January 28, 2016

4.4 Properties of Rational

Functions Day 2

Objective:Find vertical, horizontal/oblique asymptotes.Find holes.Long divide polynomials.

January 28, 2016

3 Kinds of Asymptotes:

vertical

horizontal

oblique

KEY CONCEPT: Asymptotes are LINESWatch for application of this in Ex 5!!!

Vertical Asymptotes use the denominator!

January 28, 2016

FACTOR FIRST!!!vertical asymptote: A rational function will have a vertical

asymptote x = r if q(r) = 0. (meaning, vertical asymptotes happen where the function is undefined...DENOMINATOR = 0)

Finding Asymptotes

Example 1: State the domain and any vertical asymptotes.

a. b. c.

vert

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You MUST REDUCE BEFORE determining the vertical asymptotes.

The graph will never touch the vertical asymptote.

d.Holes

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3 Kinds of Asymptotes:

vertical

horizontal

obliqueWe will need to use long division to find horizontal/oblique asymptotes.

KEY CONCEPT: Asymptotes are LINES

{ or

January 28, 2016

Finding Horizontal Asymptotes:

If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0

If the degree of the numerator equals the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients:

y =

If the degree of the numberator is greater than the degree of the denominator, then there is NO horizontal asymptote.

leading coefficient of top

leading coefficient of bottom

But.....there may be an oblique asymptote

January 28, 2016

horizontal or oblique (need to know how values behave for )

Horiz/obl

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Horiz/obl

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Example 5:

not proper...do long division

This will NOT give us a horizontal or oblique asymptote....why? What's different about this

one verses the other examples we completed?

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no horiz or oblique

Summary of Horiz/Oblique ExamplesAsymptote

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vertical: x = #, where denominator = 0 (reduce first)

holes: factor, top and bottom expression cancels.

horizontal/oblique: (compare degrees)

num < denom y = 0 (proper)

num = denom y = a (reduce leading coeffts)

num > denom by 1 long div to find oblique asymptote

num > denom by more than 1 no horiz/obl asymptote

ASYMPTOTE SUMMARY

Can you have a horizontal AND an oblique asymptote for the same function?

January 28, 2016

Homework:p.225 #43-53 odd

State the domain of each problem too!!!!!!

Objective:Find vertical, horizontal/oblique asymptotes.Find holes.Long divide polynomials.