Section 1.6Powers, Polynomials, and Rational

Functions

• Often in this class we will deal with functions of the form

• Functions of this form are called power functions– Notice the variable is being raised to an exponent– Contrast this with an exponential function where

the variable is in the exponent

constantsareand, nkkxy n

• Which of the following are power functions and identify the k and the n (recall ) nkxy

x

xy

xxy

xy

xy

3

23

.4

.3

/.2

3.1

• Power functions can be odd, even or neither– How can we decide?– What about the following?

• What about the end behavior of a power function versus an exponential– Which grows faster?

4/1

5

.2

25.0.1

xy

xy

• What happens if we add or subtract power functions?

• A polynomial is a sum (or difference) of power functions whose exponents are nonnegative integers

• What determines the degree of a polynomial?

• For example

• What is the leading term in this polynomial?

1103 2 xxy

• We have the general form of a polynomial which can be written as

• Where n is a positive integer called the degree of p– Each power function is called a term

– The constants an , an-1,… a0,are called coefficients

– The term a0 is called the constant term

– The term anxn is called the leading term

11 1 0( ) n n

n np x a x a x a x a

End Behavior

The shape of the graph of a polynomial function depends on the degree.

Degree EVEN Degree ODD

an>0

an<0an>0 an<0

• What are the zeros (or roots) of a polynomial?– Where the graph hits the x-axis– The input(s) that make the polynomial equal to 0

• How can we find zeros of a polynomial?

• For example, what are the zeros of

• Notice this polynomial is in its factored form– It is written as a product of its linear factors

• A polynomial of degree n can have at most n real zeros

)5)(3()( xxxh

Behavior of Polynomials

)3()( 2 xxxm 32 )3()( xxxn

What behavior do you notice at the zeros of these functions?

xx

What is the significance of this point?

What is the significance of this point?

• When a polynomial, p, has a repeated linear factor, then it has a multiple root– If the factor (x - k) is repeated an even number of

times, the graph does not cross the x-axis at x = k. It ‘bounces’ off. The higher the (even) exponent, the flatter the graph appears around x = k.

– If the factor (x - k) is repeated an odd number of times, the graph does cross the x-axis at x = k. It appears to flatten out. The higher the (odd) exponent, the flatter it appears around x = k.

• If r can be written as the ratio of polynomial functions p(x) and q(x),

then r is called a rational function

• The long-run behavior is determined by the leading terms of both p and q– These functions often have horizontal asymptotes

which define their long run behavior

)(

)()(

xq

xpxr

• We have three cases

• The degree of p < the degree of q– The horizontal asymptote is the line y = 0

• The degree of p > the degree of q– There is no horizontal asymptote

• The degree of p = the degree of q– The horizontal asymptote is the ratio of the

coefficients of the leading terms of p and q

• Let’s consider the following functions

• How do we find their x-intercepts?• What are they?• What happens if the denominators equal 0?• What are their horizontal asymptotes?

4

5)(

2

1)(

)3)(1(

3)(

2

3)(

2

2

2

x

xxk

x

xxh

xx

xxg

x

xxf