Pre – Calculus Math 12 Characteristics of Polynomial Functions

Lesson Focus: To identify and analyze polynomial functions.

a polynomial function has the form 0

1

1

2

2

2

2

1

1 ... axaxaxaxaxaxf n

n

n

n

n

n

where

n is a whole number

x is a variable

the coefficients 0 to aan are real numbers

the degree of the polynomial function is n, the exponent of the greatest power of x

the leading coefficient is na , the coefficient of the greatest power of x

the constant is 0a

e.g. Which of these functions are not polynomial? Justify your answer. State the degree, the leading

coefficient and the constant term of each function.

Function Polynomial? Degree Leading

coefficient

Constant

term

a. xxg 4

b. 356 xxh

c. 12 xy

d. 34 52 xxy

e. 23 xy

the type of function relates to the degree of the polynomial function (the largest exponent on any of the

variables)

a function is odd or even depending upon its degree

the end behaviour of a function is the sign of the y-values as you move from left to right on the function

i.e. there are four possible combinations: /or / ,/ ,/

the number of x-intercepts (zeroes) relates to the degree of the polynomial

i.e. an odd function can only have an odd number of zeroes while an even function can only have an even

number of zeroes (up to and including the degree of the function)

a function has either a maximum or minimum depending upon the sign of the leading coefficient

a maximum or minimum can be either global (for the entire function) or local/relative (for a particular

section of the function)

the y-intercept occurs when 0f

e.g. For each polynomial function, identify the following characteristics.

242 23 xxxxf 6510 24 xxxxg 152 35 xxxxh 1652 xxxp

Type

Odd/even

End behaviour

Number of

x-intercepts

Max. or min.

y-intercept

Corresponding

graph

we can solve problems that are based upon polynomial functions

e.g. An antibacterial spray is tested on a bacterial culture. The population, P, of bacteria t minutes after the

spray is applied is modelled by the function 800322 23 ttttP .

a) What is the population of the bacterial 3 minutes after the spray is applied?

b) How many bacteria were in the culture before the spray was applied?

c) What is the population of the bacteria 8 minutes are the spray is applied? Why is this not realistic

for this situation? Explain.