1. Polynomial Functions What you’ll learn
To classify polynomials. To graph polynomial
functions and describe end behavior.
Vocabulary
Monomial, degree of a monomial, polynomial,
degree of a polynomial, polynomial function,
standard form of a polynomial function,
turning point, end behavior.

2. Take a note
You can classify a polynomial it its degree or by
its number of terms. Polynomial with a degree
0 to 5 have specific names.
Remember: monomial is a number, a variable, or a
product of a real number with one or more variables
with whole-number exponents. Polynomial is a monomial or a sum of monomial
The degree of a monomial in one variable is the
exponent of the variable.
The degree of a polynomial in one variable is the
greatest degree among its monomials terms.

3. For example: Polynomial in one variable: X Degree of the polynomial:4
Not a polynomial in one variable
Organize the expression
Polynomial in one variable X
Degree of the polynomial 5
Not a polynomial, the term

4. Quartic polynomial of 4 terms
Your turn: Classifying Polynomials
What is the classification of each polynomial by degree?
By number of terms?
Answers:
Degree 2 and has 3 terms
Quadratic trinomial
Degree 4 and has 4 terms.
Quartic polynomial of 4 terms
Note:
A polynomial with the variable x defines a polynomial function
of x. The standard form of a polynomial function arranges the
terms by degree in descending order.
Where and

5. Take a note:
A polynomial function has distinguishing behaviors.You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form.
Constant function
Degree 0
Linear function
Degree 1
Quadratic function
Degree 2
Cubic function
Degree 3
Quartic funcion
Degree 4
Quintic function
Degree 5

6. The degree of a polynomial function affects the shape of its graph and determines the maximum numbers of turning points. TURNING POINTS are the points where the graphs changes directions. It also affects the end behavior. END BEHAVIOR is the directions of the graph to the far left and the far right.
End behavior of a
Polynomial Function
of Degree
n with Leading Term
n even
n odd
a positive
up and up
down and up
a negative
down and down
up and down
In general the graph of a polynomial function of degree
Has at most turning points. The graph of a polynomial function of odd degree has an even number of turning points. The graph of a polynomial function of even degree has an odd number of turning points.

7. Your turn: Describing end behavior of polynomial functions
Consider the leading term of each polynomial function
What is the end behavior of the graph? Check your
answer with a graphing calculator.
The leading term is
n is even and a is negative
The end behavior is
down and down
The leading term is
n is odd and a is positive
The end behavior is
down and up.

8. Your turn: Graphing cubic functions
What is the graph of each cubic function? Describe
the graph. If you have a graphing calculator use it.
Step 1: a) x
f(x) -2 -4 -1
-0.5 1
0.5 2 4
Step 2:
Step 3: the end behavior
is down and up. No turning points

9. b c d The end behavior is up and down.
There are two turning points
The end behavior is up and down;
two turning points d
The end behavior is down and up;
No turning points

10. Your turn: Using differences to determine degree
What is the degree of the polynomial function that generates the data show? a) X
f(x) -3 -1 -2 -7 5 1 11 2 9 3
y-value -1 -7 -3 5 11 9
First
Difference -6 4 8 6 -2
-16
Second
Difference 10 4 -2 -8
-14
Third
Difference -6 x
f(x) -3 23 -2
-16 -1
-15
-10 1
-13 2
-12 3 29 b)
The third difference is a constant so
the degree of the polynomial function is 3
b)Degree 4

11. Classwork odd Homework even
TB pg 285 exercises 8-49