Question 193021: f(x)=x^2-x-6/x-6
What is the slant asymptote of the graph f? I need a full explanation as the dividing is what is giving me problems.
Found 2 solutions by RAY100, scott8148:
Answer by RAY100(1637)   (Show Source):
You can put this solution on YOUR website! Division of polynomials is one of THE hardest skills to overcome. You are NOT alone.
The technique I recommend to students to get more comfortable is as follows.
Review division of integers only. Algorithm,steps,is divide, multiply, subtract,bring down,repeat.
please humor me and practice this on numbers like (75678)/2. no kidding it helps.
Now use the same algorithm on polynomials. Again practice just a few. Make life easy by
picking a FOIL set, like (x+2)(x+3)=x^2+5x=6. no kidding
This might take 10 minutes. Please try.
Now getting to Asymptotes. Let's review options:
vertical asymptotes: Find zero's of den. Factor and set =0. They are vertical asymptotes.
horizontal asymptotes:
basic form y=a(n) x^n +----- /b(m)x^m +-----
1) if n less than m,,,,y=0
2) if n=m,,,,y=a(n)/b(m)
3) if n>m,,,, no horiz asy
4) if n=(m+1),,,, slant asymptote
to solve slant asymptote, divide function . Use only integer answer (not remainder),,
let y="integer answer" ,,,,,see example following
Your problem y=(x^2-x-6)/(x-6)
1) den is (x-6), zero(root) is x=6,,,this is vertical asymptote
2) n=2, m=1,,,,therefore n>m,,,,no horizontal asymptote
3) but n=(m+1) ,,,,therefore there is a slant asymptote
divide function,,9x^2-6x-6)/(x-6) = x+5 +(25/(x-6)
let y=x=5,,,,this is slant asymptote
now lets try a rough plot
setup x - y cordinate axis
draw a vertical line at x=5, this is vertical asymptote
draw ,,,,y=x=5,,,, this is slant asymptote
NOW lets put in some points
Factoring numerator,,,(x-3)(x+2),,,therefore x intercepts are x=+3,,and(-2)
Plot these as (3,0) and (-2,0)
to see how the asymptotes work,,,do a brief "t" box
x=,,,,,y=
0,,,,1
1,,,,+6/5,,, so curve goes up from (-2),,,and then down thru (+3)
5,,,,(-14) ,,,,so curve goes down to asymptote
(-5),,,,(-2.2)
(-10),,,(-6.5),,,left side of curve goes toward slant asymptote
To Review Division
,,,,,,,,,,,,,,,,,______x+5___ 25/(x-6)
,,,,,,,(x-6),,!x^2-x-6
,,,,,,,,,,,,,,,,,,x^2-6x
,,,,,,,,,,,,,,,,,,_______
,,,,,,,,,,,,,,,,,,,,,,,,,+5x-6
,,,,,,,,,,,,,,,,,,,,,,,,,+5x-30
,,,,,,,,,,,,,,,,,,,,,,,,,,_______
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,+25
Sorry about commas,,need spacers,,just erase
Answer by scott8148(6628)  (Show Source):
You can put this solution on YOUR website! factoring the denominator gives ___ [(x+2)(x-3)]/(x-6)
this results in zeroes at -2 and 3; and a vertical asymptote at 6
dividing (by long division) gives a quotient of x+5 (IGNORING the remainder)
___ this results in a slant asymptote of y=x+5
the division: x-6 into x^2-x-6
___ x goes into x^2, x times
___ x times x-6 is x^2-6x
___ x^2-x minus x^2-6x is 5x
___ x goes into 5x, 5 times
___ so the quotient (ignoring the remainder) is x+5
this graph shows both the vertical asymptote and the slant asymptote
___ you can see the ends taking the y=x+5 shape
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