Higher Level only. The guide asks explicitly that students be made aware of the significance of the order.
Give them the two instructions and ask for the equation, without saying the order matters. "Start with y = x². Stretch vertically by 3 and translate up 2." Most of the room will write 3x² + 2 whichever order you say it in.
Then step to the second view, with both curves and the gap marked at two different x values. The gap being the same at both is what stops this feeling like a curiosity at the vertex. Ask why the gap is 4: the 2 got multiplied by 3, giving 6, so it is 4 more than the 2 that did not.
Views three and four are the signs. Ask before showing: which way does y = f(x − 3) move the graph? The room will say left. Then substitute x = 3 together and watch it give f(0).
| Question | Answer |
|---|---|
| 1. Stretch then shift, at x = 2 | 12 + 2 = 14. |
| 2. Shift then stretch, at x = 2 | 3 × 6 = 18. |
| 3. Maximum after f(x − 3) | x = 3. |
| 4. Feature at 6 after f(2x) | x = 3. |
| 5. Why the orders differ | A. The stretch multiplies the translation too. |
1 markEach transformation named, with its direction or scale factor.
1 markThe intermediate expression.
1 markThe final equation, or the image of a named point.
Insist on the middle line. A question that specifies an order is testing the order, and the only way to show you followed it is the expression you had halfway through.
| They wrote | What happened |
|---|---|
| 18 on question 1 | Did the orders the other way round. The expected error. |
| 12 | Stopped after the stretch. |
| 8 | Did not square: 3 × 2 + 2. Worth catching early, because it hides inside every later transformation question. |
| 38 | Stretched the x rather than the output: (3 × 2)² + 2. Also what you get from stretching twice. |
| 14 on question 2 | The other order again, from the other side. |
| 6 | Stopped after the translation. |
| −3 on question 3 | The commonest answer in the sub-topic. The minus sign read as a direction. |
| 12 on question 4 | Multiplied by 2 instead of halving. The scale factor is 1/q. |
| 1.5 | Halved twice. |
"Why does the minus mean right?" Because it is inside the function, so it changes the input rather than the output. The new graph at x = 3 is asked for f(0), so whatever f did at 0 now appears at 3. Substitute in front of them rather than giving a mnemonic: a mnemonic will be misremembered and a substitution can be redone.
"Is a horizontal stretch of 2 the same as f(2x)?" No, that is the reciprocal trap. f(2x) has scale factor one half. The clean way to say it: f(qx) divides every x-coordinate by q.
"Does the order matter for two translations?" No, and that is worth saying so the rule does not become "order always matters". Two translations commute; so do two stretches. A stretch and a translation in the same direction do not. Ask them to find a pair that commutes and a pair that does not, which is a better exercise than a list.
| Stage | What to do |
|---|---|
| Demonstrate | Graph 3x² + 2 and 3(x² + 2) together, then trace along and read both y values at the same x. The gap holding at 4 while the values change is the whole argument, and it takes twenty seconds. |
| Where they stick | The slider, or Dyna Graph on the Casio, is the one tool worth the setup time here: animating a in (x − a)² settles the direction question permanently. Students who have seen it animate stop writing f(x + 3) for a rightward shift. |
| The check | Test one point. If the original had a vertex at x = 0 and the image has it at x = 3, substitute x = 3 into both candidate formulas and see which gives the old output. It is faster than reasoning about signs and it never misfires. |
Equal scales on the window, or a vertical stretch looks like a horizontal squeeze and students will tell you the transformation is the wrong one.
| Step | What |
|---|---|
| 1 | Two instructions, no warning about order. Collect the equation. |
| 2 | Second view. Both curves, the gap marked twice. |
| 3 | Ask why the gap is 4. Get to "the 2 was stretched". |
| 4 | "Which way does f(x − 3) move it?" Let them say left, then substitute. |
| 5 | f(2x), and the reciprocal scale factor. |
| 6 | The table of six transformations, built from what they have just seen. |
| 7 | Translation vectors, and why only the horizontal component flips sign in the formula. |
Do not say "inside is opposite, outside is normal". It is true and it is a mnemonic, so it will come back inverted. Say "inside changes the input, so ask which x the new graph needs", which is a method rather than a rule.
Do not combine two transformations into one step on the board. If you write 3x² + 2 straight down from the two instructions, you have modelled skipping the line that carries the mark.