Section 7-1 Polynomial Functions
Algebra II
Learning Targets:
1. Evaluate polynomial functions
2. Identify general shapes of graphs of polynomial functions
I.
Polynomial Vocabulary
 A polynomial is a monomial or a sum of monomials (the coefficients must all be REAL
and the exponents must be a non-negative integer (1, 2, 3, 4…))
One example is:

3x 3  7 x 2  8x  5
The degree of a polynomial is the greatest exponent
For the polynomial 4 x 5  2 x 3  8x 2  5 , the degree is______

The leading coefficient is the number of the term with the highest degree
For the polynomial 7 x 4  2 x 3  8x  5 , the leading coefficient is ______
You try a couple:
1) 8 x 4  3x  2
3)
II.
2) 4
Degree:
Degree:
Leading coefficient:
Leading coefficient:
1
x
2
4) 18 y 2  3 y  y 4 Careful!!!!
Degree:
Degree:
Leading coefficient:
Leading coefficient:
Evaluating a polynomial function
a) Evaluating at a value
Example: Given the general polynomial f(x) = 8 x 2  3x  2 , find f(-2).
YOU TRY:
1) g(x)=12-x
2) p(c) 
Find g(-3)
Find p(-1)
1 4
c  2c 2  4
2
b) Evaluating an expression
Look at the polynomial from the previous example f(x) = 8 x 2  3x  2
Evaluate f(2a)
You try:
Given p( x)  4 x 2  2 x  3 and r ( x)  2 x 3  5x  7
1) Find r(-a)
2) Find p(2a)
Section 7-1 Polynomial Functions
Day 2: Definition of a Polynomial Function of degree n can be described by an equation of the form
P( x)  a x n  a1 x n 1  an  2 x 2  an 1 x  an
where the coefficients a  , a1 , a 2 ,..., a n
and a is not zero and n represents a nonnegative number.
Sketch the graphs:
yx
y  x3
y  x 3  4x 2  5
y  x2
y  x4
y  x 4  x 3  4x 2  1
represent real numbers

Look at the table below for a summary of end behavior
Leading Coefficient is:
POSITIVE
EVEN
Degree
ODD
Degree
NEGATIVE
x   , f(x)  
x   , f(x)  
x   , f(x) x  
x   , f(x) x  
x   , f(x)  
x   , f(x)  
x   , f(x) x  
x   , f(x) x  
End Behavior of Polynomial Functions
The end behavior of a function is the behavior as the graph goes to  or - 
Example:
This is the graph for f(x)

The end behavior for f(x)
x
x
, f ( x) 
, f ( x) 

How many real zeros?

Degree is odd or even?
1)
sketch:
2)
End behavior:
End behavior:
Number of real solutions:
Number of real solutions:
Degree of polynomial: even or odd
Degree of polynomial: even or odd
y  x 4  x 3  4x 2  4x
y  x 3  5x 2  3x  2
End behavior:
End behavior:
Number of real solutions:
Number of real solutions:
Degree of polynomial: even or odd
Degree of polynomial: even or odd