Quadratics, Trigonometry and 3D Geometry
MYP Unit Framework
Key Concept: FORM Related Concepts: Representation. Space. Measurement. Global Context: Scientific and Technical Innovation (How does mathematics describe the PHYSICAL world?) Statement of Inquiry: Mathematical FORMS — from the quadratic curve to the trigonometric ratio to three-dimensional space — provide ELEGANT MODELS that describe and predict phenomena in the natural and built worlds.
Inquiry Questions
| Type | Question |
|---|---|
| Factual | What is the quadratic formula? What are sin, cos, and tan? How do you find the volume of a cylinder? |
| Conceptual | Why do quadratic functions produce PARABOLAS — and why are parabolas so COMMON in nature and engineering? How are trigonometric ratios EXTENDED beyond right triangles? |
| Debatable | Is mathematics DISCOVERED (exists independently) or INVENTED (a human creation)? |
1. Quadratic Equations and Functions
Standard Form: ax² + bx + c = 0 (a ≠ 0)
Solving Methods
| Method | When to Use |
|---|---|
| Factorisation | Roots are rational. Split the middle term. |
| Completing the Square | Always works. Leads to the quadratic formula. |
| Quadratic Formula | x = [−b ± √(b²−4ac)] / 2a. ALWAYS works. |
The Discriminant — Δ = b² − 4ac
- Δ > 0: TWO distinct real roots. Δ = 0: ONE real root (repeated). Δ < 0: NO real roots (complex conjugates).
The Graph — The Parabola
y = ax² + bx + c is a PARABOLA. a > 0: OPENS UPWARD (∪ — minimum). a < 0: OPENS DOWNWARD (∩ — maximum). Vertex: x = −b/(2a).
Why Parabolas Matter
'The path of a THROWN BALL is a parabola. The cables of a SUSPENSION BRIDGE form a parabola. SATELLITE DISHES and HEADLIGHTS use parabolic reflectors to FOCUS signals and light. The quadratic function — simple as it is — describes some of the most important shapes in engineering and nature.'
2. Trigonometry — Triangles and Beyond
Right Triangle Trigonometry (Review)
sin θ = Opposite/Hypotenuse. cos θ = Adjacent/Hypotenuse. tan θ = Opposite/Adjacent.
The Unit Circle — Extending to ANY Angle
Place a circle of RADIUS 1 centred at the origin. Point P(x,y) on the circle at angle θ: cos θ = x. sin θ = y. tan θ = y/x. 'The Unit Circle extends trigonometry BEYOND the right triangle. It defines sin and cos for ANY angle — including negative angles and angles greater than 90°.'
The ASTC Rule (Signs by Quadrant)
I: All positive. II: Sin only. III: Tan only. IV: Cos only.
Trigonometric Identities
sin²θ + cos²θ = 1 (THE fundamental identity). tan θ = sin θ / cos θ.
Angles of Elevation and Depression
- Elevation: Looking UP from the horizontal.
- Depression: Looking DOWN from the horizontal. 'The angle of depression FROM a cliff TO a boat EQUALS the angle of elevation FROM the boat TO the cliff. This is a beautiful SYMMETRY — and it's often the KEY to solving the problem.'
3. 3D Geometry — Surface Area and Volume
Key Solids
| Solid | Surface Area | Volume |
|---|---|---|
| Prism | 2B + Ph (B=base area, P=perimeter) | Bh |
| Cylinder | 2πr² + 2πrh | πr²h |
| Pyramid | B + ½Pl | ⅓Bh |
| Cone | πr(r+l) (l=slant height) | ⅓πr²h |
| Sphere | 4πr² | (4/3)πr³ |
Why ⅓?
'The volume of a PYRAMID is ⅓ of the PRISM with the same base and height. The volume of a CONE is ⅓ of the CYLINDER. This ⅓ factor appears because the solid TAPERS — it has LESS volume at the top than at the bottom. The derivation requires calculus — but the PATTERN is elegant and worth noticing.'
4. Pythagoras in 3D
Space Diagonal of a Cuboid: d = √(l² + b² + h²)
'The diagonal runs from one corner to the OPPOSITE corner — through the interior.' 'Pythagoras in 3D is a DOUBLE application: first, diagonal of the base (√(l²+b²)). Then, with the height, the space diagonal (√((base diagonal)² + h²)).'
Your Summative Assessment
Task: 'The Roller Coaster Design Challenge' Design a section of a ROLLER COASTER using quadratic functions. Your track segment must include: a PARABOLIC DROP (y = −ax² + c). A PARABOLIC CLIMB. The equations for both. The coordinates where they MEET. A calculation of the MAXIMUM HEIGHT and the STEEPEST SLOPE (using trigonometry). 'Show that mathematics is not just about solving equations — it's about DESIGNING exciting, SAFE experiences.'
ATL Skills
| Skill | Focus |
|---|---|
| Critical Thinking | Applying abstract mathematics to real-world design. |
| Creative Thinking | Designing with mathematical functions. |
