Back to the lesson plan builder
Lesson length30 min45 min60 min
Questions🔀 New questions

Print the plan for yourself; the student sheet prints on its own page after it.

Math lesson plan · 12th grade (Year 13 in England) · 45 minutes

Find vertical and horizontal asymptotes

Algebra

CCSS HSF.IF.C.7.DEngland: A-level Pure Mathematics

Objective and success criteria

Students will: Find vertical and horizontal asymptotes.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Rational functions and asymptotes; Find limits of rational functions as x → ∞

Leads to: the next skill in this area.

Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 5 min

  • Quick retrieval from earlier learning (Rational functions and asymptotes; Find limits of rational functions as x → ∞). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Find the vertical asymptote of f(x) = (5x + 5)/(x + 5). Give the x-value. → -5Prompt: The function has no value where the denominator is zero.
  2. Find the limit as x → ∞ of (9x² + 3x + 3)/(6x² − 5x − 1). Give a fraction if needed. → 3/2 (1 1/2)Prompt: For huge x, only the highest powers matter.
  3. What happens to (x² + 1)/(2x² − 3) as x → ∞? (0 / 1/2 / No finite limit (it grows without bound)) → 1/2Prompt: Compare the degree of the top with the degree of the bottom.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Find vertical and horizontal asymptotes. Read the success criteria together.

I do: model it · 10 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: f(x) = (4x − 9)/(x − 1). Find the vertical asymptote x = ?
  • Step 1: x = 1
  • Answer: 1
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 10 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. f(x) = (−5x + 8)/(x − 3). Find the vertical asymptote x = ? → 3Prompt: A vertical asymptote is where the denominator is zero (and the top isn't).
  2. What are the vertical asymptotes of f(x) = (4x² + 1)/(2x² − 32)? (There are none / x = 4 and x = −4 / y = 2 / x = 4) → x = 4 and x = −4Prompt: Set the denominator equal to zero.

You do: independent practice · 10 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 5 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. What are the vertical asymptotes of f(x) = (5x² + 1)/(x² − 36)? (There are none / x = 6 and x = −6 / x = 36 and x = −36 / x = 6) → x = 6 and x = −6Prompt: Set the denominator equal to zero.
  2. f(x) = (−4x − 1)/(x + 6). Find the vertical asymptote x = ? → -6Prompt: A vertical asymptote is where the denominator is zero (and the top isn't).

Watch out for

  • Doing an operation to one side of an equation but not the other.
  • Reading 3x as 3 + x, or thinking x always stands for one particular number.

Support and stretch

Support

  • Review first: Rational functions and asymptotes.
  • Use concrete materials: balance scales or a balance drawing.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.

Free lesson plan from eurekami.com/lesson-plan-builder

eurekami

Find vertical and horizontal asymptotes

Name

Date

Score/ 8

  1. Question 1

    What are the vertical asymptotes of f(x) = (4x² + 1)/(x² − 25)?

    • A. x = 5 and x = −5
    • B. y = 4
    • C. There are none
    • D. x = 25 and x = −25

  2. Question 2

    f(x) = (6x + 4)/(x − 5). Find the horizontal asymptote y = ?

    Answer

  3. Question 3

    What are the vertical asymptotes of f(x) = (x² + 1)/(x² − 9)?

    • A. x = 3
    • B. x = 3 and x = −3
    • C. y = 1
    • D. There are none

  4. Question 4

    f(x) = (−5x + 4)/(x − 6). Find the vertical asymptote x = ?

    Answer

  5. Question 5

    What are the vertical asymptotes of f(x) = (2x² + 1)/(4x² − 64)?

    • A. There are none
    • B. x = 4 and x = −4
    • C. x = 16 and x = −16
    • D. y = 1/2

  6. Question 6

    f(x) = (−4x + 9)/(x − 1). Find the vertical asymptote x = ?

    Answer

  7. Question 7

    What are the vertical asymptotes of f(x) = (4x² + 1)/(4x² − 4)?

    • A. There are none
    • B. x = 1 and x = −1
    • C. x = 1
    • D. y = 1

  8. Question 8

    f(x) = (−4x − 8)/(x + 3). Find the horizontal asymptote y = ?

    Answer

Free from eurekami.com

for the adult only

Answer key: Find vertical and horizontal asymptotes

  1. 1. A. x = 5 and x = −5

  2. 2. 6

  3. 3. B. x = 3 and x = −3

  4. 4. 6

  5. 5. B. x = 4 and x = −4

  6. 6. 1

  7. 7. B. x = 1 and x = −1

  8. 8. -4