4.2 Polynomial Functions and Models
f(x) = x2 + 4x + 4
f(x) = 2x + 3

y






   






x


   




f(x) = -x3 + 4x - 1


x

f(x) = -x4 + x3 + 3x2 - x - 2
y


y








   





y





x

   





x








f(x) = .01x5 + .03x4 - .63x3 - .67x2 + 8.46x - 7.2

y



   




x
 




4.2-1
The anatomy of a polynomial
f(x) = 5x2 - 2x3 + x - 7
leading term:
- 2x3
leading coefficient:
-2
degree of leading term:
3
degree of polynomial:
3
End behavior of a polynomial
With respect to the graph of a polynomial:
left tail – appearance at extreme left of the graph
right tail – appearance at extreme right of graph
This graph:
 rises on the left
 falls on the right
 instead of writing it out, notation you may use:

Relationship of degree to x-intercepts, turning points
For a polynomial:
 degree > #turning points
 degree  #x-intercepts
4.2-2
End (tail) behavior of polynomials (of degree > 0)
It is determined by the leading term:
even
Degree
of
leading
term
odd
Leading coefficient
+
example: 3x4
example: -3x4
left tail  +
left tail  -
right tail  +
right tail  -
picture:
picture:
example: 3x3
left tail  -
right tail  +
picture:
example: -3x3
left tail  +
right tail  -
picture:
Summary – look at degree and coeff of leading term:
even: tails go same direction ( for +,  for -)
odd: tails go opposite ( for +,  for -)
Example:
f(x) =
-7x2 - 3x4 + 7
leading term:
-3x4
even: go same direction coeff(-): direction is down
4.2-3
Analyzing the graph or the formula of a polynomial
For a polynomial, we can deduce

things about the formula from the graph

things about the graph from the formula

y



   




x
 




tails:  polynomial is of odd degree, leading coeff +
degree > #turning points  degree > 4
degree  #x-intercepts  degree  5
remember the formula for this graph from the first page?
f(x) = .01x5 + .03x4 - .63x3 - .67x2 + 8.46x - 7.2
we can determine from the formula alone:
tails: , 5 > # turning points, 5  # x-intercepts
Another one: f(x) = -x4 + x3 + 3x2 - x - 2 (see page 1)
tails: 
degree > #turning points  4 > #turning points
degree  #x-intercepts  4  #x-intercepts
4.2-4