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Cambridge IGCSE0607

Asymptotes

International Mathematics 0607 Chapter Notes

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Asymptotes
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In this chapter

  1. 1What an asymptote is
  2. 2Asymptotes parallel to the y-axis
  3. 3Writing an asymptote as an equation
  4. 4Asymptotes of tan x

1. What an asymptote is

An asymptote is a straight line that a curve gets closer and closer to. A vertical asymptote is a line x = a, parallel to the y-axis. A horizontal asymptote is a line y = b, parallel to the x-axis.

You identify asymptotes parallel to the axes on a graph. You do not derive them by algebra.

A curve has no point on a vertical asymptote, so an inequality that ends there uses < or >.

Examiner insight

“The answer –4 < x ≤ –3.63 is incorrect as x cannot equal the value at the asymptote.”

Worked example 12 marks

Use your graphic display calculator to sketch y = 1/(x-3) + 4. The graph has one vertical asymptote and one horizontal asymptote. Write down their equations.

  1. 1

    Enter the function and draw the graph. The two branches run alongside two straight lines.

  2. 2

    Vertical asymptote: x = 3 [1]

  3. 3

    Horizontal asymptote: y = 4 [1]

2. Asymptotes parallel to the y-axis

Read the x-value of each asymptote parallel to the y-axis from the calculator sketch.

On paper, draw the asymptotes first so that the branches do not overlap.

Examiner insight

“Candidates who drew both asymptotes parallel to the y-axis first ensured that the branches of the graph did not overlap.”

Worked example 12 marks

f(x) = 1/(x-2) + 3/(2x-1). Write down the equations of the asymptotes parallel to the y-axis.

  1. 1

    Sketch y = f(x) on the calculator. The branches run alongside two vertical lines.

  2. 2

    x = 0.5 [1]

  3. 3

    x = 2 [1]

3. Writing an asymptote as an equation

An answer must be an equation. A vertical asymptote is written x = a. A horizontal asymptote is written y = b.

The x-axis can be an asymptote. Its equation is y = 0. The y-axis has equation x = 0.

Check the number of asymptotes the question asks for. One curve can have two asymptotes parallel to the y-axis and one parallel to the x-axis.

x-axis: y = 0

y-axis: x = 0

Examiner insight

“Some candidates wrote down the relevant numbers without equating them to the associated variable.” “Many candidates used y instead of x for their equations and some omitted x altogether just giving the values of –1 and 2.”

Worked example 13 marks

f(x) = 2x/(x² + x - 12). Write down the equations of the 3 asymptotes to the graph.

  1. 1

    Sketch the graph on the calculator. Two asymptotes are parallel to the y-axis and one is the x-axis.

  2. 2

    x = -4 [1]

  3. 3

    x = 3 [1]

  4. 4

    y = 0 [1]

4. Asymptotes of tan x

The graph of f(x) = tan x has asymptotes at 90^(°), 270^(°) and so on. Each one is parallel to the y-axis.

Write each one as an equation that starts x =.

asymptotes of y = tan x: x = 90^(°), x = 270^(°) and so on

Examiner insight

“Most candidates clearly knew what the asymptotes were, but some could not write them as equations.”

Worked example 12 marks

Write down the equations of the asymptotes of y = tan x for 0^(°) ≤ x ≤ 360^(°).

  1. 1

    Asymptote at 90^(°): x = 90^(°) [1]

  2. 2

    Asymptote at 270^(°): x = 270^(°) [1]

Recap

  • An asymptote is a straight line that a curve gets closer and closer to.
  • A vertical asymptote is x = a. A horizontal asymptote is y = b.
  • Write each asymptote as an equation, never as a bare number.
  • The x-axis is y = 0. The y-axis is x = 0.
  • Draw the asymptotes on your sketch first, then the branches.
  • y = tan x has asymptotes x = 90^(°), x = 270^(°), and so on.
  • At a vertical asymptote an inequality uses < or >, never ≤ or ≥.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A curve gets closer and closer to the x-axis. Write down the equation of this asymptote.1 mark
  2. Do not use a calculator. A curve has an asymptote along the y-axis. Write down its equation.1 mark
  3. Do not use a calculator. A graph has a vertical asymptote that crosses the x-axis at -1 and a horizontal asymptote that crosses the y-axis at 2. Write down the equation of each asymptote.2 marks
  4. Do not use a calculator. A curve has asymptotes x = 3 and y = -4. Which asymptote is parallel to the y-axis? Which is parallel to the x-axis?2 marks
  5. Do not use a calculator. The asymptotes of y = tan x are 180^(°) apart. Write down the equations of the asymptotes of y = tan x for 180^(°) ≤ x ≤ 540^(°).2 marks
  6. Use your graphic display calculator to sketch y = 1/(x+2) + 3 for -9 ≤ x ≤ 5. Write down the equations of the asymptotes.2 marks
  7. Use your graphic display calculator to sketch f(x) = 1/x + 1/(x-4) for -4 ≤ x ≤ 8. Write down the equations of all the asymptotes.3 marks
  8. A student says the graph of y = x/(x-3) has asymptotes x = 3 and y = 0. Use your graphic display calculator to check. Write down the correct equations.3 marks
  9. f(x) = 5/(x² - 4). Use your graphic display calculator to sketch y = f(x) for -6 ≤ x ≤ 6. (a) Write down the equations of the three asymptotes. (b) Solve the inequality f(x) > 1.6 marks
  10. A model for the mass, M grams, of a plant t weeks after planting is M = (120t + 30)/(t + 3) for t ≥ 0. Use your graphic display calculator to sketch M against t for 0 ≤ t ≤ 100. (a) Write down the equation of the horizontal asymptote. (b) A student says the plant reaches a mass of 120 g in week 100. Show that the student is wrong. (c) Find the first whole number of weeks for which the model gives a mass of more than 115 g.5 marks

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