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) =

a0xn a1x

n 1 a2xn 2 ... an 2x

2 an 3x1 an ,

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 (

a0) is called the leading coefficient. For example, f(x) =

4x 3 5x 2 x 8 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 =

a0x1 a1, where

a0 is the slope of the line and

a1 is the y-intercept. (Recall: y = mx + b; here m is replaced by

a0 and b is replaced by

a1.)

Also, you have learned about quadratic functions, which are 2nd degree polynomial functions They can be expressed as y =

a0x2 a1x

1 a2.

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

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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 Monomial22 3x Quadratic Binomial3x Cubic

4 23x x Quartic53 4 2x x Quintic Trinomial

x y-3 -3-1 -10 02 25 5

10 10

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. G:\Polynomials\Handout.docxf(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)

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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

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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 symmetry about the y -axis .

Describe in your own words what it means to havesymmetry 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 symmetry about the origin.

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) = + 4l(x) = + 4)

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m(x) = n(x) =

)

d) Now, sketch a higher order polynomial e) Now, sketch a higher order polynomial function (an equation is not needed) with function (an equation is not needed) with symmetry about the y -axis . symmetry about the origin.

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)

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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 zeros in the next column.

Function Degree # of Zeros x-

intercepts Zeros

f(x) = g(x) =

h(x) = j(x) =k(x) = + 4l(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:

, , .

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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 the following polynomial function will have:

8. End Behavior

In determining the range of the polynomial functions, you had to consider the end 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. y x 7. 2y x2. 2y x 8. 43y x3. 3y x 9. 3y x4. 45y x 10. 52y x5. 3y x 11. 63y x6. 52y x 12. 37y x

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,

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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 an absolute maximum?Which have an absolute 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 positive leading coefficient. function with a negative leading 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)

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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 or relative maximum values.

Function

Degree

Turning

PointsIntervals of

IncreaseIntervals of Decrease

Relative Minimu

m

Relative Maximu

mf(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 functions can be used anytime one wants to describe and/or sketch graphs of polynomial functions.

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