Topic 1 of 8 · Form 5
Pure 1: quadratics and functions
Completing the square, the discriminant (b² − 4ac), solving quadratic equations and inequalities, equations reducible to quadratics (e.g. x⁴ − 5x² + 4 = 0), and intersections of lines and curves.
Notes for this topic 1 file, in order
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Quadratics
Completing the square, the discriminant (b² − 4ac), solving quadratic equations and inequalities, equations reducible to quadratics (e.g. x⁴ − 5x² + 4 = 0), and intersections of lines and curves.
Worked example: Find the set of values of k for which y = x² + kx + 4 lies entirely above the x-axis.
No real roots: k² − 16 < 0 → −4 < k < 4
Functions
Domain and range; one-one functions and inverses; composite functions; graphs of f and f⁻¹ reflected in y = x; transformations y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and reflections.
Worked example: f(x) = (x − 2)² + 1 for x ≥ 2. Find f⁻¹(x) and state its domain.
y − 1 = (x − 2)² → x = 2 + √(y − 1) → f⁻¹(x) = 2 + √(x − 1), x ≥ 1
Check yourself
- Write 2x² − 12x + 7 in the form a(x + b)² + c.
- Describe the transformation from y = f(x) to y = f(2x).
- Solve x⁴ − 13x² + 36 = 0.
Show answers
- 2(x − 3)² − 11
- Stretch parallel to the x-axis, scale factor ½.
- x² = 4 or 9 → x = ±2, ±3
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