Year 12: Algebra

6 dot points across 6 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we decide whether a real situation is best modelled by a linear, quadratic, exponential or reciprocal function?

Compare linear and non-linear models of real-world data and select the most appropriate model

Selecting an appropriate model. The four model shapes shown side by side, the difference-ratio-product diagnostic tests, when a non-linear model fits better than a straight line, and worked Australian examples for choosing the right model.

12 min read →

How are exponential functions used to model growth, decay and compound processes in the real world?

Model practical problems with exponential functions of the form

y = a b^x

and interpret growth, decay and asymptotes

Exponential modelling. Growth and decay, the base b and rate, the horizontal asymptote, identifying an exponential from a constant ratio, sketching the curve stage by stage, and applications including compound interest, population growth, half-life and depreciation with worked examples.

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How are quadratic functions used to model real-world situations such as projectile motion and maximum revenue?

Model practical situations with quadratic functions and find maximum or minimum values, intercepts and zeros

Quadratic models. Standard form, finding the vertex, intercepts and zeros, identifying a quadratic from a constant second difference, sketching the parabola stage by stage, and applying quadratics to projectile motion, maximum revenue and area problems with worked examples.

14 min read →

How are reciprocal functions used to model inverse variation, and what does the graph look like?

Model practical problems involving reciprocal functions and inverse variation of the form

y = \frac{k}{x}

Reciprocal functions and inverse variation. Finding the constant of proportionality, graphing y = k/x stage by stage, identifying inverse variation from a constant product, the two asymptotes, and applying to speed-time, pressure-volume and shared-cost problems with worked examples.

12 min read →

How are linear cost and revenue functions used to model break-even points and profit?

Model practical problems with linear cost and revenue functions and find the break-even point

Linear modelling and break-even analysis. Fixed and variable costs, the revenue and profit functions, the break-even point built up stage by stage on labelled axes, the loss and profit regions, the round-up edge case, and worked Australian small-business examples.

13 min read →

How are simultaneous linear equations solved algebraically and graphically, and how are they used to model practical situations?

Solve a pair of simultaneous linear equations graphically and algebraically, and use simultaneous equations to model practical situations

Simultaneous linear equations. Substitution and elimination, the graphical intersection method shown stage by stage, what the crossing point means, the parallel and identical edge cases, and worked Australian comparison and break-even examples.

13 min read →