FREE LEARNING SERIES
Functions, models and change
Move from domains and quadratic models through growth and trigonometry to derivatives and optimisation.
5 lessons · Learn at your own pace
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Try the exercise before reading its explanation. Answer the self-check questions in your own words, then compare your reasoning with the answers. Revisit any difficult section before moving on. No account or payment is needed.
- 01
Functions, domains and composition
A formula becomes useful when you know which inputs it accepts, what its outputs mean, and how it connects to another rule. This lesson develops those three ideas together, using algebra and an invented printing example.
Learning outcome: Find restrictions before simplifying, calculate compositions in the correct order, and explain when an inverse function exists.
Open lesson → - 02
Quadratic equations and models
Products of changing lengths and models of accelerating motion lead to squared terms. Learn to solve the resulting equations, connect their roots to a graph, and decide which mathematical answers fit the original situation.
Learning outcome: Choose a solution method, explain the discriminant, find an extreme value, and reject a root only for a stated mathematical or contextual reason.
Open lesson → - 03
Sequences and compound growth
Adding a fixed amount and multiplying by a fixed factor produce very different patterns. Learn to model both, track when counting begins, and calculate totals without confusing a final term with a sum.
Learning outcome: Build recursive and explicit rules, sum finite sequences, compare linear and compound change, and check the assumptions behind a projection.
Open lesson → - 04
Trigonometry and measurement
A distance and an angle can reveal a height that cannot be measured directly. The calculation works because similar triangles preserve ratios, but a correct answer also requires a clear diagram, consistent angle units and realistic measurement precision.
Learning outcome: Choose a trigonometric ratio, solve one- and two-position height problems, convert degrees and radians, and explain how measurement errors affect a result.
Open lesson → - 05
Derivatives and optimisation
An average rate describes change across an interval. A derivative describes the limiting rate near one input. This connection lets us model motion, describe a curve locally, and find the best feasible dimensions of a design.
Learning outcome: Derive a simple derivative from first principles, differentiate polynomials, interpret rates with units, and solve a constrained optimisation problem with a justified maximum.
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