirm your
answer by graphing.
f(x) = - 0.01x 7
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
The behavior of the graph of
a polynomial as x gets large is
similar to that of the graph of
the leading term.
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 nx n
EXAMPLE
DO EXAMPLES 9 AND 10
CONCLUSION OF SECTION 3.1