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Lesson Worksheet: Horizontal and Vertical Asymptotes of a Function Mathematics • 10th Grade

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?

  • Aπ‘₯=34,𝑦=35
  • Bπ‘₯=14,𝑦=53
  • Cπ‘₯=14,𝑦=13
  • Dπ‘₯=43,𝑦=53
  • Eπ‘₯=34,𝑦=53

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?

  • Aπ‘₯=βˆ’π‘Žπ‘,𝑦=𝑑𝑐
  • Bπ‘₯=βˆ’π‘‘π‘,𝑦=π‘Žπ‘
  • Cπ‘₯=π‘Žπ‘,𝑦=𝑑𝑐
  • Dπ‘₯=𝑑𝑐,𝑦=π‘Žπ‘‘
  • Eπ‘₯=βˆ’π‘‘π‘,𝑦=βˆ’π‘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‘₯.

  • Aπ‘₯=12, π‘₯=3, 𝑦=βˆ’1
  • Bπ‘₯=3, π‘₯=βˆ’12, 𝑦=βˆ’1
  • Cπ‘₯=14, π‘₯=5, 𝑦=βˆ’2
  • Dπ‘₯=13, π‘₯=2, 𝑦=2
  • Eπ‘₯=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‘₯)?

  • Aο€Ό0,23
  • Bο€Όβˆ’43,23
  • Cο€Όβˆ’43,14
  • Dο€Ό43,23
  • Eο€Όβˆ’43,0

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

  • A𝑝=βˆ’43, π‘ž=23, π‘˜=βˆ’359
  • B𝑝=43, π‘ž=βˆ’23, π‘˜=βˆ’19
  • C𝑝=43, π‘ž=23, π‘˜=βˆ’59
  • D𝑝=43, π‘ž=23, π‘˜=59
  • E𝑝=βˆ’43, π‘ž=βˆ’23, π‘˜=βˆ’79

Q6:

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

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

  • Aπ‘₯=52,𝑦=βˆ’74
  • Bπ‘₯=25,𝑦=1
  • Cπ‘₯=52,𝑦=2
  • Dπ‘₯=2, 𝑦=52
  • Eπ‘₯=52,𝑦=βˆ’75

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

  • A8π‘₯+194π‘₯βˆ’5, π‘₯=54, 𝑦=βˆ’198
  • B8π‘₯+194π‘₯βˆ’5, π‘₯=54, 𝑦=2
  • C4π‘₯+172π‘₯, π‘₯=0, 𝑦=2
  • D4π‘₯+172π‘₯, π‘₯=0, 𝑦=βˆ’174
  • E4π‘₯+172π‘₯, 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?

  • A172π‘₯, π‘₯=0, 𝑦=172
  • B172π‘₯, π‘₯=1, 𝑦=172
  • C172π‘₯, π‘₯=0, 𝑦=0
  • D294π‘₯βˆ’5, π‘₯=54, 𝑦=0
  • E294π‘₯βˆ’5, π‘₯=54, 𝑦=294

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?

  • A𝑔(π‘₯)=π‘₯+4π‘₯+1
  • B𝑔(π‘₯)=βˆ’6π‘₯βˆ’52π‘₯+2
  • C𝑔(π‘₯)=6π‘₯+53π‘₯+2
  • D𝑔(π‘₯)=π‘₯βˆ’1π‘₯+2
  • E𝑔(π‘₯)=βˆ’π‘₯+3π‘₯+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

Q7:

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

  • Aπ‘₯=34,𝑦=53
  • Bπ‘₯=34,𝑦=35
  • Cπ‘₯=14,𝑦=53
  • Dπ‘₯=43,𝑦=53
  • Eπ‘₯=14,𝑦=13

Q8:

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.

Q9:

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.

Q10:

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.

This lesson includes 6 additional questions and 45 additional question variations for subscribers.

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