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).
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).
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.