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Lesson: Determining Horizontal and Vertical Asymptotes

Sample Question Videos

Worksheet • 13 Questions • 1 Video

Q1:

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

Q2:

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

Q3:

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 .
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