Stage – 5, Equations & Non linear Relationships

Module 2 – Equations & Non‑linear Relationships

This module moves students beyond linear algebra into the world of quadratics, parabolas, and exponential growth, laying the foundation for higher mathematics and real‑world applications.

1. Quadratic Equations

• Factorisation method: Solving quadratics by expressing them as products of two binomials, e.g., x2+5x+6=(x+2)(x+3).

• Completing the square: Rewriting quadratics into perfect square form, useful for finding vertex coordinates.

• Quadratic formula: Applying −b±b2−4ac2a to solve any quadratic equation.

• Applications: Projectile motion, optimisation problems, and geometric contexts.

2. Graphing Parabolas & Interpreting Quadratic Functions

• Plotting quadratic functions to visualise their U‑shaped graphs.

• Identifying key features: vertex, axis of symmetry, intercepts.

• Understanding how coefficients affect the graph’s shape (wider/narrower, upward/downward).

• Real‑world links: modelling profit curves, trajectories, and design structures.

3. Simultaneous Equations

• Revisiting substitution, elimination, and graphical methods.

• Extending to cases where one equation is linear and the other quadratic.

• Graphical interpretation: intersection points represent solutions.

• Applications: solving real‑life problems with two constraints (e.g., cost vs. revenue, geometry intersections).

4. Exponential & Logarithmic Functions (Stage 5.3 Pathway)

• Exponential functions: Growth and decay models, e.g., population growth, radioactive decay.

• Logarithmic functions: Inverse of exponentials, solving equations like 2x=16 using logs.

• Properties: Laws of exponents and logarithms, base changes.

• Applications: Compound interest, scientific modelling, data analysis.

Learning Outcome

By the end of this Module , students will:

• Solve quadratic equations using multiple methods.

• Interpret and sketch parabolas with confidence.

• Apply simultaneous equations to both linear and non‑linear contexts.

• Understand exponential growth/decay and logarithmic relationships, preparing them for advanced pathways in senior mathematics.