Outside moves it up/down; inside moves it left/right: Picture the graph of a temperature model y = f(t) drawn on tracing paper. A translation slides the whole sheet without bending it.
y = f(x) + b slides it up by b (the b is outside f, so it changes the output/height).
y = f(x − a) slides it right by a (the a is inside f, with the curve where x − a behaves like the old x — counter-intuitively, minus a moves it right).
Why the inside is 'backwards': to get the same height the new graph had before, x must now be a units bigger, so every feature shifts to a larger x — i.e. to the right.
IB-style question — translate a cost model
A workshop's running cost is modelled by C = f(n) for n items made. A new tax adds a fixed $50 per day and the machine now needs 3 extra setup items before output counts.
Write the new cost g(n) as a transformation of f, and describe the slide.
Step by step
- Fixed $50 added to the cost is an outside change (height): + 50.
- Needing 3 extra items first shifts the input: replace n by (n − 3).
- Combine into one transformation.
Final answer
g(n) = f(n − 3) + 50 — a translation by the vector (3, 50): right 3, up 50.
IB-style question — find a point after a translation
The curve y = f(x) passes through the maximum point (2, 7).
The graph is translated to y = f(x + 1) − 4. Find the new position of that maximum.
Step by step
- Inside (x + 1) means a = −1, so slide LEFT 1. Outside −4 slides DOWN 4.
- Apply to the point (2, 7).
Final answer
The maximum moves to (1, 3).
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Outside multiplier = vertical; inside multiplier = horizontal; minus = flip: Now instead of sliding, we scale the tracing paper.
y = p·f(x) stretches it vertically by factor p — every height is multiplied by p (the x-axis stays fixed; points on it don't move).
y = f(qx) stretches it horizontally by factor 1/q — the curve is squashed toward the y-axis when q > 1 (the y-axis stays fixed).
y = −f(x) reflects it in the x-axis (flip up↔down). y = f(−x) reflects it in the y-axis (flip left↔right).
Think of −1 as a stretch of factor −1: it scales and flips.
IB-style question — scale a population model
A bacterial count is modelled by y = f(t). In a richer broth the count is everywhere 3 times higher, and the colony reacts twice as fast (events happen in half the time).
Write the new model and name the two transformations.
Step by step
- '3 times higher' scales the output → vertical stretch, factor 3 (outside).
- 'Twice as fast' compresses time by ½ → replace t by 2t (inside).
- Combine.
Final answer
y = 3 f(2t): a vertical stretch of factor 3 and a horizontal stretch of factor ½ (a squash).
Animated graph
Watch the graph build step by step in study mode.
IB-style question — effect on a key point
y = f(x) has a y-intercept at (0, 6).
Where is the y-intercept of y = −½ f(x)?
Step by step
- An outside factor (here −½) scales the height; the x-coordinate of the y-intercept (x = 0) is unchanged.
- Multiply the height by −½.
Final answer
The y-intercept moves to (0, −3) — reflected and halved.