Example 1: Write the cubic function in standard form by determining the product:
 x

2
     x

A common type of polynomial function is a power function . A power function is a function of the form where n is a non-negative integer.
( )
 ax n
,
Example 2:
Consider each power function and its graph in the sequence shown. Complete the final two graphs in the sequence.

The end behavior of a graph of a function is the behavior of the graph as x approaches infinity and as x approaches negative infinity.
Example 3: Consider the sequence of graphs shown.
Write each function in terms of x , and then sketch it.
Example 4: Complete the table to describe the end behavior for any polynomial function.
Odd Degree Power Function Even Degree Power Function a > 0 a < 0

C.)
B.)
Example 5:
A.)

Example 1: Analyze the graph shown.
a.
Identify 2 symmetric points. b.
If one point is ( , ) what are the coordinates of the other symmetric point? c.
What do you notice about the y -values when you replace x with
 x ? d.
Write a general statement to explain the relationship between any two points symmetrical to the line x

0 .
Example 2: Analyze the graph shown.
a.
Identify 2 symmetric points. b.
If one point is ( , ) what are the coordinates of the other symmetric point? c.
What do you notice about the y -values when you replace x with
 x ? d.
Write a general statement to explain the relationship between any two points symmetrical to the line x

0 .
An even function has a graph symmetric about the y -axis, thus ( )
 f (
 x )
An odd function has a graph symmetric about the origin, thus
   x )
Example 3: State whether the graph of each function shown is even, odd, or neither.

Example 4: Determine algebraically whether the functions are even, odd, or neither. a.
f x

2 x
3 
3 x b.
g x

6 x
2 
10 c.
( ) c.
Example 5:
 x
3 
3 x
2
2 x

7

Example 6: c.
Example 7:
Example 8:
The equation for a polynomial function 𝑝(𝑥) is given. The equation for the transformed function 𝑡(𝑥) in terms of 𝑝(𝑥) is also given. Describe the transformation(s) performed on 𝑝(𝑥) that produced 𝑡(𝑥) . Then, write an equation for 𝑡(𝑥) in terms of 𝑥 .
A.) B.)

Example 1: Find the number of relative maximums and relative minimums in each (Extrema)

Example 2: Power functions vs.
Polynomial functions:
Example 3 : Find the zeros of the polynomials. Multiplicity occurs when a zero occurs more than one time. State the multiplicity of each zero. a.
𝑓(𝑥) = 𝑥
3
+ 2𝑥
2
+ 𝑥 b. 𝑓(𝑥) = 2𝑥
4
− 2𝑥
2
If you are given the zeros (or x -intercepts) of a polynomial function, you can build a possible polynomial function by building a factor from each zero.
Example 4. Write a polynomial function having zeros of 𝑥 = 3 , and 𝑥 = −3 ( multiplicity 2)
Convert your polynomial equation to standard form.
Example 5 . Write a cubic function with the given characteristics. You do not need to convert to standard form, but do not leave imaginary numbers in your answers (multiply enough to get rid of them).
A. Zeros: 𝑥 = 2, 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −4
B. Zeros: 𝑥 = 0, 𝑥 = 2𝑖, 𝑎𝑛𝑑 𝑥 = −2𝑖 C. Zeros: 𝑥 = 6 (𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑦 2), 𝑎𝑛𝑑 𝑥 = −5

Example 6: a.
Consider the graph shown: a.
Is the a -value of this function positive or negative? Is the degree even or odd? b.
Can this function be a cubic function? Explain why or why not. c.
State the domain and range of this function. d.
Determine the number of relative extrema (maxs and mins) in this graph. b.
Consider the graph shown: a.
Is the a -value of this function positive or negative? Is the degree even or odd? b.
Can this function be a cubic function? Explain why or why not. c.
State the domain and range of this function. d.
Determine the number of relative extrema in this graph.
** Change ‘Zeros’ on the next page to either ‘Real Zeros’ or ‘X-Intercepts”

Example 7: Use the coordinate plan to sketch a graph with the characteristics given. a.
Characteristics:

Degree 4

Starts in quadrant III

Ends in quadrant IV

Relative maximum at x
 
4 b.
Characteristics:

As x
As x
 
, ( )
 
, ( )
 
 

4 x -intercepts

Relative maximum at y

3

Characteristics:
 x -intercepts at -2, 2 and 5

Negative a value

Degree 3

Unit 3 Assn 4 Study Notes
Example 1 . Find the x -intercepts (zeros) and their multiplicity for each of the polynomial functions.
A. 𝑓(𝑥) = 𝑥
3
− 𝑥
2
− 2𝑥 B. 𝑔(𝑥) = 𝑥
2
− 𝑥 − 3 C. ℎ(𝑥) = (𝑥 + 3)(𝑥 − 2)
2
(𝑥 + 1)
3
Multiplicity is how many times a particular number is a zero for a given polynomial function. If the multiplicity is even the graph will “Bounce” at that zero and if the multiplicity is odd the graph will “Cross” at that zero .
** In Example 1, include whether the graph ‘bounces’ or ‘crosses’ at each x intercept.
Polynomial functions
2.
Polynomials are continuous (there are no breaks in their graphs) and have only smooth turns (there are no sharp turns).
Example 2 . Are the following Polynomials? If not, why not?
Example 3 . Graph the following functions (rough graphs)
A. 𝑔(𝑥) = −3𝑥
3
− 12𝑥
2
End Behavior y -intercept: (0, ) x -intercepts
B. 𝑓(𝑥) = (𝑥
2
− 1)
2
End Behavior y -intercept: (0, )
   
    y y






 x

 x


x -intercepts
C. ℎ(𝑥) = −(𝑥 + 1)
2 (𝑥 − 2) 4
(𝑥 + 3)
Degree y
End Behavior x -intercepts
        x

The Closure Property
When an operation is performed on any number or expression in a set and the result is in the same set, it is said to closed under that operation.
Determine whether each set within the Real Number System is closed under addition, subtraction, multiplication, and division.
Example 1 . Complete the table. If the set is not closed under a given operation, provide a counterexample .
Multiplication Division
Natural Numbers
{𝟏, 𝟐, 𝟑, 𝟒, … . }
Whole Numbers
{𝟎, 𝟏, 𝟐, 𝟑, 𝟒, … . }
Integers
{… . , −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, … . }
Rational Numbers
Can be represented as the ratio 𝑎 of two integers 𝑏 ≠ 0 𝑏
Where a and b are integers.
Irrational Numbers
Cannot be represented as the ratio of two integers: 𝜋
Addition Subtraction
Example 2 . Is the set of even integers closed under addition?
Example 3 . Is the set of odd integers closed under addition?
Example 4 . Is the set of even integers closed under division?
Example 5 . Is the set of odd integers closed under division?

Example 6 . Is the set of even integers closed under multiplication?
Example 7.
Is the set of odd integers closed under multiplication?
Example 8 . Write 5 polynomials with various degrees.
𝑦
1= 𝑦
2
=
𝑦
4
= 𝑦
5
=
Determine whether polynomials are closed under the following properties:
Addition
Subtraction
Multiplication
Division 𝑦
3
=