jdlogo

jdlogo

jdlogo

jdlogo

jdlogo

Home

Polynomial Functions Introduction

Investigating Polynomial Functions

Polynomial Division

Remainder & Factor Theorems

Polynomial Equations

Polynomial Inequalities

Rational Functions

Review&Test

 

jdsmathnotes

 

 


  UNIT 4  : POLYNOMIAL AND RATIONAL FUNCTIONS

 LESSON 1: POLYNOMIAL FUNCTIONS INTRODUCTION

 

 

Example 1:

f(x) = x3 – 4x2 + x + 6 is a cubic polynomial function because the largest exponent of the variable is 3.  Make a table of values or use

a graphing calculator to draw it’s graph [see below].

 

Note:  The function has 3 zeros (x – intercepts), x = -1, 2, 3 and two

turning  points A and B.  Turning point A is a local maximum point as

the function changes from increasing to decreasing at A.  Turning point B

is a local minimum point as the function changes from decreasing to increasing

at B.  The function has a local maximum value of y = 6.1 at point A and a

local minimum value of y = -0.9 at point B.

 

Text Box: In general a cubic function will have at most 3 zeros and at most 2 turning points.  There could be fewer. (see example 2 below).
A quartic function (degree 4) will have at most 4 zeros and 3 turning points.  Again there could be fewer. (see example 3 below).

 

 

 

 

 

 


Example 2:

Use your graphing calculator to graph and study the cubic functions below.

 

       a)  f(x) = – x3 – x2 + 5x – 3                                                            b)  f(x) = (x – 3)3 + 1

 

     

 

 

c)      f(x) = x3 + 3x2 + 3x + 2

 

 

 

Note: From the examples above we can make the following observations.

If k < 0, the function rises to the left and falls to the right (ex. 2a).  This characteristic is true for all polynomial functions of odd degree (1, 3, 5, …).

 

Example 3:

Use your graphing calculator to graph and study the quartic functions below.

 

a)      f(x) = x4 + x3 –5x2 – 3x                                                 b) y = x4 + x3 – 2x2 – 3x

 

       

 

 

     c)  y = -0.5x4 – x3 + 2x2 – 5                                                             d)  y = x4 + x3 – 2x2 – 3x + 3

 

   

 

 

Note: From the examples above we can make the following observations.

In (a,b,d) the leading coefficient is positive and the function rises to the left and right

If k < 0, the function falls to the left and right (ex. 3c).  This characteristic is true for all polynomial functions of even degree (2, 4, 6, …).

 

 

Polynomial Functions in Factored Form  f(x) = k(x – a)(x – b)(x – c) . . . etc.

 

Linear Functions in factored Form:  f(x) = k(x – a)

                                                         

 

   

 

Text Box:

 

 


Quadratic Functions in factored Form:  f(x) = k(x – a)(x – b)

 

 

 

 

 

Text Box:

 

 

 


Cubic Functions in factored Form:  f(x) = k(x – a)(x – b)(x – c)

 

 

 

x

y

-2

6

0

-6

 

 

 

Text Box:

 

 

 


Quartic Functions in factored Form:  f(x) = k(x – a)(x – b)(x – c)(x – d)

 

 

 

x

y

-1

16

2

16

 

 

Text Box:

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Return to top of page

Click here to go to homework questions