4.4 Properties of Rational Functions Day 2...4.4 Properties of Rational Functions Day 2 Objective:...
Transcript of 4.4 Properties of Rational Functions Day 2...4.4 Properties of Rational Functions Day 2 Objective:...
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
January 28, 2016
You MUST REDUCE BEFORE determining the vertical asymptotes.
The graph will never touch the vertical asymptote.
d.Holes
January 28, 2016
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
January 28, 2016
Horiz/obl
January 28, 2016
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?
January 28, 2016
no horiz or oblique
Summary of Horiz/Oblique ExamplesAsymptote
January 28, 2016
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.