Stretch y = x² by 3 and then lift it by 2 and you get 3x² + 2. Lift it first and then stretch and you get 3x² + 6. Same two instructions, and the two curves are 4 apart at every x.
On the first view, move the slider anywhere: the gap does not change, which is what makes the order a real difference rather than a near miss. The second view has nothing to slide, so the slider rests there.
| Written | What it does | Watch out |
|---|---|---|
| y = f(x) + b | Translate up by b. | Outside the function, so it does what it says. |
| y = f(x − a) | Translate right by a. | The minus moves it the POSITIVE way. Inside the function, everything is backwards. |
| y = −f(x) | Reflect in the x-axis. | Outside: the y values flip. |
| y = f(−x) | Reflect in the y-axis. | Inside: the x values flip. |
| y = p f(x) | Vertical stretch, factor p. | Outside: stretches by p, as expected. |
| y = f(qx) | Horizontal stretch, factor 1/q. | Inside: q = 3 SQUASHES by three. |
The pattern is worth more than the six rows: outside the function means vertical and behaves normally; inside the function means horizontal and behaves backwards. Every one of the "watch out" entries is that sentence applied.
Take y = x², a vertical stretch of 3, and a translation of 2 up.
Stretch first, then translate:
x² → 3x² → 3x² + 2, vertex (0, 2)
Translate first, then stretch:
x² → x² + 2 → 3(x² + 2) = 3x² + 6, vertex (0, 6)
and the difference is 4 at every x, because the second order stretches the translation as well:
3 × 2 − 2 = 4
At x = 1 the two are 5 and 9; at x = 2 they are 14 and 18. The gap never closes, which is how you know it is not a rounding or a special case.
So read the final equation, not the list of instructions. 3x² + 2 says: multiply the function by 3, then add 2. The outermost operation is the LAST one applied. Work from the outside in and the order comes off the page rather than out of memory.
And the reverse: if a question gives you the instructions and asks for the equation, apply them in the order stated and build the expression from the inside out.
y = (x − 3)² is y = x² moved 3 to the right, not the left. Its vertex is at x = 3.
Why. The curve is at its lowest where the bracket is zero, and x − 3 is zero at x = 3. That is the whole argument. At x = 0 the curve is at 9, and at x = −3 it is at 36, both a long way up.
The reason it feels wrong is the pattern above: inside the function, everything is backwards. To get the same output as before you now have to feed in a bigger x, so the picture moves the positive way.
For an even function, one of the two reflections does nothing. x² is even, so f(−x) = f(x) and reflecting in the y-axis leaves it exactly where it was: f(−2) and f(2) are both 4. The other reflection does plenty: −f(2) = −4. If a question's answer looks identical to the original, check whether the function was even before assuming you have made a mistake.
Not required at Standard Level: f(ax + b). A combination of a horizontal stretch and a horizontal translation in one bracket is Higher Level material, at AHL 2.16. At Standard Level you will be given the transformations one at a time, in a stated order, which is exactly why the order is what gets examined.
Typing both orders as two functions and drawing them together turns the question "does the order matter" into something you can see, and it takes two entries.
When you may use it. Analysis Paper 1 is non-calculator, and sketching a transformed graph from a given one is Paper 1 work. Use the machine to check that your two orders really are different, then put it away.
The mark people lose. Doing the translation before the stretch when the equation says otherwise. 3x² + 2 has the 3 applied to the function and the 2 added afterwards, so the stretch is first. Read the expression from the outside in: the outermost operation was done last. The other half of the same mark is the sign inside the bracket, where (x − 3)² moves RIGHT; check it by asking where the bracket is zero, which takes four words and never misleads.
1. For y = 3x² + 2, find y when x = 1.
2. For y = 3(x² + 2), find y when x = 1.
3. Where is the vertex of y = (x − 3)²? Give the x value.
4. For f(x) = x², find −f(2).
5. Why do the two orders give different curves?
1 markEach transformation named, with its direction and factor.
1 markThe order, where more than one is involved.
1 markThe resulting equation, or the sketch with the image of one named point marked.
On "describe the transformation" questions the words matter: a translation needs a vector or a direction and a distance, and a stretch needs a direction and a factor. "It moves and gets bigger" earns nothing.
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