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Math lesson plan · 12th grade (Year 13 in England) · 60 minutes

Relate a function's graph to its inverse

Algebra

CCSS HSF.BF.B.4.CEngland: A-level Pure Mathematics

Objective and success criteria

Students will: Relate a function's graph to its inverse.

  • 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: Inverse functions (non-linear)

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 · 6 min

  • Quick retrieval from earlier learning (Inverse functions (non-linear)). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. f(x) = (x − 6)/6. Find f⁻¹(x). → 6x+6Prompt: Write y = f(x), swap x and y, then make y the subject.
  2. What is the inverse of f(x) = 3ˣ? (log_3(x) / x/3 / 3⁻ˣ / x³) → log_3(x)Prompt: Exponential and logarithmic functions undo each other.
  3. f(x) = 3/(x − 2). Find f⁻¹(x). → 3/x+2Prompt: Write y = f(x), swap x and y, then make y the subject.

Hook · 6 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Relate a function's graph to its inverse. Read the success criteria together.

I do: model it · 12 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: The point (−1, −4) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1).
  • Step 1: (−4, −1)
  • Answer: (−4, −1)
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 12 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. The point (1, −3) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1). → (−3, 1)Prompt: An inverse swaps inputs and outputs.
  2. f(x) = x³ + 9. Find f⁻¹(36). → 3Prompt: f⁻¹(36) is the x that makes f(x) = 36.

You do: independent practice · 17 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 · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. The graph of y = f(x) passes through (3, −6). Which transformation maps it onto the graph of y = f⁻¹(x)? (Reflection in the line y = x / Reflection in the line y = −x / Reflection in the y-axis / Rotation of 180° about the origin) → Reflection in the line y = xPrompt: Where does (3, −6) go on the inverse?
  2. The point (8, −5) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1). → (−5, 8)Prompt: An inverse swaps inputs and outputs.

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: Inverse functions (non-linear).
  • 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

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Relate a function's graph to its inverse

Name

Date

Score/ 8

  1. Question 1

    The graph of y = f(x) passes through (3, 6). Which transformation maps it onto the graph of y = f⁻¹(x)?

    • A. Reflection in the line y = −x
    • B. Reflection in the x-axis
    • C. Reflection in the y-axis
    • D. Reflection in the line y = x

  2. Question 2

    The point (5, 4) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1).

    Answer

  3. Question 3

    f(x) = x³ − 1. Find f⁻¹(0).

    Answer

  4. Question 4

    The graph of y = f(x) passes through (5, 2). Which transformation maps it onto the graph of y = f⁻¹(x)?

    • A. Reflection in the line y = −x
    • B. Rotation of 180° about the origin
    • C. Reflection in the line y = x
    • D. Reflection in the y-axis

  5. Question 5

    The point (5, 1) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1).

    Answer

  6. Question 6

    f(x) = x³ − 2. Find f⁻¹(62).

    Answer

  7. Question 7

    The graph of y = f(x) passes through (−4, 4). Which transformation maps it onto the graph of y = f⁻¹(x)?

    • A. Reflection in the x-axis
    • B. Rotation of 180° about the origin
    • C. Reflection in the y-axis
    • D. Reflection in the line y = x

  8. Question 8

    The point (1, −8) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1).

    Answer

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Answer key: Relate a function's graph to its inverse

  1. 1. D. Reflection in the line y = x

  2. 2. (4, 5)

  3. 3. 1

  4. 4. C. Reflection in the line y = x

  5. 5. (1, 5)

  6. 6. 4

  7. 7. D. Reflection in the line y = x

  8. 8. (−8, 1)