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.
- f(x) = √(x − 1), x ≥ 1. Find f⁻¹(x). → x^2+1Prompt: Write y = f(x), swap x and y, then make y the subject.
- What is the inverse of f(x) = 2ˣ? (log_2(x) / 1/2ˣ / x/2 / x²) → log_2(x)Prompt: Exponential and logarithmic functions undo each other.
- f(x) = 3/(x − 4). Find f⁻¹(x). → 3/x+4Prompt: 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.
- The point (2, −7) 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). → (−7, 2)Prompt: An inverse swaps inputs and outputs.
- f(x) = x³ + 9. Find f⁻¹(1). → -2Prompt: f⁻¹(1) is the x that makes f(x) = 1.
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.
- The graph of y = f(x) passes through (2, 1). Which transformation maps it onto the graph of y = f⁻¹(x)? (Reflection in the y-axis / Reflection in the line y = x / Reflection in the x-axis / Rotation of 180° about the origin) → Reflection in the line y = xPrompt: Where does (2, 1) go on the inverse?
- f(x) = x³ + 8. Find f⁻¹(0). → -2Prompt: f⁻¹(0) is the x that makes f(x) = 0.
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