Move the function.
Run all four transformations of y = a f(b(x + c)) + d at once, shrink h until a chord becomes a tangent, then shade probability under a normal curve.
Transformations · y = a f(b(x + c)) + d
FIG. 01 · PURE 2.9 / 2.10Base function f(x)
Where every point goes
(x, y) ↦ (x, y)
Exam technique. Inside the bracket the transformations are the ones that feel backwards: b stretches x by scale factor 1/b, and +c moves the curve left c. Outside the bracket they behave: ×a stretches y by a, +d moves it up d. Do the inside ones first, then the outside — and if a or b is negative, note the reflection separately (b < 0 in the y-axis, a < 0 in the x-axis).
Differentiation · watch h → 0
FIG. 02 · PURE 7.1 / 7.2Curve
Slide right to shrink h from 2 down to 0.001. The gold chord runs from (p, f(p)) to (p + h, f(p + h)); the dashed line is the true tangent.
Exam technique. "From first principles" means writing the quotient [f(x + h) − f(x)] / h, expanding, cancelling the h, then letting h → 0 — you must cancel before you take the limit, or you are dividing by zero. For y = x² the quotient tidies to 2x + h, and the h you are watching shrink here is exactly the term that vanishes.
Normal distribution · shade the area
FIG. 03 · STATISTICS 4.1 / 4.2Total area stays 1. A bigger σ does not add probability — it spreads the same probability thinner.
Jump the bounds to
Exam technique. Standardise with z = (x − μ) / σ and the numbers stop mattering: μ ± σ always holds about 68.3% of the data, μ ± 2σ about 95.4%, μ ± 3σ about 99.7%. Sketch the curve and shade before you touch the calculator — most lost marks here are the wrong tail, not the wrong arithmetic.