2.3 Polynomial and 2.3 Polynomial and Rational FunctionsRational Functions

• Identify a Polynomial Function• Identify a Rational Function• Find Vertical and Horizontal Asymptotes for

Rational Functions• Review finding x and y intercepts of graphs

Polynomial and rational functions are often used to express relationships in application problems.

Scary Math Definition for Polynomial

Be sure to know the end behavior properties (2 and 3 below).

Scary Math Definition for Rational Function

Go forward a few slides to see the easy-to-understand explanation.

DEFINITION:

The line x = a is a vertical asymptote if any of the following limit statements are true:

We will learn about limits in section 3.1

limx a

f x limx a

f x

limx a

f x .limx a

f x

•If c makes the denominator zero, but doesn’t make the numerator zero, then x = c is a vertical asymptote.

•If c makes both the denominator and the numerator zero, then there is a hole at x=c

Example of holeExample of hole

Example 2: Determine the vertical asymptotes of the function given by

( 2)( )

( 1)( 1)

x xf x

x x x

Example 2

There are Vertical Asymptotes atx = 1 and x = -1.

There isn‘t a vertical asymptote at x = 0.Since 0 makes both the numerator and denominator equal zero, there is a hole where x = 0.

• Since x = 1 and x = –1 make the denominator 0, but don’t make the numerator 0, x = 1 and x = –1 are vertical asymptotes.

• x=0 is not a vertical asymptote since it makes both the numerator and denominator 0.

The line y = b is a horizontal asymptote if either or both of the following limit statements are true:

or

We will learn about limits in section 3.1.

limx

f x b limx

f x b.

The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes are found by comparing the degree of the numerator to the degree of the denominator. 3 cases

Same: y = leading coefficient/leading coefficientBOB: y = 0 (bottom degree bigger)TUB: undefined-no H.A. (top degree bigger

Bob and tub are not in the textbook.

f (x) 3x2 2x 4

2x2 x 1.

Determine the horizontal asymptote of the function given by

Example of vertical and Example of vertical and horizontal asymptoteshorizontal asymptotes

3x-5

8y of intercepts theFind

Intercepts •The x-intercepts occur at values for which y = 0. For a fraction to = 0, the numerator must equal 0. Since 8 ≠ 0, there are no x-intercepts.

•To find the y-intercept, let x = 0.

y-intercept (0, 8/5)5

8y

Suppose the average cost per unit in dollars, to produce x units of a product is given by

30

500

x

xC

(10),C (50),C )100(C(a) find (b) How much would 10 units cost? (c) Identify any intercepts & asymptotes. Graph the function to verify your answers.

C

(a) $12.50, $6.25, $3.85(b) $12.50 x 10 = $125.00(c) V.A. x = -30 H.A. y = 0 no x-intercepts y-intercept (0, 50/3)