Functions (ME-F1, ME-F2)

5 dot points across 5 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does the multiplicity of a root affect the graph of a polynomial near that root?

Sketch polynomial functions using leading-term behaviour, intercepts and the multiplicity of each root

Graphing polynomials. End behaviour from the leading term, the role of root multiplicity (cross, touch, inflection), y-intercept and turning points, with worked sketches.

14 min read →

How do parametric equations describe a curve, and how do we convert between parametric and Cartesian forms?

Sketch curves given parametrically, eliminate the parameter to obtain Cartesian equations, and use parametric form for circles, parabolas and lines

Parametric equations. Eliminating the parameter, sketching parametric curves, and standard parametrisations of lines, circles and parabolas, with worked examples.

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How do we solve polynomial and rational inequalities using sign analysis?

Solve polynomial and rational inequalities by factoring and analysing the sign of each factor across critical values

Solving polynomial and rational inequalities. Sign tables, critical values, multiplicity behaviour, and the special care needed when a denominator is involved, with worked examples.

14 min read →

How do we divide one polynomial by another and use the remainder and factor theorems to find roots?

Apply the division algorithm for polynomials and use the remainder and factor theorems to identify and verify factors and roots

Polynomial division. The division algorithm, the remainder theorem, the factor theorem, and using these to factorise cubics and quartics, with worked examples.

15 min read →

How do the coefficients of a polynomial relate to the sum and product of its roots?

Use the relationships between roots and coefficients (Vieta's formulas) for polynomials of degree two, three and four

Relationships between roots and coefficients. Sum and product of roots, sum of roots taken in pairs, and applications to building polynomials from given root conditions, with worked examples.

13 min read →