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(π‘₯)=3π‘₯βˆ’15π‘₯+3.

  • AThe function has no vertical asymptote and a horizontal asymptote at 𝑦=35.
  • BThe function has no vertical asymptote and a horizontal asymptote at 𝑦=βˆ’13.
  • CThe function has a vertical asymptote at π‘₯=35 and no horizontal asymptote.
  • DThe function has no vertical asymptote and a horizontal asymptote at 𝑦=53.
  • EThe function has a vertical asymptote at π‘₯=βˆ’13 and no horizontal asymptote.

Q2:

What are the two asymptotes of the hyperbola 𝑦=5π‘₯+13π‘₯βˆ’4?

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

Q3:

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

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

Q4:

By writing the expression π‘Žπ‘₯+𝑏𝑐π‘₯+𝑑 in the form 𝐴𝑃π‘₯+𝑄+𝑅, determine the asymptotes of 5π‘₯βˆ’13π‘₯βˆ’3+2+12π‘₯1βˆ’2π‘₯.

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

Q5:

On the left is the graph of 𝑓(π‘₯)=2π‘₯+13π‘₯+4 and on the right is the graph of 𝑦=1π‘₯.

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

  • A ο€Ό 0 , 2 3 
  • B ο€Ό βˆ’ 4 3 , 2 3 
  • C ο€Ό βˆ’ 4 3 , 1 4 
  • D ο€Ό 4 3 , 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 = βˆ’ 4 3 , π‘ž = 2 3 , π‘˜ = βˆ’ 3 5 9
  • B 𝑝 = 4 3 , π‘ž = βˆ’ 2 3 , π‘˜ = βˆ’ 1 9
  • C 𝑝 = 4 3 , π‘ž = 2 3 , π‘˜ = βˆ’ 5 9
  • D 𝑝 = 4 3 , π‘ž = 2 3 , π‘˜ = 5 9
  • E 𝑝 = βˆ’ 4 3 , π‘ž = βˆ’ 2 3 , π‘˜ = βˆ’ 7 9

Q6:

Which of the following lines is a vertical asymptote of the graph of the function 𝑓(π‘₯)=π‘₯βˆ’8π‘₯+2π‘₯βˆ’15?

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

Q7:

Consider the function 𝑓(π‘₯)=4π‘₯+72π‘₯βˆ’5.

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

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

Write 𝑓π‘₯+52 in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦=𝑓π‘₯+52?

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

Write 𝑓π‘₯+52οˆβˆ’2 in a simplified form. What are the vertical and horizontal asymptotes of the graph 𝑦=𝑓π‘₯+52οˆβˆ’2?

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

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 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 𝑦=𝑓π‘₯+52οˆβˆ’2 onto the hyperbola 𝑦=1π‘₯? Write this in the form 𝐴𝑓π‘₯+52οˆβˆ’2=1π‘₯.

  • Aa dilation by a factor of 217 so 217𝑓π‘₯+52οˆβˆ’2=1π‘₯
  • Ba dilation by a factor of 35 so 35𝑓π‘₯+52οˆβˆ’2=1π‘₯
  • Ca dilation by a factor of 17 so 17𝑓π‘₯+52οˆβˆ’2=1π‘₯
  • Da dilation by a factor of 49 so 49𝑓π‘₯+52οˆβˆ’2=1π‘₯
  • Ea dilation by a factor of 917 so 917𝑓π‘₯+52οˆβˆ’2=1π‘₯

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‘Žπ‘₯+𝑏𝑐π‘₯+𝑑 produces the graph of 𝑦=1π‘₯. What is 𝑔?

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

What sequence of transformations maps the graph of 𝑔(π‘₯)=5π‘₯βˆ’32π‘₯+1 onto the hyperbola 𝑦=1π‘₯?

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

Q8:

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

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

Q9:

What are the two asymptotes of the hyperbola 𝑦=84π‘₯βˆ’3+53?

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

Q10:

Determine the vertical and horizontal asymptotes of the function 𝑓(π‘₯)=βˆ’1+3π‘₯βˆ’4π‘₯.

  • AThe vertical asymptote is at π‘₯=3, and the horizontal asymptote is at 𝑦=βˆ’4.
  • BThe vertical asymptote is at π‘₯=0, and the horizontal asymptote is at 𝑦=βˆ’1.
  • CThe vertical asymptote is at 𝑦=βˆ’1, and the horizontal asymptote is at π‘₯=0.
  • DThe function has no vertical asymptote, and the horizontal asymptote is at 𝑦=0.
  • EThe vertical asymptote is at π‘₯=βˆ’1, and the function has no horizontal asymptote.

Q11:

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

  • AThe function has vertical asymptotes at π‘₯=βˆ’15 and π‘₯=1π‘’οŠ« and a horizontal asymptote at 𝑦=βˆ’5.
  • BThe function has vertical asymptotes at π‘₯=0 and π‘₯=π‘’οŠ« and no horizontal asymptotes.
  • CThe function has vertical asymptotes at π‘₯=5 and π‘₯=π‘’οŠ« and a horizontal asymptote at 𝑦=βˆ’5.
  • DThe function has vertical asymptotes at π‘₯=0 and π‘₯=1π‘’οŠ« and no horizontal asymptotes.
  • EThe function has vertical asymptotes at π‘₯=0 and π‘₯=π‘’οŠ« and a horizontal asymptote at 𝑦=βˆ’5.

Q12:

Find the vertical and horizontal asymptotes of the function 𝑓(π‘₯)=βˆ’2π‘₯3+2π‘₯βˆ’15π‘₯ln.

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

Q13:

Find the vertical and horizontal asymptotes of the function 𝑓(π‘₯)=3π‘’οŠ±οŠ¨ο—οŽ‘.

  • AThe horizontal asymptote is 𝑦=0, and there are no vertical asymptotes.
  • BThere are no horizontal asymptotes, and the vertical asymptote is π‘₯=0.
  • CHorizontal asymptote at 𝑦=3, and there are no vertical asymptotes.
  • DThe horizontal asymptote is 𝑦=βˆ’2, and there are no vertical asymptotes.
  • EThere are no horizontal asymptotes, and the vertical asymptote is π‘₯=3.

Q14:

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

What is the value of 𝐴?

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

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

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

Q15:

Find 𝐴 and 𝐡 so that the function 𝑓(π‘₯)=𝐴4(1βˆ’3)+π΅οŠ±ο— has the single vertical asymptote π‘₯=0 and the two horizontal asymptotes 𝑦=βˆ’1 and 𝑦=1. What is the range of this function?

  • A 𝐴 = 8 , 𝐡 = βˆ’ 1 , and the range is all values of 𝑦 satisfying βˆ’βˆž<𝑦<βˆ’1 together with 1<𝑦<∞.
  • B 𝐴 = 8 , 𝐡 = βˆ’ 1 , and the range is all values of 𝑦 satisfying βˆ’βˆž<𝑦<∞.
  • C 𝐴 = 4 , 𝐡 = βˆ’ 1 2 , and the range is all values of 𝑦 satisfying βˆ’βˆž<𝑦<βˆ’1 together with 1<𝑦<∞.
  • D 𝐴 = 1 6 , 𝐡 = βˆ’ 4 , and the range is all values of 𝑦 satisfying βˆ’βˆž<𝑦<βˆ’1 together with 1<𝑦<∞.
  • E 𝐴 = 1 6 , 𝐡 = βˆ’ 4 , and the range is all values of 𝑦 satisfying βˆ’βˆž<𝑦<∞.

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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