2.11. Oblique Asymptotes http://www.ck12.org
Ahorizontal asymptoteis a flat dotted line that indicates where a function goes asxget infinitely large or infinitely
small.
End behavioris a term that asks you to describe the horizontal asymptotes.
Avertical asymptoteis a dashed vertical line that indicates that as a function approaches, it shoots off to positive or
negative infinity without ever actually touching the line.
Arational functionis a function with at least one rational expression.
Arational expressionis a ratio of two polynomial expressions.
Guided Practice
- Find the asymptotes and intercepts of the function:
f(x) =x 2 x−^34 - Create a function with an oblique asymptote aty= 3 x−1, vertical asymptotes atx= 2 ,−4 and includes a hole
wherexis 7. - Identify the backbone of the following function and explain why the function does not have an oblique asymptote.
f(x) =^5 x^5 x+ 327
Answers: - The function has vertical asymptotes atx=±2.
After long division, the function becomes:
f(x) =x+x (^24) − 4
This makes the oblique asymptote aty=x
- While there are an infinite number of functions that match these criteria, one example is:
f(x) = 3 x− 1 +(x− 2 )((xx+−^74 ))(x− 7 )- While polynomial long division is possible, it is also possible to just divide each term byx^3.
f(x) =^5 x^5 x+ 327 =^5 xx 35 +x^273 = 5 x^2 +^27 x 3
The backbone of this function is the parabolay= 5 x^2. This is not an oblique asymptote because it is not a line.
Practice
- What is an oblique asymptote?
- How can you tell by looking at the equation of a function if it will have an oblique asymptote or not?
- Can a function have both an oblique asymptote and a horizontal asymptote? Explain.
For each of the following graphs, sketch the graph and then sketch in the oblique asymptote if it exists. If it doesn’t
exist, explain why not.