Spot each piece: y = a·f(x) + k combines a vertical stretch by a with a vertical translation k.
Inside changes (x − h, or bx) handle horizontal moves and stretches.
Identify each piece separately.
IB-style question — name the parts
Describe the transformations in y = 2f(x) + 3.
Step by step
- Outside ×2 then + 3.
Final answer
A vertical stretch of factor 2, followed by a translation 3 up.
Combined transformations stack up: do the stretch / reflection (the a) first, then the translation (h, k). Each chip shows the finished curve against the dashed base.
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Outside acts on y, inside on x: Read the outside operations (stretch/reflect, then shift) on the y-values, and inside operations on the x-values.
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Stretch/reflect before translate: For the y-side (a·f(x) + k), apply the stretch/reflection first, then the translation: multiply the y-coordinate by a, then add k.
Doing it the other way gives the wrong image.
IB-style question — apply in order
y = f(x) passes through (1, 4).
Find its image on y = 2f(x) − 1.
Step by step
- Stretch first: ×2.
- Then translate: − 1.
Final answer
(1, 7).
Don't add before you stretch: 2f(x) − 1 is not 2(f(x) − 1).
Multiply by 2 first, then subtract 1.
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Transform the coordinates: Track a point by applying each rule to the right coordinate: horizontal pieces change x, vertical pieces change y (stretch/reflect, then shift).
IB-style question — full combination
y = f(x) passes through (2, 3).
Find its image on y = f(x − 1) + 5.
Step by step
- x − 1 → right 1 (add 1 to x).
- + 5 → up 5 (add 5 to y).
Final answer
(3, 8).
Vertical stretch scales the y first: If there's also a stretch, e.g. 3f(x − 1) + 5, do ×3 on the y before the + 5.
Name each step clearly: When asked to describe a combined transformation, name each move precisely: e.g. "a reflection in the x-axis, then a translation 4 up" — using the right word (stretch / reflection / translation) and direction/factor.
IB-style question — describe it
Describe fully the transformations in y = f(x − 2) + 5.
Step by step
- Inside x − 2 → right 2.
- Outside + 5 → up 5.
Final answer
A translation 2 to the right and 5 up (vector (2, 5)).
Use the exam vocabulary: Marks need the precise words: translation, stretch (scale factor …), reflection (in the x-/y-axis) — plus the amount/direction.