Lesson Worksheet: Graphs of Rational Functions Mathematics

In this worksheet, we will practice graphing rational functions whose denominators are linear, determining the types of their asymptotes, and describing their end behaviors.

Q1:

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

  • A(d)
  • B(a)
  • C(b)
  • D(c)

Q2:

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

  • A𝑎=5, 𝑏=1, 𝑘=4
  • B𝑎=3, 𝑏=2, 𝑘=3
  • C𝑎=4, 𝑏=3, 𝑘=1
  • D𝑎=3, 𝑏=3, 𝑘=1
  • E𝑎=2, 𝑏=3, 𝑘=12

Q3:

What function is represented in the figure below?

  • A𝑓(𝑥)=1𝑥3
  • B𝑓(𝑥)=1𝑥3
  • C𝑓(𝑥)=1𝑥3
  • D𝑓(𝑥)=1𝑥3

Q4:

The graph shows 𝑦=𝑘(𝑥3)2. We can see that the intersection of its asymptotes is at (3,2) and that the points (0.5,1.5) and (1.5,1) are below and above the graph respectively. Determine the interval in which 𝑘 lies.

  • A2<𝑘<0.5
  • B0.5<𝑘<3
  • C1.25<𝑘<1
  • D2<𝑘<1
  • E1.5<𝑘<1.25

Q5:

Consider the graph of the function 𝑦=1𝑥.

By looking at the graph and substituting a few successively larger values of 𝑥 into the function, what is the end behavior of the graph as 𝑥 increases along the positive 𝑥-axis?

  • AThe value of 𝑦 approaches infinity as 𝑥 increases.
  • BThe value of 𝑦 approaches zero as the value of 𝑥 increases.
  • CThe value of 𝑦 approaches negative infinity as 𝑥 increases.

Similarly, what is the end behavior of the graph as 𝑥 decreases?

  • AThe value of 𝑦 approaches zero.
  • BThe value of 𝑦 approaches .
  • CThe value of 𝑦 approaches .

Finally, by interpreting the graph, what is happening to the function when the value of 𝑥 approaches zero?

  • AThe value of 𝑦 approaches positive infinity when 𝑥 gets closer to zero from the negative direction and approaches negative infinity when 𝑥 gets closer to zero from the positive direction.
  • BThe value of 𝑦 approaches negative infinity when 𝑥 gets closer to zero from the negative direction or from the positive direction.
  • CThe value of 𝑦 approaches positive infinity when 𝑥 gets closer to zero from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches negative infinity when 𝑥 gets closer to zero from the negative direction and approaches positive infinity when 𝑥 gets closer to zero from the positive direction.

Q6:

What is the 𝑦-intercept of the function represented by the given graph?

  • A(10,2)
  • B(0,2)
  • C(2,0)
  • D(10,0)
  • E(0,10)

Q7:

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

  • A𝑦=𝑥2𝑥2𝑥10
  • B𝑦=𝑥2𝑥2𝑥10
  • C𝑦=𝑥2𝑥2𝑥10
  • D𝑦=𝑥2𝑥42𝑥10𝑥

Q8:

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

  • A𝑦=𝑥+3𝑥𝑥+4
  • B𝑦=𝑥3𝑥2𝑥4
  • C𝑦=𝑥3𝑥2𝑥4
  • D𝑦=𝑥2𝑥3𝑥+4

Q9:

Sketch a graph of the function 𝑓(𝑥)=𝑥3𝑥2𝑥4 and determine which graph represents it.

  • A
  • B
  • C
  • D

Q10:

Sketch the graph of the function 𝑓(𝑥)=1𝑥+2, and then find the 𝑦-intercept of 𝑓(𝑥).

  • A0,12
  • B0,12
  • C(0,2)
  • D(0,2)
  • E(0,1)

Q11:

Sketch a graph of the function 𝑓(𝑥)=12𝑥+1, then find the 𝑦-intercept of 𝑓(𝑥).

  • A(0,3)
  • B(0,1)
  • C0,12
  • D0,32
  • E0,32

Q12:

Sketch the graph of the function 𝑓(𝑥)=1𝑥+2+2, then find the 𝑥-intercept of 𝑓(𝑥).

  • A𝑥=12,0
  • B𝑥=32,0
  • C𝑥=52,0
  • D𝑥=52,0
  • E𝑥=32,0

Q13:

Sketch the graph of the function 𝑓(𝑥)=1𝑥+21, and then find the horizontal asymptote of 𝑓(𝑥).

  • A𝑦=1
  • B𝑦=1
  • C𝑦=2
  • D𝑦=2

Find the vertical asymptote of 𝑓(𝑥).

  • A𝑥=1
  • B𝑥=2
  • C𝑥=1
  • D𝑥=2

Q14:

Which function is represented by the graph shown?

  • A𝑓(𝑥)=1𝑥2+3
  • B𝑓(𝑥)=1𝑥23
  • C𝑓(𝑥)=12𝑥+3
  • D𝑓(𝑥)=12𝑥3
  • E𝑓(𝑥)=1𝑥+2+3

Q15:

Consider the graph of the function 𝑓(𝑥)=11𝑥+2. What is the end behavior of the graph as 𝑥 approaches 1?

  • AThe value of 𝑦 approaches when 𝑥 gets closer to 1 from the positive direction and approaches when 𝑥 gets closer to 1 from the negative direction.
  • BThe value of 𝑦 approaches when 𝑥 gets closer to 1 from the negative direction or from the positive direction.
  • CThe value of 𝑦 approaches when 𝑥 gets closer to 1 from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches when 𝑥 gets closer to 1 from the positive direction and approaches when 𝑥 gets closer to 1 from the negative direction.

Q16:

Which function is represented in the following graph?

  • A11𝑥+2
  • B11𝑥2
  • C1𝑥2+1
  • D1𝑥+21
  • E12𝑥+1

Q17:

Consider the graph of the function 𝑓(𝑥)=1𝑥+2. What happens to the function when the value of 𝑥 approaches 2?

  • AThe value of 𝑦 approaches when 𝑥 gets closer to 2 from the negative direction or from the positive direction.
  • BThe value of 𝑦 approaches when 𝑥 gets closer to 2 from the positive direction and approaches when 𝑥 gets closer to 2 from the negative direction.
  • CThe value of 𝑦 approaches when 𝑥 gets closer to 2 from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches when 𝑥 gets closer to 2 from the positive direction and approaches when 𝑥 gets closer to 2 from the negative direction.

Q18:

Which of the following graphs represents 𝑓(𝑥)=1𝑥2+1?

  • AGraph 2
  • BGraph 1
  • CGraph 3
  • DGraph 4

Q19:

Which of the following graphs represents a function 𝑓(𝑥) whose asymptotes are 𝑥=1 and 𝑦=2?

  • AGraph 2
  • BGraph 4
  • CGraph 1
  • DGraph 3

Q20:

Which function is represented by the graph shown?

  • A𝑓(𝑥)=1𝑥3+2
  • B𝑓(𝑥)=1𝑥+23
  • C𝑓(𝑥)=1𝑥+2+3
  • D𝑓(𝑥)=1𝑥23
  • E𝑓(𝑥)=1𝑥+3+2

Q21:

Which of the following is the equation of the graphed function 𝑓(𝑥) whose asymptotes are 𝑥=1 and 𝑦=2?

  • A𝑓(𝑥)=𝑥+1𝑥2
  • B𝑓(𝑥)=𝑥+1𝑥+2
  • C𝑓(𝑥)=2𝑥+1𝑥+1
  • D𝑓(𝑥)=𝑥+1𝑥1
  • E𝑓(𝑥)=2𝑥+1𝑥1

Q22:

Consider the graph of the function 𝑓(𝑥)=𝑥+1𝑥2. What is the end behavior of the graph as 𝑥 approaches ?

  • AThe value of 𝑦 approaches 1.
  • BThe value of 𝑦 approaches .
  • CThe value of 𝑦 approaches 2.
  • DThe value of 𝑦 approaches 2.
  • EThe value of 𝑦 approaches 1.

Q23:

Which of the following represents the function 𝑔(𝑥) whose asymptotes are the same as those of the graphed function 𝑓(𝑥)?

  • A𝑔(𝑥)=1𝑥3+2
  • B𝑔(𝑥)=1𝑥2+3
  • C𝑔(𝑥)=1𝑥23
  • D𝑔(𝑥)=1𝑥+2+3
  • E𝑔(𝑥)=1𝑥+3+2

Q24:

Consider the graph of the function 𝑓(𝑥)=1𝑥+2+2.

What is the end behavior of the graph as 𝑥 increases along the positive 𝑥-axis?

  • AThe value of 𝑦 approaches 2.
  • BThe value of 𝑦 approaches 2.
  • CThe value of 𝑦 approaches .
  • DThe value of 𝑦 approaches .

What is the end behavior of the graph as 𝑥 decreases along the negative 𝑥-axis?

  • AThe value of 𝑦 approaches 2.
  • BThe value of 𝑦 approaches .
  • CThe value of 𝑦 approaches 2.
  • DThe value of 𝑦 approaches .

Q25:

Which of the following is the graph of the function 𝑓(𝑥)=12+𝑥?

  • A
  • B
  • C
  • D
  • E

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