Lesson Worksheet: Graphs of Rational Functions Mathematics

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


The graph of 𝑦=1(𝑥1)(𝑥+2)(𝑥3) has vertical asymptotes at 𝑥=2, 1, and 3. Which of the following is this graph?

  • A(b)
  • B(a)
  • C(d)
  • D(c)


What are the 𝑥-intercepts of the function represented by the given graph?

  • A(1,0) and (0,2)
  • B(1,0) and (2,0)
  • C(0,1) and (0,2)
  • D1,12 and 2,12
  • E(0,1) and (2,0)


The figure shows the graph of (𝑥2)(𝑥4)𝑔(𝑥).

Which of the following could 𝑔(𝑥) be?

  • A3518𝑥(𝑥3)
  • B3554𝑥(𝑥3)
  • C35108𝑥(𝑥3)
  • D3518𝑥(𝑥3)
  • E35324𝑥(𝑥3)


A team of scientists have been working on the growth of metal oxide nanowires, that is, metal oxide in the form of wires (cylinders) with dimensions in the order of nanometers. They observed that when the nanowires had reached a critical size, namely, a diameter of 50 nm and a length of 250 nm, the diameter increased at a rate of 1 nm/min and the length at a rate of 15 nm/min.

Write the function 𝑓(𝑡) that gives the aspect ratio of the nanowires, the ratio of their lengths to their diameters, as a function of the growth duration 𝑡, in minutes, after the nanowires have reached the critical size.

  • A𝑓(𝑡)=50+𝑡250+15𝑡
  • B𝑓(𝑡)=25015𝑡50𝑡
  • C𝑓(𝑡)=50+15𝑡250+𝑡
  • D𝑓(𝑡)=5015𝑡250𝑡
  • E𝑓(𝑡)=250+15𝑡50+𝑡

The scientists want to get nanowires with an aspect ratio of 10. Use the graph to find the corresponding growth duration after the nanowires have reached the critical size.

Assuming the growth mechanism remains the same, what would the aspect ratio of the nanowires be after a very long growing time?


True or False: The graph of a rational function must have a vertical asymptote.

  • AFalse
  • BTrue


True or False: If a rational function has a horizontal asymptote, then the degree of the numerator is at most the degree of the denominator.

  • ATrue
  • BFalse


Here are the graphs of 𝑦=1𝑥1+1𝑥2+1𝑥3 and 𝑦=1(𝑥1)(𝑥2)(𝑥3), which have all the same asymptotes.

Which one is 𝑦=1𝑥1+1𝑥2+1𝑥3?

  • A((a))
  • B((b))


Consider the following graph of the rational function 36𝑃(𝑥) for some polynomial 𝑃(𝑥).

Suppose we are told that the degree of 𝑃(𝑥) is at most 7. What is the degree of 𝑃(𝑥)?

  • A7
  • B6
  • C4
  • D3
  • E5

Where are the zeros of 𝑃(𝑥)?

  • A3, 1, and 2
  • B1, 2, and 3
  • C2, 1, and 3
  • D3, 2, and 1
  • E2, 1, and 3

By considering the values of 𝑃(𝑥) at points near the zeros, which of the following is 𝑃(𝑥)?

  • A(𝑥+2)(𝑥1)(𝑥3)
  • B(𝑥+2)(𝑥1)(𝑥3)
  • C(𝑥+2)(𝑥1)(𝑥3)
  • D(𝑥+2)(𝑥+1)(𝑥+3)
  • E(𝑥+2)(𝑥1)(𝑥3)


Consider the square prism shown in the diagram.

Write its surface-area-to-volume ratio in terms of 𝑥. Give your answer in standard form.

  • A16𝑥+20𝑥+64𝑥+8𝑥+5𝑥+1
  • B20𝑥+22𝑥+64𝑥+8𝑥+5𝑥+1
  • C4𝑥+8𝑥+5𝑥+116𝑥+20𝑥+6
  • D12𝑥+18𝑥+64𝑥+8𝑥+5𝑥+1
  • E4𝑥+8𝑥+5𝑥+120𝑥+22𝑥+6

The diagram shows the graph of the surface-area-to-volume ratio of the prism as a function of 𝑥. Which of the following is an approximate value of 𝑥 for which the surface-area-to-volume ratio is 1?

  • A6
  • B3.3
  • C2.3
  • D1.3
  • E1.5


Which of the following is the graph of 𝑦=1𝑥1+1𝑥2+1𝑥3?

  • A(d)
  • B(c)
  • C(a)
  • D(b)

This lesson includes 6 additional questions for subscribers.

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