Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Lesson: Asymptotes of Rational Functions

Sample Question Videos

Worksheet • 13 Questions • 1 Video

Q1:

Find the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 1 5 π‘₯ + 3 2 2 .

  • A The function has no vertical asymptote and a horizontal asymptote at 𝑦 = 3 5 .
  • B The function has no vertical asymptote and a horizontal asymptote at 𝑦 = 5 3 .
  • C The function has a vertical asymptote at π‘₯ = 3 5 and no horizontal asymptote.
  • D The function has a vertical asymptote at π‘₯ = βˆ’ 1 3 and no horizontal asymptote.
  • E The function has no vertical asymptote and a horizontal asymptote at 𝑦 = βˆ’ 1 3 .

Q2:

What are the two asymptotes of the hyperbola 𝑦 = 5 π‘₯ + 1 3 π‘₯ βˆ’ 4 ?

  • A π‘₯ = 4 3 , 𝑦 = 5 3
  • B π‘₯ = 1 4 , 𝑦 = 1 3
  • C π‘₯ = 3 4 , 𝑦 = 5 3
  • D π‘₯ = 1 4 , 𝑦 = 5 3
  • E π‘₯ = 3 4 , 𝑦 = 3 5

Q3:

The graph of equation 𝑦 = π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 is a hyperbola only if 𝑐 β‰  0 . In that case, what are the two asymptotes?

  • A π‘₯ = βˆ’ 𝑑 𝑐 , 𝑦 = π‘Ž 𝑐
  • B π‘₯ = π‘Ž 𝑐 , 𝑦 = 𝑑 𝑐
  • C π‘₯ = 𝑑 𝑐 , 𝑦 = π‘Ž 𝑑
  • D π‘₯ = βˆ’ 𝑑 𝑐 , 𝑦 = βˆ’ 𝑏 π‘Ž
  • E π‘₯ = βˆ’ π‘Ž 𝑐 , 𝑦 = 𝑑 𝑐

Q4:

By writing the expression π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 in the form 𝐴 𝑃 π‘₯ + 𝑄 + 𝑅 , determine the asymptotes of 5 π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 3 + 2 + 1 2 π‘₯ 1 βˆ’ 2 π‘₯ .

  • A π‘₯ = 1 2 , π‘₯ = 3 , 𝑦 = βˆ’ 1
  • B π‘₯ = 3 , π‘₯ = βˆ’ 1 2 , 𝑦 = βˆ’ 1
  • C π‘₯ = 1 3 , π‘₯ = 2 , 𝑦 = 2
  • D π‘₯ = 2 , π‘₯ = βˆ’ 3 , 𝑦 = 1
  • E π‘₯ = 1 4 , π‘₯ = 5 , 𝑦 = βˆ’ 2

Q5:

Find the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = 3 𝑒 βˆ’ 2 π‘₯ 2 .

  • A The horizontal asymptote is 𝑦 = 0 , and there are no vertical asymptotes.
  • B The horizontal asymptote is 𝑦 = βˆ’ 2 , and there are no vertical asymptotes.
  • C There are no horizontal asymptotes, and the vertical asymptote is π‘₯ = 0 .
  • D There are no horizontal asymptotes, and the vertical asymptote is π‘₯ = 3 .
  • E Horizontal asymptote at 𝑦 = 3 , and there are no vertical asymptotes.

Q6:

Find the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ 3 + 2 π‘₯ βˆ’ 1 5 π‘₯ 2 l n .

  • AThe function has a vertical asymptote at π‘₯ = 0 and no horizontal asymptote.
  • BThe function has a vertical asymptote at 𝑦 = 3 and a horizontal asymptote at π‘₯ = 0 .
  • CThe function has a vertical asymptote at 𝑦 = 0 and no horizontal asymptote.
  • DThe function has a vertical asymptote at 𝑦 = 0 and a horizontal asymptote at π‘₯ = 3 .
  • EThe function has a vertical asymptote at π‘₯ = 3 and a horizontal asymptote at 𝑦 = 0 .

Q7:

Find the vertical and horizontal asymptotes of the function 𝑓 ( π‘₯ ) = 4 ( βˆ’ π‘₯ + 5 ) l n l n .

  • AThe function has vertical asymptotes at π‘₯ = 0 and π‘₯ = 𝑒 5 and no horizontal asymptotes.
  • BThe function has vertical asymptotes at π‘₯ = 0 and π‘₯ = 𝑒 5 and a horizontal asymptote at 𝑦 = βˆ’ 5 .
  • CThe function has vertical asymptotes at π‘₯ = 5 and π‘₯ = 𝑒 5 and a horizontal asymptote at 𝑦 = βˆ’ 5 .
  • DThe function has vertical asymptotes at π‘₯ = 0 and π‘₯ = 1 𝑒 5 and no horizontal asymptotes.
  • EThe function has vertical asymptotes at π‘₯ = βˆ’ 1 5 and π‘₯ = 1 𝑒 5 and a horizontal asymptote at 𝑦 = βˆ’ 5 .

Q8:

On the left is the graph of 𝑓 ( π‘₯ ) = 2 π‘₯ + 1 3 π‘₯ + 4 and on the right is the graph of 𝑦 = 1 π‘₯ .

What are the coordinates of the intersection of the asymptotes of 𝑦 = 𝑓 ( π‘₯ ) ?

  • A ο€Ό βˆ’ 4 3 , 2 3 
  • B ο€Ό 4 3 , 2 3 
  • C ο€Ό βˆ’ 4 3 , 1 4 
  • D ο€Ό 0 , 2 3 
  • E ο€Ό βˆ’ 4 3 , 0 

Find 𝑝 , π‘ž , and π‘˜ so that with 𝑔 ( π‘₯ ) = π‘˜ π‘₯ , we have 𝑓 ( π‘₯ ) = 𝑔 ( π‘₯ + 𝑝 ) + π‘ž .

  • A 𝑝 = 4 3 , π‘ž = 2 3 , π‘˜ = βˆ’ 5 9
  • B 𝑝 = 4 3 , π‘ž = 2 3 , π‘˜ = 5 9
  • C 𝑝 = βˆ’ 4 3 , π‘ž = 2 3 , π‘˜ = βˆ’ 3 5 9
  • D 𝑝 = βˆ’ 4 3 , π‘ž = βˆ’ 2 3 , π‘˜ = βˆ’ 7 9
  • E 𝑝 = 4 3 , π‘ž = βˆ’ 2 3 , π‘˜ = βˆ’ 1 9

Q9:

Which of the following lines is a vertical asymptote of the graph of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 π‘₯ + 2 π‘₯ βˆ’ 1 5 3 2 ?

  • A π‘₯ = 3
  • B π‘₯ = 2
  • C π‘₯ = 8
  • D π‘₯ = 5

Q10:

The function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 3 3 π‘₯ + 8 0 ( π‘₯ βˆ’ 3 ) ( π‘₯ βˆ’ 5 ) ( π‘₯ βˆ’ 7 ) 2 is found to have a vertical asymptote at π‘₯ = 3 . To which of { βˆ’ ∞ , + ∞ } do the values of 𝑓 ( π‘₯ ) go as π‘₯ approaches 3 from the left and the right?

  • A βˆ’ ∞ , + ∞
  • B βˆ’ ∞ , βˆ’ ∞
  • C + ∞ , βˆ’ ∞
  • D + ∞ , + ∞

Q11:

On the left is the graph of 𝑓 ( π‘₯ ) = 7 ( 1 βˆ’ 2 ) βˆ’ π‘₯ , which has a horizontal asymptote. On the right is the graph of 𝑔 ( π‘₯ ) = 1 𝑓 ( π‘₯ ) .

What is the value of 𝐴 ?

  • A 𝐴 = 7
  • B 𝐴 = 7
  • C 𝐴 = 2
  • D 𝐴 = βˆ’ 2
  • E 𝐴 = βˆ’ 7

List all the asymptotes of 𝑔 ( π‘₯ ) = 1 𝑓 ( π‘₯ ) .

  • A π‘₯ = 0 , 𝑦 = 0 , 𝑦 = 1 7
  • B π‘₯ = 1 7 , 𝑦 = 0
  • C π‘₯ = 0 , 𝑦 = 0 , 𝑦 = 1 2
  • D π‘₯ = 0 , 𝑦 = 0 , 𝑦 = βˆ’ 1 2
  • E π‘₯ = 0 , 𝑦 = 0 , 𝑦 = βˆ’ 1 7

Q12:

Consider the function 𝑓 ( π‘₯ ) = 4 π‘₯ + 7 2 π‘₯ βˆ’ 5 .

What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 ( π‘₯ ) ?

  • A π‘₯ = 5 2 , 𝑦 = 2
  • B π‘₯ = 5 2 , 𝑦 = βˆ’ 7 4
  • C π‘₯ = 2 , 𝑦 = 5 2
  • D π‘₯ = 2 5 , 𝑦 = 1
  • E π‘₯ = 5 2 , 𝑦 = βˆ’ 7 5

Write 𝑓 ο€Ό π‘₯ + 5 2  in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 ο€Ό π‘₯ + 5 2  ?

  • A 4 π‘₯ + 1 7 2 π‘₯ , π‘₯ = 2 , 𝑦 = 2
  • B 8 π‘₯ + 1 9 4 π‘₯ βˆ’ 5 , π‘₯ = 5 4 , 𝑦 = βˆ’ 1 9 8
  • C 4 π‘₯ + 1 7 2 π‘₯ , π‘₯ = 0 , 𝑦 = 2
  • D 8 π‘₯ + 1 9 4 π‘₯ βˆ’ 5 , π‘₯ = 5 4 , 𝑦 = 2
  • E 4 π‘₯ + 1 7 2 π‘₯ , π‘₯ = 0 , 𝑦 = βˆ’ 1 7 4

Write 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2 in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2 ?

  • A 1 7 2 π‘₯ , π‘₯ = 0 , 𝑦 = 0
  • B 2 9 4 π‘₯ βˆ’ 5 , π‘₯ = 5 4 , 𝑦 = 2 9 4
  • C 1 7 2 π‘₯ , π‘₯ = 0 , 𝑦 = 1 7 2
  • D 2 9 4 π‘₯ βˆ’ 5 , π‘₯ = 5 4 , 𝑦 = 0
  • E 1 7 2 π‘₯ , π‘₯ = 1 , 𝑦 = 1 7 2

What combination of horizontal and vertical shifts moves the intersection of the asymptotes of the graph 𝑦 = 𝑓 ( π‘₯ ) to the origin ( 0 , 0 ) ?

  • Aa shift of 5 2 to the left and a shift of 2 downward
  • Ba shift of 1 3 to the left and a shift of 1 downward
  • Ca shift of 2 5 to the left and a shift of 1 downward
  • Da shift of 5 3 to the left and a shift of 4 downward
  • Ea shift of 1 2 to the left and a shift of 3 downward

What is the dilation factor A required to map the graph of 𝑦 = 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2 onto the hyperbola 𝑦 = 1 π‘₯ ? Write this in the form 𝐴 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯ .

  • Aa dilation by a factor of 2 1 7 so 2 1 7 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯
  • Ba dilation by a factor of 9 1 7 so 9 1 7 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯
  • Ca dilation by a factor of 1 7 so 1 7 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯
  • Da dilation by a factor of 4 9 so 4 9 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯
  • Ea dilation by a factor of 3 5 so 3 5 ο€Ό 𝑓 ο€Ό π‘₯ + 5 2  βˆ’ 2  = 1 π‘₯

Applying a shift of 1 to the right, a shift of 3 upward, and then a dilation by a factor of 2 to the graph of 𝑔 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 produces the graph of 𝑦 = 1 π‘₯ . What is g?

  • A 𝑔 ( π‘₯ ) = βˆ’ 6 π‘₯ βˆ’ 5 2 π‘₯ + 2
  • B 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 1 π‘₯ + 2
  • C 𝑔 ( π‘₯ ) = βˆ’ π‘₯ + 3 π‘₯ + 1
  • D 𝑔 ( π‘₯ ) = π‘₯ + 4 π‘₯ + 1
  • E 𝑔 ( π‘₯ ) = 6 π‘₯ + 5 3 π‘₯ + 2

What sequence of transformations maps the graph of 𝑔 ( π‘₯ ) = 5 π‘₯ βˆ’ 3 2 π‘₯ + 1 onto the hyperbola 𝑦 = 1 π‘₯ ?

  • Aa shift of 1 2 to the right, a shift of 5 2 downward, and then a dilation by a factor of βˆ’ 4 1 1
  • Ba shift of 1 4 to the right, a shift of 5 2 downward, and then a dilation by a factor of βˆ’ 1 7
  • Ca shift of 1 4 to the right, a shift of 2 5 downward, and then a dilation by a factor of βˆ’ 4 7
  • Da shift of 1 2 to the right, a shift of 2 5 downward, and then a dilation by a factor of βˆ’ 4 7
  • Ea shift of 1 3 to the right, a shift of 1 2 downward, and then a dilation by a factor of βˆ’ 1 7

Q13:

By sketching a graph, find the vertical asymptotes of the function 𝑓 ( π‘₯ ) = 2 π‘₯ + 6 π‘₯ βˆ’ 2 π‘₯ βˆ’ 3 2 2 .

  • A π‘₯ = βˆ’ 1 , π‘₯ = 3
  • B π‘₯ = 1 , π‘₯ = βˆ’ 3
  • C π‘₯ = 2 , π‘₯ = 3
  • D π‘₯ = 0 , π‘₯ = βˆ’ 3
Preview