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‘₯)=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.

Q2:

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.

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

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=4π‘₯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‘₯  
  • D 𝑃 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯ + 4 
  • 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‘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‘Ž βˆ’ 2 ) π‘₯ + ( 𝑏 βˆ’ 1 ) π‘₯ + ( 𝑐 βˆ’ 1 ) π‘₯ + 𝑑 βˆ’ 3  
  • B ( π‘Ž + 2 ) π‘₯ + ( 𝑏 βˆ’ 1 ) π‘₯ + ( 𝑐 βˆ’ 1 ) π‘₯ + 𝑑 βˆ’ 3  
  • C ( π‘Ž + 2 ) π‘₯ + ( 𝑏 + 1 ) π‘₯ + ( 𝑐 + 1 ) π‘₯ + 𝑑 + 3  
  • D ( π‘Ž βˆ’ 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‘₯).

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

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

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