At a glance. One lesson of about 60 minutes. It assumes the previous lesson. Students write the rule for a described combination, say when order matters, read f(bx − h), and explain an equivalence of two transformations of a power function.
Calculator: no calculator. No calculator is needed. The lesson is about structure and algebra, and a GDC would only draw single sketches.
What students will learn
Know
- Order can matter only when operations act on the same coordinate, as a stretch and a lift do.
- f(bx − h) = f(b(x − h ÷ b)) shifts by h ÷ b.
- For f(x) = xⁿ, f(bx) = bⁿf(x).
Understand
- Each operation replaces x or y in the rule so far.
- A rule that holds for all f is different from one that holds for one f.
- A counterexample such as x² + 5 shows an equivalence is not general.
Do
- Write the rule for a described combination.
- Decide whether two orders agree and justify it.
- Prove an equivalence for a power function and disprove it for another.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying combinations and writing rules (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if and the power-function equivalence. Students find the pattern bⁿ and test it.
- Criterion C. Communicating: justifying the order result algebraically, and writing a counterexample (worked example 3).
- Criterion D. Applying mathematics in real-life contexts: the ramp, where two ways of building the same thing give different results.
The one thing to do with the figure
Ask for a prediction, then press the order button. With a = 2 and k = −3 on the screen, ask: “Will the ghost lie on the graph?” Then press Translate first and let students see the ghost swap.
In figure 2, ask: “If I set b = 2 and h = 6, how far right does the graph move?” Many say 6. Then read the shift.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the ramp. Collect predictions about the two orders. |
| 12 min | Both figures with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to taking the b out. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 2x + 6; (7, 3); (2, 5). |
|---|---|
| Hook | No: 2f(x) + 4 and 2(f(x) + 4) = 2f(x) + 8 differ; they agree only if the stretch is 1 or the lift is 0. |
| Worked examples | Left 1, stretch 2, reflect, up 3; y = f(2x − 3); (3x)⁴ = 81x⁴, but x² + 5 fails. |
| Faded example | y = f(3(x + 2)) = f(3x + 6). |
| 1. Reading f(bx − h) | C. A shifts by 10 after squashing; B shifts 5 before; D stretches. |
| 2. Horizontal squash of a point | 4. 2x = 8. |
| 3. Stretch then translate a point | 17. 3 × 5 + 2. |
| 4. Reflection, stretch and shift of a vertex | −7. −8 + 1. |
| 5. Writing an equation from a description | 15. 3 × 5. |
| 6. Two orders, one graph | 1.5. 2t = 3. |
| 7. Squash equals stretch for a power | 8. 2³. |
| 8. Which transformations commute | A. B stretches vertically; C lifts 6; D stretches horizontally. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Shifted by the whole 10, shifted before squashing, or stretched. | Take the 2 out of 2x − 10. |
| 16 or 10 in question 2 | Doubled x, or added 2. | Solve 2x = 8. |
| 21 or 7 in question 3 | Added the 2 inside, or forgot the 3. | Multiply, then add. |
| −10, 9 or −8 in question 4 | Put the +1 inside, forgot the reflection, or forgot the +1. | Do the −2 on 4, then +1. |
| 5 or 8 in question 5 | Did not stretch the lift, or added the factor. | Write the rule after the lift, then multiply it all. |
| 3 or 6 in question 6 | Used the same lift, or doubled it. | Expand 2(f(x) + t). |
| 2, 6 or 9 in question 7 | Used the factor, the product, or swapped base and power. | Expand (2x)³ fully. |
| B, C or D in question 8 | Stretched vertically instead of squashing, doubled the lift, or stretched instead of squashing. | Write each option as a rule and compare with f(2x) + 3. |
Questions worth asking
- Why do a horizontal and a vertical move commute? One changes only x-coordinates and the other only y-coordinates, so neither affects the other.
- How far does f(3x − 12) shift? 4 to the right, because it is f(3(x − 4)).
- Does f(bx) = bⁿf(x) work for f(x) = x² + 5? No: the constant breaks the pattern. It needs a pure power.
- When are af(x) + k and a(f(x) + k) the same? Only when a = 1 or k = 0, because the second is af(x) + ak; for 2 and 3 they differ by 3 for every f.
Differentiation
A way in for anyone stuck
- Give a three-step recipe: apply the operations in the order said, replacing x or y each time.
- Use a table with columns for the operation, what it replaces, and the new rule.
- Start with numbers: track one point through each order before writing the rule.
Stretch
- Show that f(−x) = f(x) for even n and f(−x) = −f(x) for odd n, and use it to say when a reflection in the y-axis gives the same graph as a reflection in the x-axis.
- For which functions f is f(2x) = 2f(x) for every x? Find a family and explain.
Students learning in English as an additional language
“Order” means which operation comes first. “Combination” means two or more transformations together. “Equivalent” means giving the same graph. “Counterexample” means one case that shows a rule is not always true.
| Word | Plain meaning | Example |
|---|---|---|
| combination | two or more transformations applied one after another | stretch, then lift |
| order of transformations | which transformation is done first | f(2x − 6) versus f(2(x − 6)) |
| equivalent transformations | different transformations giving the same graph | squash by 1 ÷ 3 and stretch by 81 for x⁴ |
| counterexample | one case showing a statement is not always true | x² + 5 for f(bx) = bⁿf(x) |
| power function | a function f(x) = xⁿ with n a positive whole number | x⁴ |
| commute | give the same result in either order | a lift and a shift sideways |
Sentence starters for explaining aloud or in writing:
- I did the … first, so the rule became …
- The two orders agree because …
- This is a counterexample because …
Two things not to say
“The order never matters.” It can matter when both act on the same coordinate, as a stretch and a lift do.
“f(2x − 6) moves 6 to the right.” Take out the 2: it moves 3.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ApplyThe point (3, 4) is on f. Find its image on y = 2f(x) − 1.
(3, 7). - ShiftHow far right does y = f(5x − 20) move the graph?
4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA zigzag fence is drawn by y = f(x). A copy is made by stretching it by 3 and then lifting it 2. The copy's corner at x = 4 has height 11. What was the original height at x = 4?
3f + 2 = 11 gives f = 3. - Earlier work (Lesson 6)The zero of f is at x = 8. Where is the zero of y = f(−x)?
x = −8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that for f(x) = x⁵, a squash by 1 ÷ 2 is a stretch by 32.
f(2x) = 32x⁵ = 32f(x). - CounterexampleShow that f(2x) = 4f(x) fails for f(x) = x² + 2.
f(2x) = 4x² + 2, but 4f(x) = 4x² + 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed. The whole lesson is exact algebra and every answer is reached by hand.