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Math lesson plan · 11th grade (Year 12 in England) · 30 minutes

Transformations of functions

Algebra

CCSS HSF.BF.B.3England: GCSE Higher Algebra

Objective and success criteria

Students will: Transformations of functions.

  • 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: Graphs of quadratics: roots, vertex, intercepts; Translations, reflections and rotations

Leads to: Rational functions and asymptotes; Graphs of sine and cosine: amplitude and period; Describe the transformation from y = f(x)

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

Warm-up · 4 min

  • Quick retrieval from earlier learning (Graphs of quadratics: roots, vertex, intercepts; Translations, reflections and rotations). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Does the parabola y = (x − 1)² − 4 have a maximum or a minimum point? (Maximum / Minimum) → MinimumPrompt: Look at the sign of the x² coefficient.
  2. Does the parabola y = (x − 1)² + 3 have a maximum or a minimum point? (Maximum / Minimum) → MinimumPrompt: Look at the sign of the x² coefficient.
  3. Does the parabola y = (x + 4)² + 2 have a maximum or a minimum point? (Maximum / Minimum) → MinimumPrompt: Look at the sign of the x² coefficient.

Hook · 3 min

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

I do: model it · 7 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: Which transformation maps y = f(x) to y = f(x + 2)?
  • Step 1: translate left 2
  • Answer: translate left 2
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 6 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. Which transformation maps y = f(x) to y = f(x) + 5? (translate up 5 / translate left 5 / reflect in the x-axis / reflect in the y-axis) → translate up 5Prompt: Changes outside f affect y; changes inside f affect x (and act the opposite way).
  2. Which transformation maps y = f(x) to y = f(x) + 4? (translate up 4 / translate left 4 / reflect in the x-axis / reflect in the y-axis) → translate up 4Prompt: Changes outside f affect y; changes inside f affect x (and act the opposite way).

You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Which transformation maps y = f(x) to y = f(x) + 3? (translate up 3 / translate left 3 / reflect in the x-axis / reflect in the y-axis) → translate up 3Prompt: Changes outside f affect y; changes inside f affect x (and act the opposite way).
  2. Which transformation maps y = f(x) to y = f(x + 2)? (translate up 2 / translate left 2 / reflect in the x-axis / reflect in the y-axis) → translate left 2Prompt: Changes outside f affect y; changes inside f affect x (and act the opposite way).

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: Graphs of quadratics: roots, vertex, intercepts.
  • 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

  • Move on to: Rational functions and asymptotes; Graphs of sine and cosine: amplitude and period; Describe the transformation from y = f(x).
  • 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-11.

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

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Transformations of functions

Name

Date

Score/ 6

  1. Question 1

    Which transformation maps y = f(x) to y = f(x) + 4?

    • A. translate up 4
    • B. translate left 4
    • C. reflect in the x-axis
    • D. reflect in the y-axis

  2. Question 2

    The graph of y = x² is transformed to y = (x − 5)² + 1. Where is the new vertex? Type it like (2, −1).

    xy−10−8−6−4−2246810−10−8−6−4−22468100O

    Answer

  3. Question 3

    Which transformation maps y = f(x) to y = f(x + 4)?

    • A. translate up 4
    • B. translate left 4
    • C. reflect in the x-axis
    • D. reflect in the y-axis

  4. Question 4

    The graph of y = x² is transformed to y = (x + 6)² − 5. Where is the new vertex? Type it like (2, −1).

    xy−10−8−6−4−2246810−10−8−6−4−22468100O

    Answer

  5. Question 5

    Which transformation maps y = f(x) to y = f(−x)?

    • A. translate up 4
    • B. translate left 4
    • C. reflect in the x-axis
    • D. reflect in the y-axis

  6. Question 6

    The graph of y = x² is transformed to y = (x + 3)² − 6. Where is the new vertex? Type it like (2, −1).

    xy−10−8−6−4−2246810−10−8−6−4−22468100O

    Answer

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Answer key: Transformations of functions

  1. 1. A. translate up 4

  2. 2. (5, 1)

  3. 3. B. translate left 4

  4. 4. (−6, −5)

  5. 5. D. reflect in the y-axis

  6. 6. (−3, −6)