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

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=1π‘₯.

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

  • AThe value of 𝑦 approaches infinity as π‘₯ increases.
  • BThe value of 𝑦 approaches zero as the value of π‘₯ increases.
  • CThe value of 𝑦 approaches negative infinity as π‘₯ increases.

Similarly, what is the end behavior of the graph as π‘₯ decreases?

  • AThe value of 𝑦 approaches zero.
  • BThe value of 𝑦 approaches βˆ’βˆž.
  • CThe value of 𝑦 approaches ∞.

Finally, by interpreting the graph, what is happening to the function when the value of π‘₯ approaches zero?

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

Q2:

Which of the following graphs represents 𝑓(π‘₯)=1π‘₯+1?

  • A(d)
  • B(a)
  • C(b)
  • D(c)

Q3:

What function is represented in the figure below?

  • A𝑓(π‘₯)=βˆ’1π‘₯βˆ’3
  • B𝑓(π‘₯)=1π‘₯βˆ’3
  • C𝑓(π‘₯)=βˆ’1π‘₯βˆ’3
  • D𝑓(π‘₯)=1π‘₯βˆ’3

Q4:

The graph shows 𝑦=π‘˜(π‘₯βˆ’π‘Ž)+𝑏. A single point is marked on the graph. What are the values of the constants π‘Ž, 𝑏, and π‘˜?

  • Aπ‘Ž=5, 𝑏=1, π‘˜=4
  • Bπ‘Ž=3, 𝑏=βˆ’2, π‘˜=3
  • Cπ‘Ž=4, 𝑏=βˆ’3, π‘˜=βˆ’1
  • Dπ‘Ž=βˆ’3, 𝑏=3, π‘˜=βˆ’1
  • Eπ‘Ž=βˆ’2, 𝑏=3, π‘˜=βˆ’12

Q5:

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

  • Aβˆ’2<π‘˜<βˆ’0.5
  • B0.5<π‘˜<3
  • Cβˆ’1.25<π‘˜<βˆ’1
  • Dβˆ’2<π‘˜<βˆ’1
  • Eβˆ’1.5<π‘˜<βˆ’1.25

Q6:

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

  • A𝑓(π‘₯)=π‘₯+1π‘₯βˆ’2
  • B𝑓(π‘₯)=π‘₯+1π‘₯+2
  • C𝑓(π‘₯)=2π‘₯+1π‘₯+1
  • D𝑓(π‘₯)=π‘₯+1π‘₯βˆ’1
  • E𝑓(π‘₯)=2π‘₯+1π‘₯βˆ’1

Q7:

Consider the graph of the function 𝑓(π‘₯)=1π‘₯+2. What happens to the function when the value of π‘₯ approaches βˆ’2?

  • AThe value of 𝑦 approaches ∞ when π‘₯ gets closer to βˆ’2 from the negative direction or from the positive direction.
  • BThe value of 𝑦 approaches βˆ’βˆž when π‘₯ gets closer to βˆ’2 from the positive direction and approaches ∞ when π‘₯ gets closer to βˆ’2 from the negative direction.
  • CThe value of 𝑦 approaches βˆ’βˆž when π‘₯ gets closer to βˆ’2 from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches ∞ when π‘₯ gets closer to βˆ’2 from the positive direction and approaches βˆ’βˆž when π‘₯ gets closer to βˆ’2 from the negative direction.

Q8:

Consider the graph of the function 𝑓(π‘₯)=11βˆ’π‘₯+2. What is the end behavior of the graph as π‘₯ approaches 1?

  • AThe value of 𝑦 approaches βˆ’βˆž when π‘₯ gets closer to 1 from the positive direction and approaches ∞ when π‘₯ gets closer to 1 from the negative direction.
  • BThe value of 𝑦 approaches ∞ when π‘₯ gets closer to 1 from the negative direction or from the positive direction.
  • CThe value of 𝑦 approaches βˆ’βˆž when π‘₯ gets closer to 1 from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches ∞ when π‘₯ gets closer to 1 from the positive direction and approaches βˆ’βˆž when π‘₯ gets closer to 1 from the negative direction.

Q9:

Which of the following is the graph of the function 𝑓(π‘₯)=π‘₯+54βˆ’π‘₯?

  • A
  • B
  • C
  • D
  • E

Q10:

Sketch the graph of the function 𝑓(π‘₯)=1π‘₯+2βˆ’1, and then find the horizontal asymptote of 𝑓(π‘₯).

  • A𝑦=1
  • B𝑦=βˆ’1
  • C𝑦=βˆ’2
  • D𝑦=2

Find the vertical asymptote of 𝑓(π‘₯).

  • Aπ‘₯=1
  • Bπ‘₯=βˆ’2
  • Cπ‘₯=βˆ’1
  • Dπ‘₯=2

This lesson includes 21 additional questions and 205 additional question variations for subscribers.

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