Lesson Worksheet: Graphs of Rational Functions Mathematics • 10th Grade

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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:

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.

Q2:

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

  • A(d)
  • B(a)
  • C(b)
  • D(c)

Q3:

What function is represented in the figure below?

  • A𝑓(𝑥)=−1𝑥−3
  • B𝑓(𝑥)=1𝑥−3
  • C𝑓(𝑥)=−1𝑥−3
  • D𝑓(𝑥)=1𝑥−3

Q4:

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?

  • 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

Q5:

The graph shows 𝑦=𝑘(𝑥−3)−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.

  • A−2<𝑘<−0.5
  • B0.5<𝑘<3
  • C−1.25<𝑘<−1
  • D−2<𝑘<−1
  • E−1.5<𝑘<−1.25

Q6:

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

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

Q7:

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.

Q8:

Consider the graph of the function 𝑓(𝑥)=11−𝑥+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.

Q9:

Which of the following is the graph of the function 𝑓(𝑥)=𝑥+54−𝑥?

  • A
  • B
  • C
  • D
  • E

Q10:

Sketch the graph of the function 𝑓(𝑥)=1𝑥+2−1, 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

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