Overview of Finding Asymptotes Slant Asymptotes: Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. Slant asymptotes can be found by dividing the numerator of the equation by the denominator. original function: graph of the original function: long division: rearranged function: polynomial part: graph of
polynomial part: y = –3x – 3
Note the similarity between the two graphs.
Except for where the vertical asymptote causes a
break in the middle, the two graphs are practically
the same, as you can see from the overlay.
Vertical Asymptotes: A vertical asymptote is the x-­‐value at which x does not exist but gets infinitely close. A vertical asymptote can be found be setting the denominator of the equation to zero and solving for x. Original Function: Graph of Original Function: Set Denominator to Zero and Solve for x: x2 – 5x – 6 = 0 (x – 6)(x + 1) = 0 x = 6 or –1 Note that x comes infinitely close to but never equals the roots of the polynomial of the function’s denominator (6 and -­‐1). Horizontal Asymptote: A horizontal asymptote occurs at the y-­‐value at which y does not exist but comes infinitely close. A horizontal asymptote can be found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. Original Function and Graph: The Y-­‐Value Gets Infinitely Close to 2: x
horizontal asymptote:
y=2
In the example above, the degrees on the numerator
and denominator were the same, and the horizontal
asymptote turned out to be the horizontal line
whose y-value was equal to the value found by
dividing the leading coefficients of the two
polynomials. This is always true: When the degrees
of the numerator and the denominator are the same,
then the horizontal asymptote is found by dividing
the leading terms, so the asymptote is given by:
–100 000
1.9999999...
–10 000
1.9999997...
–1 000
1.9999710...
–100
1.9971026...
–10
1.7339449...
–1
–0.9
0
–1.2222222...
1
–0.9
10
1.7339449...
100
1.9971026...
1 000
1.9999710...
10 000
1.9999997...
100 000
1.9999999...
y = (numerator's leading coefficient) /
(denominator's leading coefficient)