Worksheet: Oblique Asymptotes

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

Q1:

The figure shows the graph of 𝑓 ( π‘₯ ) = 6 π‘₯ βˆ’ 3 π‘₯ + 1 0 π‘₯ βˆ’ 2 π‘₯ + 1 3 π‘₯ + 4 π‘₯ βˆ’ 1 οŠͺ    and an oblique asymptote 𝑦 = π‘š π‘₯ + 𝑐 .

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

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

Q2:

The figure shows the graph of 𝑓 ( π‘₯ ) = 2 βˆ’ 6 π‘₯ + 4 π‘₯ + 6 π‘₯ βˆ’ 5 π‘₯ 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‘š π‘₯ + 𝑐 .

  • A 𝑦 = βˆ’ 5 π‘₯ + 1 1
  • B 𝑦 = βˆ’ 3 π‘₯ + 1
  • C 𝑦 = βˆ’ 5 π‘₯ + 1
  • D 𝑦 = βˆ’ 3 π‘₯ + 2
  • E 𝑦 = βˆ’ 5 π‘₯ + 2

Q3:

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 = 4 π‘₯ βˆ’ 4 .

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

  • A 𝑃 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯ βˆ’ 4  
  • B 𝑃 ( π‘₯ ) = 4 π‘₯ βˆ’ 8 π‘₯ βˆ’ 4 
  • C 𝑃 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯ + 4 
  • D 𝑃 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯  
  • E 𝑃 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯ βˆ’ 4 

Q4:

We consider how the line of an oblique asymptote depends on the numerator of the rational function. Consider 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 π‘₯ + 𝑑 π‘₯ + π‘₯ + 1    .

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

  • A ( π‘Ž βˆ’ 2 ) π‘₯ + ( 𝑏 + 1 ) π‘₯ + ( 𝑐 + 1 ) π‘₯ + 𝑑 + 3  
  • B ( π‘Ž βˆ’ 2 ) π‘₯ + ( 𝑏 βˆ’ 1 ) π‘₯ + ( 𝑐 βˆ’ 1 ) π‘₯ + 𝑑 βˆ’ 3  
  • C ( π‘Ž + 2 ) π‘₯ + ( 𝑏 βˆ’ 1 ) π‘₯ + ( 𝑐 βˆ’ 1 ) π‘₯ + 𝑑 βˆ’ 3  
  • D ( π‘Ž + 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 = 2 π‘₯ βˆ’ 3 is an oblique asymptote to the graph 𝑦 = 𝑓 ( π‘₯ ) .

  • A π‘Ž = 3 , 𝑏 = βˆ’ 1 , 𝑐 = 2 , 𝑑 = 0
  • B π‘Ž = 1 , 𝑏 = βˆ’ 1 , 𝑐 = 0 , 𝑑 = 0
  • C π‘Ž = 2 , 𝑏 = βˆ’ 1 , and 𝑐 and 𝑑 can take any values.
  • D π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 3 , 𝑑 = βˆ’ 2
  • E π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 0 , 𝑑 = 0

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

  • A π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 1 , 𝑑 = βˆ’ 4
  • B π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = βˆ’ 4 , 𝑑 = βˆ’ 1
  • C π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 1 , 𝑑 = 4
  • D π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = βˆ’ 1 , 𝑑 = βˆ’ 4
  • E π‘Ž = 2 , 𝑏 = βˆ’ 1 , 𝑐 = 4 , 𝑑 = 1

Q5:

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

  • A 𝑦 = π‘₯
  • B 𝑦 = βˆ’ π‘₯ 2
  • C 𝑦 = π‘₯ 2
  • D 𝑦 = π‘₯ 4
  • E 𝑦 = 2 π‘₯

Q6:

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

  • A 𝑦 = π‘₯ βˆ’ 1
  • B 𝑦 = 1 2 π‘₯ βˆ’ 2
  • C 𝑦 = 1 2 π‘₯ + 1
  • D 𝑦 = 1 2 π‘₯ βˆ’ 1
  • E 𝑦 = 1 2 π‘₯

Q7:

Determine the oblique asymptote to the curve 𝑦 = 6 π‘₯ βˆ’ 1 3 π‘₯ + 5 π‘₯ βˆ’ 1 2 π‘₯ βˆ’ 3 π‘₯ βˆ’ 2    .

  • A 𝑦 = 5 π‘₯ βˆ’ 5
  • B 𝑦 = 3 π‘₯
  • C 𝑦 = 6 π‘₯ βˆ’ 4
  • D 𝑦 = 3 π‘₯ + 2
  • E 𝑦 = 3 π‘₯ βˆ’ 2

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