Worksheet: Horizontal and Vertical Asymptotes of a Function

In this worksheet, we will practice finding the horizontal and vertical asymptotes of a function.

Q1:

Find the vertical and horizontal asymptotes of the function 𝑓 ( 𝑥 ) = 3 𝑥 1 5 𝑥 + 3 2 2 .

  • A The function has no vertical asymptote and a horizontal asymptote at 𝑦 = 1 3 .
  • B The function has a vertical asymptote at 𝑥 = 3 5 and no horizontal asymptote.
  • C The function has a vertical asymptote at 𝑥 = 1 3 and no horizontal asymptote.
  • D The function has no vertical asymptote and a horizontal asymptote at 𝑦 = 3 5 .
  • E The function has no vertical asymptote and a horizontal asymptote at 𝑦 = 5 3 .

Q2:

What are the two asymptotes of the hyperbola 𝑦 = 5 𝑥 + 1 3 𝑥 4 ?

  • A 𝑥 = 3 4 , 𝑦 = 3 5
  • B 𝑥 = 3 4 , 𝑦 = 5 3
  • C 𝑥 = 1 4 , 𝑦 = 5 3
  • D 𝑥 = 4 3 , 𝑦 = 5 3
  • E 𝑥 = 1 4 , 𝑦 = 1 3

Q3:

The graph of equation 𝑦 = 𝑎 𝑥 + 𝑏 𝑐 𝑥 + 𝑑 is a hyperbola only if 𝑐 0 . In that case, what are the two asymptotes?

  • A 𝑥 = 𝑎 𝑐 , 𝑦 = 𝑑 𝑐
  • B 𝑥 = 𝑑 𝑐 , 𝑦 = 𝑎 𝑑
  • C 𝑥 = 𝑑 𝑐 , 𝑦 = 𝑏 𝑎
  • D 𝑥 = 𝑑 𝑐 , 𝑦 = 𝑎 𝑐
  • E 𝑥 = 𝑎 𝑐 , 𝑦 = 𝑑 𝑐

Q4:

By writing the expression 𝑎 𝑥 + 𝑏 𝑐 𝑥 + 𝑑 in the form 𝐴 𝑃 𝑥 + 𝑄 + 𝑅 , determine the asymptotes of 5 𝑥 1 3 𝑥 3 + 2 + 1 2 𝑥 1 2 𝑥 .

  • A 𝑥 = 1 4 , 𝑥 = 5 , 𝑦 = 2
  • B 𝑥 = 1 3 , 𝑥 = 2 , 𝑦 = 2
  • C 𝑥 = 2 , 𝑥 = 3 , 𝑦 = 1
  • D 𝑥 = 1 2 , 𝑥 = 3 , 𝑦 = 1
  • E 𝑥 = 3 , 𝑥 = 1 2 , 𝑦 = 1

Q5:

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

What are the coordinates of the intersection of the asymptotes of 𝑦 = 𝑓 ( 𝑥 ) ?

  • A 4 3 , 0
  • B 4 3 , 1 4
  • C 0 , 2 3
  • D 4 3 , 2 3
  • E 4 3 , 2 3

Find 𝑝 , 𝑞 , and 𝑘 so that with 𝑔 ( 𝑥 ) = 𝑘 𝑥 , we have 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 + 𝑝 ) + 𝑞 .

  • A 𝑝 = 4 3 , 𝑞 = 2 3 , 𝑘 = 5 9
  • B 𝑝 = 4 3 , 𝑞 = 2 3 , 𝑘 = 7 9
  • C 𝑝 = 4 3 , 𝑞 = 2 3 , 𝑘 = 1 9
  • D 𝑝 = 4 3 , 𝑞 = 2 3 , 𝑘 = 3 5 9
  • E 𝑝 = 4 3 , 𝑞 = 2 3 , 𝑘 = 5 9

Q6:

Which of the following lines is a vertical asymptote of the graph of the function 𝑓 ( 𝑥 ) = 𝑥 8 𝑥 + 2 𝑥 1 5 3 2 ?

  • A 𝑥 = 5
  • B 𝑥 = 2
  • C 𝑥 = 8
  • D 𝑥 = 3

Q7:

Consider the function 𝑓 ( 𝑥 ) = 4 𝑥 + 7 2 𝑥 5 .

What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 ( 𝑥 ) ?

  • A 𝑥 = 5 2 , 𝑦 = 7 5
  • B 𝑥 = 2 , 𝑦 = 5 2
  • C 𝑥 = 2 5 , 𝑦 = 1
  • D 𝑥 = 5 2 , 𝑦 = 2
  • E 𝑥 = 5 2 , 𝑦 = 7 4

Write 𝑓 𝑥 + 5 2 in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 𝑥 + 5 2 ?

  • A 4 𝑥 + 1 7 2 𝑥 , 𝑥 = 2 , 𝑦 = 2
  • B 8 𝑥 + 1 9 4 𝑥 5 , 𝑥 = 5 4 , 𝑦 = 2
  • C 4 𝑥 + 1 7 2 𝑥 , 𝑥 = 0 , 𝑦 = 1 7 4
  • D 4 𝑥 + 1 7 2 𝑥 , 𝑥 = 0 , 𝑦 = 2
  • E 8 𝑥 + 1 9 4 𝑥 5 , 𝑥 = 5 4 , 𝑦 = 1 9 8

Write 𝑓 𝑥 + 5 2 2 in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦 = 𝑓 𝑥 + 5 2 2 ?

  • A 2 9 4 𝑥 5 , 𝑥 = 5 4 , 𝑦 = 2 9 4
  • B 1 7 2 𝑥 , 𝑥 = 0 , 𝑦 = 1 7 2
  • C 1 7 2 𝑥 , 𝑥 = 0 , 𝑦 = 0
  • D 1 7 2 𝑥 , 𝑥 = 1 , 𝑦 = 1 7 2
  • E 2 9 4 𝑥 5 , 𝑥 = 5 4 , 𝑦 = 0

What combination of horizontal and vertical shifts moves the intersection of the asymptotes of the graph 𝑦 = 𝑓 ( 𝑥 ) 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 5 3 to the left and a shift of 4 downward
  • Da shift of 2 5 to the left and a shift of 1 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 𝑥 + 5 2 2 onto the hyperbola 𝑦 = 1 𝑥 ? Write this in the form 𝐴 𝑓 𝑥 + 5 2 2 = 1 𝑥 .

  • Aa dilation by a factor of 4 9 so 4 9 𝑓 𝑥 + 5 2 2 = 1 𝑥
  • Ba dilation by a factor of 3 5 so 3 5 𝑓 𝑥 + 5 2 2 = 1 𝑥
  • Ca dilation by a factor of 1 7 so 1 7 𝑓 𝑥 + 5 2 2 = 1 𝑥
  • Da dilation by a factor of 2 1 7 so 2 1 7 𝑓 𝑥 + 5 2 2 = 1 𝑥
  • Ea dilation by a factor of 9 1 7 so 9 1 7 𝑓 𝑥 + 5 2 2 = 1 𝑥

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 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑐 𝑥 + 𝑑 produces the graph of 𝑦 = 1 𝑥 . What is g?

  • A 𝑔 ( 𝑥 ) = 6 𝑥 + 5 3 𝑥 + 2
  • B 𝑔 ( 𝑥 ) = 𝑥 1 𝑥 + 2
  • C 𝑔 ( 𝑥 ) = 𝑥 + 3 𝑥 + 1
  • D 𝑔 ( 𝑥 ) = 6 𝑥 5 2 𝑥 + 2
  • E 𝑔 ( 𝑥 ) = 𝑥 + 4 𝑥 + 1

What sequence of transformations maps the graph of 𝑔 ( 𝑥 ) = 5 𝑥 3 2 𝑥 + 1 onto the hyperbola 𝑦 = 1 𝑥 ?

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

Q8:

By sketching a graph, find the vertical asymptotes of the function 𝑓 ( 𝑥 ) = 2 𝑥 + 6 𝑥 2 𝑥 3 2 2 .

  • A 𝑥 = 0 , 𝑥 = 3
  • B 𝑥 = 1 , 𝑥 = 3
  • C 𝑥 = 2 , 𝑥 = 3
  • D 𝑥 = 1 , 𝑥 = 3

Q9:

What are the two asymptotes of the hyperbola 𝑦 = 8 4 𝑥 3 + 5 3 ?

  • A 𝑥 = 3 4 , 𝑦 = 3 5
  • B 𝑥 = 4 3 , 𝑦 = 5 3
  • C 𝑥 = 1 4 , 𝑦 = 5 3
  • D 𝑥 = 3 4 , 𝑦 = 5 3
  • E 𝑥 = 1 4 , 𝑦 = 1 3

Q10:

Determine the vertical and horizontal asymptotes of the function 𝑓 ( 𝑥 ) = 1 + 3 𝑥 4 𝑥 2 .

  • A The vertical asymptote is at 𝑥 = 3 , and the horizontal asymptote is at 𝑦 = 4 .
  • B The vertical asymptote is at 𝑦 = 1 , and the horizontal asymptote is at 𝑥 = 0 .
  • C The function has no vertical asymptote, and the horizontal asymptote is at 𝑦 = 0 .
  • D The vertical asymptote is at 𝑥 = 0 , and the horizontal asymptote is at 𝑦 = 1 .
  • E The vertical asymptote is at 𝑥 = 1 , and the function has no horizontal asymptote.

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