Worksheet: Graphs of Rational Functions

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:

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?

  • Atrue
  • Bfalse

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:

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 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(b)
  • B(a)
  • C(d)
  • D(c)

Q6:

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

Which of the following could 𝑔(𝑥) be?

  • A3518𝑥(𝑥−3)
  • B−3554𝑥(𝑥−3)
  • C−35108𝑥(𝑥−3)
  • D−3518𝑥(𝑥−3)
  • E35324𝑥(𝑥−3)

Q7:

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

  • Afalse
  • Btrue

Q8:

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

Which one is 𝑦=1𝑥−1+1𝑥−2+1𝑥−3?

  • A((a))
  • B((b))

Q9:

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

  • A(d)
  • B(c)
  • C(a)
  • D(b)

Q10:

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

  • A
  • B
  • C
  • D

Q11:

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

  • A(c)
  • B(a)
  • C(d)
  • D(b)

Q12:

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

  • A(b)
  • B(d)
  • C(a)
  • D(c)

Q13:

Shown is the graph of the rational function 6𝑃(𝑥).

Which of the following could be the polynomial 𝑃(𝑥)?

  • A−(𝑥−1)(𝑥+2)(𝑥−3)
  • B(𝑥−1)(𝑥+6)
  • C(𝑥−1)(𝑥−2)(𝑥−3)
  • D(𝑥+1)(𝑥+2)(𝑥−3)
  • E(𝑥−1)(𝑥+2)(𝑥−3)

Q14:

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

  • A(b)
  • B(a)
  • C(c)
  • D(d)

Q15:

What function is represented in the figure below?

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

Q16:

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

Q17:

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)=250−15𝑡50−𝑡
  • C𝑓(𝑡)=50+15𝑡250+𝑡
  • D𝑓(𝑡)=50−15𝑡250−𝑡
  • 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?

Q18:

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

Q19:

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.

Q20:

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)?

  • A−3, 1, and 2
  • B1, 2, and 3
  • C−2, 1, and 3
  • D−3, −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)(𝑥−1)(𝑥−3)
  • B(𝑥+2)(𝑥−1)(𝑥−3)
  • C(𝑥+2)(𝑥−1)(𝑥−3)
  • D(𝑥+2)(𝑥+1)(𝑥+3)
  • E(𝑥+2)(𝑥−1)(𝑥−3)

Q21:

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)
  • D−1,12 and 2,12
  • E(0,−1) and (2,0)

Q22:

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)

Q23:

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

  • A𝑦=𝑥−2𝑥2𝑥−10
  • B𝑦=𝑥−2𝑥2𝑥−10
  • C𝑦=𝑥−2𝑥2𝑥−10
  • D𝑦=𝑥−2𝑥−42𝑥−10𝑥

Q24:

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

  • A𝑦=𝑥+3𝑥𝑥+4
  • B𝑦=𝑥−3𝑥−2𝑥−4
  • C𝑦=𝑥−3𝑥−2𝑥−4
  • D𝑦=𝑥−2𝑥−3𝑥+4

Q25:

Sketch a graph of the function 𝑓(𝑥)=𝑥−3𝑥−2𝑥−4 and determine which graph represents it.

  • A
  • B
  • C
  • D

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