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Achievement objective M8-7

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • form and use trigonometric, polynomial, and other non-linear equations.


  • Solves contextual problems by forming and solving equations.
  • Solves quadratic and cubic equations with complex roots. These equations are likely to involve real coefficients so that solutions are complex conjugate pairs.
  • Analyses the existence of solutions in the context of the situation.
  • Non-linear equations include:
    • exponential
    • logarithmic
    • rational
    • surd
    • hyperbolic.
  • Makes links to graphs of functions M8-2, curve fitting M8-4, manipulating expressions M7-6 and M8-6, and complex numbers M8-9.
  • See key mathematical ideas on NZmaths.


M8-7 links from M7-4, M7-7 and to M8-9.

Possible context elaborations

  • Natural phenomena that result in periodic behaviours (for example, tidal patterns, sound waves, circular motion, harmonics …).
  • Explore growth and decay situations.
  • Completely factor a third degree polynomial, using long division and/or reminder and factor theorems.
  • Given general solution for a trig equation, find the solutions that satisfy a particular context.
  • Although numerical solutions (bisection and Newton-Raphson) do not need to be included, they could form part of an inquiry into dealing with non-linear equations that can’t be solved with algebraic methods.
  • Activity: Big Bang Theory – Exact values
  • Graphing projectile paths
  • Investigating algebra: Exploring graphs
  • Practical situations involving trigonometric graphs: Includes pedal height of a bicycle, height of loaded oscillating spring.
  • Tidal estuaries: Investigating the times yachts can enter and leave tidal estuaries.
  • Use of remainder and factor theorems.

Assessment for qualifications

NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.

The following achievement standard(s) could assess learning outcomes from this AO:

Last updated September 17, 2018