Worksheet: Graphs of Rational Functions

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In this worksheet, we will practice sketching the graphs of simple rational functions and identify simple rational functions from their graphs.

Q1:

Which of the following graphs represents 𝑓 ( 𝑥 ) = 1 𝑥 + 1 ?

  • A(b)
  • B(d)
  • C(a)
  • D(c)

Q2:

If a rational function has a horizontal asymptote, then the degree of the numerator is at most the degree of the denominator. Is this true or false?

  • A false
  • B true

Q3:

Consider the following graphs.

Which of these is the graph of 𝑓 ( 𝑥 ) = ( 𝑥 + 1 ) ( 𝑥 − 1 ) ( 𝑥 + 3 ) ?

  • A (d)
  • B (a)
  • C (b)
  • D (c)

Q4:

Which of the following is the equation of the rational function graphed below?

  • A 𝑦 = 𝑥 − 2 𝑥 2 𝑥 − 1 0
  • B 𝑦 = 𝑥 − 2 𝑥 − 4 2 𝑥 − 1 0 𝑥
  • C 𝑦 = 𝑥 − 2 𝑥 2 𝑥 − 1 0
  • D 𝑦 = 𝑥 − 2 𝑥 2 𝑥 − 1 0

Q5:

The graph shows 𝑦 = 𝑘 ( 𝑥 − 𝑎 ) + 𝑏 . A single point is marked on the graph. What are the values of the constants 𝑎 , 𝑏 , and 𝑘 ?

  • A 𝑎 = − 3 , 𝑏 = 3 , 𝑘 = − 1
  • B 𝑎 = 5 , 𝑏 = 1 , 𝑘 = 4
  • C 𝑎 = 4 , 𝑏 = − 3 , 𝑘 = − 1
  • D 𝑎 = − 2 , 𝑏 = 3 , 𝑘 = − 1 2
  • E 𝑎 = 3 , 𝑏 = − 2 , 𝑘 = 3

Q6:

The graph of 𝑦 = 1 ( 𝑥 − 1 ) ( 𝑥 + 2 ) ( 𝑥 − 3 ) has vertical asymptotes at 𝑥 = − 2 , 1, and 3. Which of the following is this graph?

  • A(a)
  • B(d)
  • C(c)
  • D(b)

Q7:

The figure shows the graph of ( 𝑥 − 2 ) ( 𝑥 − 4 ) 𝑔 ( 𝑥 ) .

Which of the following could 𝑔 ( 𝑥 ) be?

  • A 3 5 3 2 4 𝑥 ( 𝑥 − 3 )
  • B − 3 5 5 4 𝑥 ( 𝑥 − 3 )
  • C − 3 5 1 0 8 𝑥 ( 𝑥 − 3 )
  • D − 3 5 1 8 𝑥 ( 𝑥 − 3 )
  • E 3 5 1 8 𝑥 ( 𝑥 − 3 )

Q8:

Which of the following is the equation of the rational function graphed below?

  • A 𝑦 = 𝑥 − 2 𝑥 − 3 𝑥 + 4
  • B 𝑦 = 𝑥 + 3 𝑥 𝑥 + 4
  • C 𝑦 = 𝑥 − 3 𝑥 − 2 𝑥 − 4
  • D 𝑦 = 𝑥 − 3 𝑥 − 2 𝑥 − 4

Q9:

Which of the following is the equation of the rational function graphed below?

  • A 𝑓 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 1 ) ( 𝑥 − 1 ) ( 𝑥 − 3 )
  • B 𝑓 ( 𝑥 ) = ( 𝑥 − 4 ) ( 𝑥 + 4 ) ( 𝑥 + 1 ) ( 𝑥 − 1 ) ( 𝑥 − 3 ) + 0 . 5
  • C 𝑓 ( 𝑥 ) = 1 1 4 ( 𝑥 + 4 ) ( 𝑥 + 1 ) ( 𝑥 − 1 ) ( 𝑥 − 3 ) + 0 . 5
  • D 𝑓 ( 𝑥 ) = ( 𝑥 + 4 ) ( 𝑥 + 1 ) ( 𝑥 − 1 ) + 0 . 5

Q10:

The graph of a rational function must have a vertical asymptote. Is this true or false?

  • A true
  • B false

Q11:

Here are the graphs of 𝑦 = 1 𝑥 − 1 + 1 𝑥 − 2 + 1 𝑥 − 3 and 𝑦 = 1 ( 𝑥 − 1 ) ( 𝑥 − 2 ) ( 𝑥 − 3 ) , which have all the same asymptotes.

Which one is 𝑦 = 1 𝑥 − 1 + 1 𝑥 − 2 + 1 𝑥 − 3 ?

  • A ((a))
  • B ((b))

Q12:

Which of the following is the graph of 𝑦 = 1 𝑥 − 1 + 1 𝑥 − 2 + 1 𝑥 − 3 ?

  • A(a)
  • B(d)
  • C(c)
  • D(b)

Q13:

Which of the following is the graph of 𝑦 = − 1 𝑥 − 1 + 1 𝑥 − 2 + 1 𝑥 − 3 ?

  • A
  • B
  • C
  • D

Q14:

Which of the following is the graph of 𝑦 = 2 𝑥 − 1 𝑥 + 9 ?

  • A (d)
  • B (a)
  • C (b)
  • D (c)

Q15:

Which of the following is the equation of the rational function graphed below?