1. in Accelerated Math 1 We Learned About a Third Degree Polynomial Function, Which Is

CHARACTERISTICS OF POLYNOMIAL FUNCTIONS

1. In Accelerated Math 1 we learned about a third degree polynomial function, which is also called

a cubic function. What do you think the word “polynomial” means?

Let’s break down the word: poly- and –nomial. What does “poly” mean?

a) A monomial is a numeral, variable, or the product of a numeral and one or more variables.

For example: -1, ½, 3x, 2xy, 5x 2. Give three examples of other monomials:

b) What is a constant?

Give three examples:

c) A coefficient is the numerical factor of the monomial, or the ______in front of

the variable in a monomial.

Give three examples of monomials and their coefficients.

d) The degree of a monomial is the sum of the exponents of its variables.

has degree 4. Explain why.

What is the degree of the monomial 3? Why?

e) Do you know what a polynomial is now? Give a definition in your own words..

A polynomial function is defined as a function,

f(x)= ,

where the coefficients are real numbers.

The degree of a polynomial (n) is equal to the greatest exponent of its variable.

The coefficient of the variable with the greatest exponent () is called the leading coefficient. For example, f(x)= is a third degree polynomial with a leading coefficient of 4.

2. Previously, you have learned about linear functions, which are first degree polynomial functions,

y=, where is the slope of the line and is the y-intercept.

(Recall: y=mx+b; here m is replaced by and b is replaced by .)

Also, you have learned about quadratic functions, which are 2nd degree polynomial functions

They can be expressed as y = .

a) To get an idea of what these functions look like, we can graph the first through fifth degree

polynomials with leading coefficients of 1. For each polynomial function, make a table of 6 points

and then plot them so that you can determine the shape of the graph. Choose points that are

both positive and negative so that you can get a good idea of the shape of the graph. Also,

include the x-intercept as one of your points. Do these five tables and graphs on your own

paper.

For example, for the first order polynomial function: . You might have the following table

and graph:

b) Compare these five graphs you just created. By looking at the graphs, describe in your own

words how is different from . Also, how is different from

c) Note any other observations you make when you compare these graphs.

3. In this unit, we will discover different characteristics of polynomial functions by looking at

patterns in their behavior. Polynomials can be classified by the number of monomials (or terms)

as well as by the degree of the polynomial. The degree of the polynomial is the same as the

term with the highest degree. Complete the following chart. Make up your own expression for

the last row.

Polynomial / Degree / Name / No. of terms / Name
2 / Constant / Monomial
/ Quadratic / Binomial
/ Cubic
/ Quartic
/ Quintic / Trinomial

4. In order to examine their characteristics in detail so that we can find the patterns that arise in

the behavior of polynomial functions, here are 8 polynomial functions (expressed both in standard

form as well as a product of linear factors) and their accompanying graphs that we will use to

refer back to throughout the task.

f(x) = or f(x)= x(x+2) k(x) = +4 or k(x) = (x-1)(x+1)(x-2)(x+2)

g(x) = or g(x)=x(-2x+1) l(x) = +4) or l(x) = -(x-1)(x+1)(x-2)(x+2)

h(x) = or h(x) = m(x) = or

m(x)= x(x-1)(x-2)(x+3)(x+4)

j(x)= or j(x) = -x(x-3)(x+1) n(x) = ) or

n(x)=- x(x-1)(x-2)(x+3)(x+4)

a) Using the graph of y=j(x), list its x-intercepts. ______

How are these x-intercepts related to the linear factors?

b) Why might it be useful to know the linear factors of a polynomial function?

c) Although we will not factor higher order polynomial functions in this unit, you have factored

quadratic functions in Math II. For review, factor the following second degree polynomials.

a)2

b)

c)

d) Using these factors, find the roots of these three equations.

a)

b)

c)

e) On your own paper, sketch a graph of the three quadratic equations above without using your

calculator, and then use your calculator to check your graphs.

f) Although you will not need to be able to find all of the roots of higher order polynomials until a

later unit, using what you already know, you can factor some polynomial equations and find their

roots in a similar way.

Try this one: . .

(HINT: Look for a Greatest Common Factor first.)

What are the roots of this fifth order polynomial function? ______

g) How many roots are there? ______

Why are there not five roots since this is a fifth degree polynomial?

h) Check the roots by generating a graph of this equation using your calculator.

i) With other polynomial functions, we will not be able to draw upon our knowledge of factoring

quadratic functions. For example, you may not be able to factor , but

can still find its zeros by graphing it in your calculator? How? What are the zeros of this

polynomial function?

5. Symmetry

The first characteristic of these 8 polynomials functions we will consider is symmetry.

a) Sketch a function below you have seen before that has symmetryaboutthey-axis.

Describe in your own words what it means to have

symmetry about the y-axis.

What do we call a function that has symmetry about

the y-axis?

b) Sketch a function below you have seen before that has symmetryabouttheorigin.

Describe in your own words what it means to have

symmetry about the origin.

What do we call a function that has symmetry about

the origin?

c) Using the table below and your handout of following eight polynomial functions, classify the

functions by their symmetry.

Function / Symmetry about the y-axis? / Symmetry about the origin? / Even, Odd, or Neither?
f(x) =
g(x) =
h(x) =
j(x)=
k(x) = +4
l(x) = +4)
m(x) =
n(x) = )

d) Now, sketcha higher order polynomial e) Now, sketch a higher order polynomial

function (an equation is not needed) with function (an equation is not needed) with

symmetryaboutthey-axis. symmetryabouttheorigin.

f) Why don’t we talk about functions that have symmetry about the x-axis? Sketch a graph that

has symmetry about the x-axis. What do you notice?

6. Domain and Range

Another characteristic of functions that you have studied is domain and range. For each polynomial function, determine the domain and range.

Function / Domain / Range
f(x) =
g(x) =
h(x) =
j(x)=
k(x) = +4
l(x) = +4)
m(x) =
n(x) = )

7. Zeroes

a) We can also describe the functions by determining some points on the functions. We could

find the x-intercepts for each function as we discussed before. Under the column labeled

“x-intercepts” write the ordered pairs (x,y) of each intercept. Also record the degree

of the polynomial, and record the number of zerosin the next column.

Function / Degree / # of Zeros / x-intercepts / Zeros
f(x) =
g(x) =
h(x) =
j(x)=
k(x) = +4
l(x) = +4)
m(x) =
n(x) = )

b) These x-intercepts are called the zeros of the polynomial functions. Why do you think they

have this name?

c) Fill in the column labeled “Zeroes” by writing the zeroes that correspond to the x-intercepts

of each polynomial function, and also record the number of zeroes each function has.

d) Make a conjecture about the relationship between the degree of the polynomial and number of

zeroes.

e) Test your conjecture by graphing the following polynomial functions using your calculator:
, , .

Function / Degree / # of Zeros / x-intercepts / Zeros
/ (0,0)
/ (0,0); (-1,-0;(-4,0)

How are these functions different from the functions in the previous table?

Now amend your conjecture about the relationship between the degree of the polynomial and the number of x-intercepts. Make a conjecture for the maximum number of x-intercepts

thefollowing polynomial function will have:

8. End Behavior

In determining the range of the polynomial functions, you had to consider theend behavior of the functions, that is the value of f(x) as x approaches infinity or negative infinity.

Polynomials exhibit patterns of end behavior that are helpful in sketching polynomial functions.

a) Graph the following equations on your calculator. Make a rough sketch next to each one and

answer the following:

• Is the degree even or odd?
• Is the leading coefficient(the coefficient on the term of highest degree) positive or negative?
• Does the graph rise or fall on the left? On the right?

1. 7.

2. 8.

3. 9.

4. 10.

5. 11.

6. 12.

b) Write a conjecture about the end behavior, whether it rises or falls at the ends, of a function of

the form for each pair of conditions below. Then test your conjectures on some of

the 8 polynomial functions graphed before on page 3 of your handout.

Condition a: When n is even and a > 0,

Condition b: When n is even and a < 0,

Condition c: When n is odd and a > 0,

Condition d: When n is odd and a < 0,

c) Which of the graphs from part (a) have anabsolute maximum?

Which have anabsolute minimum?

What do you notice about the degree of these functions?

d) Can you ever have an absolute maximum AND an absolute minimum in the same function?

If so, sketch a graph with both. If not, why not?

e) Based on your conjectures in part (b), f) Now sketch a fifth degree polynomial

sketch a fourth degree polynomial with a positiveleading coefficient.

function with a negativeleading coefficient.

g) Note we can sketch the graph with end behavior even though we cannot determine where and how

the graph behaves otherwise, and without an equation or without the zeros.

If we are given the real zeros of a polynomial function, we can combine what we know about end behavior to make a rough sketch of the function.

Sketch the graph of the following functions using what you know about end behavior and zeros:

a) b)

9. Critical Points

a) Other points of interest may also be where the graph begins or ends increasing or decreasing.

For each graph back on page 3, locate the turning points and the related intervals of increase

and decrease, as you have determined previously for linear and quadratic polynomial functions.

Then record which turning points are relative minimum orrelative maximumvalues.

Function / Degree / Turning Points / Intervals of Increase / Intervals of Decrease / Relative Minimum / Relative Maximum
f(x)
g(x)
h(x)
j(x)
k(x)
l(x)
m(x)
n(x)

b) Make a conjecture about the relationship between the degree of the polynomial and the number

of turning points that the polynomial has. Recall that this is the maximum number of turning

points a polynomial of this degree can have because these graphs are examples in which all zeros

have a multiplicity of one.

These characteristics of polynomial functionscan be used anytime one wants to describe and/or sketch graphs of polynomial functions.