Bend the curve.
Drive a parabola with a, b and c sliders, walk a point round the unit circle until it draws sin and cos, then stretch a shape and watch k² bite.
Quadratics · y = ax² + bx + c
FIG. 01 · ALGEBRA A11 / A18Three faces of the same curve
roots
vertex
factd
c.sq
Exam technique. Completing the square hands you the turning point for free: y = a(x + p)² + q has its vertex at (−p, q), and the line of symmetry is x = −b/2a either way. If the question says show that the curve never crosses the x-axis, it is asking for the discriminant — b² − 4ac < 0, one line, done.
Trigonometry · the unit circle unrolled
FIG. 02 · GEOMETRY G21 / A12Turn anticlockwise from the positive x-axis. The point on the circle is (cos θ, sin θ) — that is the whole definition.
Which one to trace
Exam technique. sin θ is a height, cos θ is a width, tan θ is the height over the width — which is why tan blows up at 90° and 270°, where the width is zero. When a question asks for all solutions in 0° to 360°, the graph tells you how many to expect before you do any algebra: sin and cos give two per cycle, tan gives one.
Similar shapes · k, k², k³
FIG. 03 · GEOMETRY G19 / R12Shape A is 3 cm by 2 cm, so 6 cm²; as a solid, take its volume as 6 cm³. B is A enlarged by k.
The classic trap. Doubling every length does not double the area. Count the tiles: k = 2 fits 4 copies of A inside B, and 8 if it were a solid. If a question gives you an area ratio and asks for a length, square-root it first — and if it gives volumes, take the cube root.