Chapter 1

Quadratics and Functions (Pure 1)

Completing the square, the discriminant, and functions with their domains, ranges, inverses and transformations — the toolkit the whole of Pure 1 is written on top of.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.4 ความสัมพันธ์และฟังก์ชัน (เพิ่มเติม) — A Level เน้นการแปลงกราฟและการระบุโดเมนที่ทำให้ฟังก์ชันผกผันมีจริง
Common Core (US): A-SSE, A-REI, F-IF, F-BF
Cambridge: 9709 Pure Mathematics 1 — Quadratics; Functions

1.1 Quadratics and the discriminant

Completed-square form

a x^{2} + b x + c = a (x + \frac{b}{2 a})^{2} + (c - \frac{b ^{2}}{4 a})

Worth deriving once rather than memorising: it is the source of the vertex, the range, the quadratic formula and the discriminant, all four.

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What the discriminant $b^{2} - 4 a c$ decides
- positive — two distinct real roots, the curve crosses the axis twice
- zero — one repeated root, the curve touches the axis
- negative — no real roots, the curve misses the axis entirely
- and for a line and a curve: cuts twice, is a tangent, or misses

ตัวอย่าง

The line $y = m x + 3$ is a tangent to the curve $y = x^{2} + 5 x + 7$ . Find the possible values of $m$ .

วิธีทำ

1.Equate: $x^{2} + (5 - m) x + 4 = 0$
2.Tangent means the discriminant is zero: $(5 - m)^{2} = 16$
3. $5 - m = \pm 4$

ตอบ $m = 1$ or $m = 9$

เคล็ดลับ

"Show that the curve and the line do not meet" is answered by producing a negative discriminant and saying so. It is not answered by trying values, and a page of substitutions earns nothing.

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