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(đ‘Ĩ)=3đ‘Ĩ−15đ‘Ĩ+3īŠ¨īŠ¨.

  • AThe function has no vertical asymptote and a horizontal asymptote at đ‘Ļ=35.
  • BThe function has no vertical asymptote and a horizontal asymptote at đ‘Ļ=−13.
  • CThe function has a vertical asymptote at đ‘Ĩ=35 and no horizontal asymptote.
  • DThe function has no vertical asymptote and a horizontal asymptote at đ‘Ļ=53.
  • EThe function has a vertical asymptote at đ‘Ĩ=−13 and no horizontal asymptote.

Q2:

What are the two asymptotes of the hyperbola đ‘Ļ=5đ‘Ĩ+13đ‘Ĩ−4?

  • Ađ‘Ĩ=34,đ‘Ļ=35
  • Bđ‘Ĩ=14,đ‘Ļ=53
  • Cđ‘Ĩ=14,đ‘Ļ=13
  • Dđ‘Ĩ=43,đ‘Ļ=53
  • Eđ‘Ĩ=34,đ‘Ļ=53

Q3:

The graph of equation đ‘Ļ=𝑎đ‘Ĩ+𝑏𝑐đ‘Ĩ+𝑑 is a hyperbola only if 𝑐≠0. In that case, what are the two asymptotes?

  • Ađ‘Ĩ=−𝑎𝑐,đ‘Ļ=𝑑𝑐
  • Bđ‘Ĩ=−𝑑𝑐,đ‘Ļ=𝑎𝑐
  • Cđ‘Ĩ=𝑎𝑐,đ‘Ļ=𝑑𝑐
  • Dđ‘Ĩ=𝑑𝑐,đ‘Ļ=𝑎𝑑
  • Eđ‘Ĩ=−𝑑𝑐,đ‘Ļ=−𝑏𝑎

Q4:

By writing the expression 𝑎đ‘Ĩ+𝑏𝑐đ‘Ĩ+𝑑 in the form 𝐴𝑃đ‘Ĩ+𝑄+𝑅, determine the asymptotes of 5đ‘Ĩ−13đ‘Ĩ−3+2+12đ‘Ĩ1−2đ‘Ĩ.

  • Ađ‘Ĩ=12, đ‘Ĩ=3, đ‘Ļ=−1
  • Bđ‘Ĩ=3, đ‘Ĩ=−12, đ‘Ļ=−1
  • Cđ‘Ĩ=14, đ‘Ĩ=5, đ‘Ļ=−2
  • Dđ‘Ĩ=13, đ‘Ĩ=2, đ‘Ļ=2
  • Eđ‘Ĩ=2, đ‘Ĩ=−3, đ‘Ļ=1

Q5:

On the left is the graph of 𝑓(đ‘Ĩ)=2đ‘Ĩ+13đ‘Ĩ+4 and on the right is the graph of đ‘Ļ=1đ‘Ĩ.

What are the coordinates of the intersection of the asymptotes of đ‘Ļ=𝑓(đ‘Ĩ)?

  • Aī€ŧ0,23īˆ
  • Bī€ŧ−43,23īˆ
  • Cī€ŧ−43,14īˆ
  • Dī€ŧ43,23īˆ
  • Eī€ŧ−43,0īˆ

Find 𝑝, 𝑞, and 𝑘 so that with 𝑔(đ‘Ĩ)=𝑘đ‘Ĩ, we have 𝑓(đ‘Ĩ)=𝑔(đ‘Ĩ+𝑝)+𝑞.

  • A𝑝=−43, 𝑞=23, 𝑘=−359
  • B𝑝=43, 𝑞=−23, 𝑘=−19
  • C𝑝=43, 𝑞=23, 𝑘=−59
  • D𝑝=43, 𝑞=23, 𝑘=59
  • E𝑝=−43, 𝑞=−23, 𝑘=−79

Q6:

Which of the following lines is a vertical asymptote of the graph of the function 𝑓(đ‘Ĩ)=đ‘Ĩ−8đ‘Ĩ+2đ‘Ĩ−15īŠŠīŠ¨?

  • Ađ‘Ĩ=3
  • Bđ‘Ĩ=8
  • Cđ‘Ĩ=2
  • Dđ‘Ĩ=5

Q7:

Consider the function 𝑓(đ‘Ĩ)=4đ‘Ĩ+72đ‘Ĩ−5.

What are the vertical and horizontal asymptotes of the graph đ‘Ļ=𝑓(đ‘Ĩ)?

  • Ađ‘Ĩ=52,đ‘Ļ=−74
  • Bđ‘Ĩ=25,đ‘Ļ=1
  • Cđ‘Ĩ=52,đ‘Ļ=2
  • Dđ‘Ĩ=2, đ‘Ļ=52
  • Eđ‘Ĩ=52,đ‘Ļ=−75

Write 𝑓ī€ŧđ‘Ĩ+52īˆ in a simplified form. What are the vertical and horizontal asymptotes of the graph đ‘Ļ=𝑓ī€ŧđ‘Ĩ+52īˆ?

  • A8đ‘Ĩ+194đ‘Ĩ−5, đ‘Ĩ=54, đ‘Ļ=−198
  • B8đ‘Ĩ+194đ‘Ĩ−5, đ‘Ĩ=54, đ‘Ļ=2
  • C4đ‘Ĩ+172đ‘Ĩ, đ‘Ĩ=0, đ‘Ļ=2
  • D4đ‘Ĩ+172đ‘Ĩ, đ‘Ĩ=0, đ‘Ļ=−174
  • E4đ‘Ĩ+172đ‘Ĩ, đ‘Ĩ=2, đ‘Ļ=2

Write 𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2 in a simplified form. What are the vertical and horizontal asymptotes of the graph đ‘Ļ=𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2?

  • A172đ‘Ĩ, đ‘Ĩ=0, đ‘Ļ=172
  • B172đ‘Ĩ, đ‘Ĩ=1, đ‘Ļ=172
  • C172đ‘Ĩ, đ‘Ĩ=0, đ‘Ļ=0
  • D294đ‘Ĩ−5, đ‘Ĩ=54, đ‘Ļ=0
  • E294đ‘Ĩ−5, đ‘Ĩ=54, đ‘Ļ=294

What combination of horizontal and vertical shifts moves the intersection of the asymptotes of the graph đ‘Ļ=𝑓(đ‘Ĩ) to the origin (0,0)?

  • Aa shift of 52 to the left and a shift of 2 downward
  • Ba shift of 25 to the right and a shift of 2 upward
  • Ca shift of 25 to the left and a shift of 2 downward
  • Da shift of 13 to the left and a shift of 1 downward
  • Ea shift of 52 to the right and a shift of 2 upward

What is the dilation factor 𝐴 required to map the graph of đ‘Ļ=𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2 onto the hyperbola đ‘Ļ=1đ‘Ĩ? Write this in the form 𝐴ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ.

  • Aa dilation by a factor of 217 so 217ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ
  • Ba dilation by a factor of 35 so 35ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ
  • Ca dilation by a factor of 17 so 17ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ
  • Da dilation by a factor of 49 so 49ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ
  • Ea dilation by a factor of 917 so 917ī€ŧ𝑓ī€ŧđ‘Ĩ+52īˆâˆ’2īˆ=1đ‘Ĩ

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(đ‘Ĩ)=𝑎đ‘Ĩ+𝑏𝑐đ‘Ĩ+𝑑 produces the graph of đ‘Ļ=1đ‘Ĩ. What is 𝑔?

  • A𝑔(đ‘Ĩ)=đ‘Ĩ+4đ‘Ĩ+1
  • B𝑔(đ‘Ĩ)=−6đ‘Ĩ−52đ‘Ĩ+2
  • C𝑔(đ‘Ĩ)=6đ‘Ĩ+53đ‘Ĩ+2
  • D𝑔(đ‘Ĩ)=đ‘Ĩ−1đ‘Ĩ+2
  • E𝑔(đ‘Ĩ)=−đ‘Ĩ+3đ‘Ĩ+1

What sequence of transformations maps the graph of 𝑔(đ‘Ĩ)=5đ‘Ĩ−32đ‘Ĩ+1 onto the hyperbola đ‘Ļ=1đ‘Ĩ?

  • Aa shift of 12 to the right, a shift of 25 downward, and then a dilation by a factor of −47
  • Ba shift of 14 to the right, a shift of 52 downward, and then a dilation by a factor of −17
  • Ca shift of 12 to the right, a shift of 52 downward, and then a dilation by a factor of −411
  • Da shift of 13 to the right, a shift of 12 downward, and then a dilation by a factor of −17
  • Ea shift of 14 to the right, a shift of 25 downward, and then a dilation by a factor of −47

Q8:

By sketching a graph, find the vertical asymptotes of the function 𝑓(đ‘Ĩ)=2đ‘Ĩ+6đ‘Ĩ−2đ‘Ĩ−3īŠ¨īŠ¨.

  • Ađ‘Ĩ=2, đ‘Ĩ=3
  • Bđ‘Ĩ=0, đ‘Ĩ=−3
  • Cđ‘Ĩ=1, đ‘Ĩ=−3
  • Dđ‘Ĩ=−1, đ‘Ĩ=3

Q9:

What are the two asymptotes of the hyperbola đ‘Ļ=84đ‘Ĩ−3+53?

  • Ađ‘Ĩ=34,đ‘Ļ=53
  • Bđ‘Ĩ=34,đ‘Ļ=35
  • Cđ‘Ĩ=14,đ‘Ļ=53
  • Dđ‘Ĩ=43,đ‘Ļ=53
  • Eđ‘Ĩ=14,đ‘Ļ=13

Q10:

Determine the vertical and horizontal asymptotes of the function 𝑓(đ‘Ĩ)=−1+3đ‘Ĩ−4đ‘ĨīŠ¨.

  • AThe vertical asymptote is at đ‘Ĩ=3, and the horizontal asymptote is at đ‘Ļ=−4.
  • BThe vertical asymptote is at đ‘Ĩ=0, and the horizontal asymptote is at đ‘Ļ=−1.
  • CThe vertical asymptote is at đ‘Ļ=−1, and the horizontal asymptote is at đ‘Ĩ=0.
  • DThe function has no vertical asymptote, and the horizontal asymptote is at đ‘Ļ=0.
  • EThe vertical asymptote is at đ‘Ĩ=−1, and the function has no horizontal asymptote.

Q11:

Find the vertical and horizontal asymptotes of the function 𝑓(đ‘Ĩ)=4(−đ‘Ĩ+5)lnln.

  • AThe function has vertical asymptotes at đ‘Ĩ=−15 and đ‘Ĩ=1𝑒īŠĢ and a horizontal asymptote at đ‘Ļ=−5.
  • BThe function has vertical asymptotes at đ‘Ĩ=0 and đ‘Ĩ=𝑒īŠĢ and no horizontal asymptotes.
  • CThe function has vertical asymptotes at đ‘Ĩ=5 and đ‘Ĩ=𝑒īŠĢ and a horizontal asymptote at đ‘Ļ=−5.
  • DThe function has vertical asymptotes at đ‘Ĩ=0 and đ‘Ĩ=1𝑒īŠĢ and no horizontal asymptotes.
  • EThe function has vertical asymptotes at đ‘Ĩ=0 and đ‘Ĩ=𝑒īŠĢ and a horizontal asymptote at đ‘Ļ=−5.

Q12:

Find the vertical and horizontal asymptotes of the function 𝑓(đ‘Ĩ)=−2đ‘Ĩ3+2đ‘Ĩ−15đ‘ĨīŠ¨ln.

  • AThe function has a vertical asymptote at đ‘Ĩ=3 and a horizontal asymptote at đ‘Ļ=0.
  • BThe function has a vertical asymptote at đ‘Ļ=3 and a horizontal asymptote at đ‘Ĩ=0.
  • CThe function has a vertical asymptote at đ‘Ļ=0 and no horizontal asymptote.
  • DThe function has a vertical asymptote at đ‘Ĩ=0 and no horizontal asymptote.
  • EThe function has a vertical asymptote at đ‘Ļ=0 and a horizontal asymptote at đ‘Ĩ=3.

Q13:

Find the vertical and horizontal asymptotes of the function 𝑓(đ‘Ĩ)=3𝑒īŠąīŠ¨ī—īŽĄ.

  • AThe horizontal asymptote is đ‘Ļ=0, and there are no vertical asymptotes.
  • BThere are no horizontal asymptotes, and the vertical asymptote is đ‘Ĩ=0.
  • CHorizontal asymptote at đ‘Ļ=3, and there are no vertical asymptotes.
  • DThe horizontal asymptote is đ‘Ļ=−2, and there are no vertical asymptotes.
  • EThere are no horizontal asymptotes, and the vertical asymptote is đ‘Ĩ=3.

Q14:

On the left is the graph of 𝑓(đ‘Ĩ)=7(1−2)īŠąī—, which has a horizontal asymptote. On the right is the graph of 𝑔(đ‘Ĩ)=1𝑓(đ‘Ĩ).

What is the value of 𝐴?

  • A𝐴=7
  • B𝐴=−7
  • C𝐴=7
  • D𝐴=2
  • E𝐴=−2

List all the asymptotes of 𝑔(đ‘Ĩ)=1𝑓(đ‘Ĩ).

  • Ađ‘Ĩ=0, đ‘Ļ=0, đ‘Ļ=−12
  • Bđ‘Ĩ=0, đ‘Ļ=0, đ‘Ļ=17
  • Cđ‘Ĩ=0, đ‘Ļ=0, đ‘Ļ=12
  • Dđ‘Ĩ=17, đ‘Ļ=0
  • Eđ‘Ĩ=0, đ‘Ļ=0, đ‘Ļ=−17

Q15:

Find 𝐴 and đĩ so that the function 𝑓(đ‘Ĩ)=𝐴4(1−3)+đĩīŠąī— has the single vertical asymptote đ‘Ĩ=0 and the two horizontal asymptotes đ‘Ļ=−1 and đ‘Ļ=1. What is the range of this function?

  • A𝐴=8, đĩ=−1, and the range is all values of đ‘Ļ satisfying −∞<đ‘Ļ<−1 together with 1<đ‘Ļ<∞.
  • B𝐴=8, đĩ=−1, and the range is all values of đ‘Ļ satisfying −∞<đ‘Ļ<∞.
  • C𝐴=4, đĩ=−12, and the range is all values of đ‘Ļ satisfying −∞<đ‘Ļ<−1 together with 1<đ‘Ļ<∞.
  • D𝐴=16, đĩ=−4, and the range is all values of đ‘Ļ satisfying −∞<đ‘Ļ<−1 together with 1<đ‘Ļ<∞.
  • E𝐴=16, đĩ=−4, and the range is all values of đ‘Ļ satisfying −∞<đ‘Ļ<∞.

Q16:

Determine whether the following statement is true: The only polynomials whose graphs have a horizontal asymptote are the constant polynomials, those of degree 0.

  • Afalse
  • Btrue

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