Worksheet: Oblique Asymptotes

In this worksheet, we will practice finding the equations of the oblique asymptotes of functions, especially rational functions.


The figure shows the graph of 𝑓(π‘₯)=6π‘₯βˆ’3π‘₯+10π‘₯βˆ’2π‘₯+13π‘₯+4π‘₯βˆ’1οŠͺ and an oblique asymptote 𝑦=π‘šπ‘₯+𝑐.

By synthetic division, determine the value of π‘š.

By considering the behavior of 𝑓(π‘₯)βˆ’2π‘₯ as π‘₯ goes to ±∞, determine the value of 𝑐.


The figure shows the graph of 𝑓(π‘₯)=2βˆ’6π‘₯+4π‘₯+6π‘₯βˆ’5π‘₯π‘₯βˆ’π‘₯βˆ’π‘₯+1οŠͺ together with its asymptotes π‘₯=1 and π‘₯=βˆ’1 and an oblique line 𝐿.

Determine the equation of 𝐿 giving your answer in the form 𝑦=π‘šπ‘₯+𝑐.

  • A𝑦=βˆ’5π‘₯+2
  • B𝑦=βˆ’3π‘₯+2
  • C𝑦=βˆ’5π‘₯+11
  • D𝑦=βˆ’3π‘₯+1
  • E𝑦=βˆ’5π‘₯+1


The figure shows the graph of the function 𝑓(π‘₯)=𝑃(π‘₯)(π‘₯βˆ’1)(π‘₯+2) with vertical asymptotes at π‘₯=1 and π‘₯=βˆ’2 and oblique asymptote 𝑦=4π‘₯βˆ’4.

Determine the polynomial 𝑃 given the points shown on the graph.

  • A𝑃(π‘₯)=4π‘₯+8π‘₯βˆ’4
  • B𝑃(π‘₯)=4π‘₯βˆ’8π‘₯βˆ’4
  • C𝑃(π‘₯)=4π‘₯+8π‘₯
  • D𝑃(π‘₯)=4π‘₯+8π‘₯+4
  • E𝑃(π‘₯)=4π‘₯+8π‘₯βˆ’4


We consider how the line of an oblique asymptote depends on the numerator of the rational function. Consider 𝑓(π‘₯)=π‘Žπ‘₯+𝑏π‘₯+𝑐π‘₯+𝑑π‘₯+π‘₯+1.

Simplify and then write the numerator of 𝑓(π‘₯)βˆ’(2π‘₯βˆ’3) as a polynomial in descending powers of π‘₯.

  • A(π‘Žβˆ’2)π‘₯+(π‘βˆ’1)π‘₯+(π‘βˆ’1)π‘₯+π‘‘βˆ’3
  • B(π‘Ž+2)π‘₯+(π‘βˆ’1)π‘₯+(π‘βˆ’1)π‘₯+π‘‘βˆ’3
  • C(π‘Ž+2)π‘₯+(𝑏+1)π‘₯+(𝑐+1)π‘₯+𝑑+3
  • D(π‘Žβˆ’2)π‘₯+(𝑏+1)π‘₯+(𝑐+1)π‘₯+𝑑+3

Using your answer above, find the conditions on π‘Ž, 𝑏, 𝑐, and 𝑑 under which the line 𝑦=2π‘₯βˆ’3 is an oblique asymptote to the graph 𝑦=𝑓(π‘₯).

  • Aπ‘Ž=2, 𝑏=βˆ’1, 𝑐=0, 𝑑=0
  • Bπ‘Ž=1, 𝑏=βˆ’1, 𝑐=0, 𝑑=0
  • Cπ‘Ž=2, 𝑏=βˆ’1, 𝑐=3, 𝑑=βˆ’2
  • Dπ‘Ž=2, 𝑏=βˆ’1, and 𝑐 and 𝑑 can take any values.
  • Eπ‘Ž=3, 𝑏=βˆ’1, 𝑐=2, 𝑑=0

Find π‘Ž, 𝑏, 𝑐, and 𝑑 so that 𝑦=2π‘₯βˆ’3 is an asymptote of 𝑦=𝑓(π‘₯), and that 𝑓(1)=2 and 𝑓(βˆ’1)=0.

  • Aπ‘Ž=2, 𝑏=βˆ’1, 𝑐=1, 𝑑=4
  • Bπ‘Ž=2, 𝑏=βˆ’1, 𝑐=1, 𝑑=βˆ’4
  • Cπ‘Ž=2, 𝑏=βˆ’1, 𝑐=βˆ’1, 𝑑=βˆ’4
  • Dπ‘Ž=2, 𝑏=βˆ’1, 𝑐=4, 𝑑=1
  • Eπ‘Ž=2, 𝑏=βˆ’1, 𝑐=βˆ’4, 𝑑=βˆ’1


The graph of 𝑦=π‘₯2+1π‘₯βˆ’1 is asymptotic to a line as π‘₯β†’Β±βˆž. What is this line?

  • A𝑦=π‘₯4
  • B𝑦=π‘₯2
  • C𝑦=2π‘₯
  • D𝑦=π‘₯
  • E𝑦=βˆ’π‘₯2


Use partial fractions to determine the line 𝑦=π‘šπ‘₯+𝑐 that is asymptotic to the curve 𝑦=π‘₯βˆ’2π‘₯+π‘₯2π‘₯+2.

  • A𝑦=12π‘₯βˆ’2
  • B𝑦=12π‘₯+1
  • C𝑦=12π‘₯
  • D𝑦=12π‘₯βˆ’1
  • E𝑦=π‘₯βˆ’1


Determine the oblique asymptote to the curve 𝑦=6π‘₯βˆ’13π‘₯+5π‘₯βˆ’12π‘₯βˆ’3π‘₯βˆ’2.

  • A𝑦=3π‘₯
  • B𝑦=6π‘₯βˆ’4
  • C𝑦=3π‘₯βˆ’2
  • D𝑦=5π‘₯βˆ’5
  • E𝑦=3π‘₯+2


Consider the function 𝑓(π‘₯)=4π‘₯+5π‘₯βˆ’102π‘₯+5.

Find the equation of the oblique asymptote of the graph of 𝑓 to decide which of the following graphs is the graph of 𝑓.

  • A
  • B
  • C
  • D

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