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SingaporeMaths

Functions and Graphs

8 dot points across 8 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

What are the key features of a curve, and how do asymptotes and symmetry guide a sketch?

Identify and use the key features of a curve - intercepts, turning points, asymptotes, symmetry and behaviour at infinity - to produce and interpret graph sketches
A focused answer to the H2 Mathematics outcome on curve features. Vertical, horizontal and oblique asymptotes, symmetry, behaviour at infinity, and how these features combine to determine a sketch.
9 min answer →

How do we combine functions and reverse them, and when does an inverse exist?

Form and find the domain of composite functions, determine when a composite is defined, find inverse functions and their domains, and use the graphical relationship between a function and its inverse
A focused answer to the H2 Mathematics outcome on composite and inverse functions. Forming composites and their domains, the condition for a composite to exist, finding inverses, and the reflection in y equals x.
10 min answer →

How do we recognise standard conic graphs and work with curves defined parametrically?

Recognise and sketch the standard conics (circle, ellipse, parabola, hyperbola) from their equations, and sketch and analyse curves defined parametrically
A focused answer to the H2 Mathematics outcome on conics and parametric curves. Recognising circles, ellipses, parabolas and hyperbolas from their equations, and sketching and converting curves given parametrically.
9 min answer →

What makes a relation a function, and how do its domain and range determine its behaviour?

Define a function and its domain and range, decide whether a relation is a function or one-to-one, and find the range of a given function over a stated domain
A focused answer to the H2 Mathematics outcome on functions. The definition of a function, the vertical and horizontal line tests, one-to-one functions, and finding the range from a stated domain.
9 min answer →

How do we sketch a rational function by finding its intercepts, asymptotes and turning points?

Sketch graphs of rational functions of the form a linear over linear and a quadratic over linear, finding intercepts, asymptotes, stationary points and the regions where the curve lies
A focused answer to the H2 Mathematics outcome on sketching rational functions. Finding intercepts, vertical and oblique asymptotes, stationary points, and assembling a correct sketch of linear-over-linear and quadratic-over-linear curves.
10 min answer →

How do we solve polynomial, rational and modulus inequalities reliably?

Solve quadratic, polynomial and rational inequalities algebraically and graphically, using a sign analysis and respecting the sign of any denominator
A focused answer to the H2 Mathematics outcome on inequalities. Solving quadratic and higher polynomial inequalities by sign analysis, handling rational inequalities without cross-multiplying carelessly, and reading solutions off a graph.
9 min answer →

How does the modulus function behave, and how do we solve equations and sketch graphs that involve it?

Define the modulus function, sketch graphs involving the modulus of a function, and solve equations and inequalities involving the modulus
A focused answer to the H2 Mathematics outcome on the modulus function. The definition, sketching the modulus of a linear and a curved function, and solving modulus equations and inequalities by cases and by squaring.
9 min answer →

How do translations, stretches and reflections change the graph and the equation of a function?

Relate the graph of y equals a f(b(x + c)) + d to the graph of y equals f(x) through translations, stretches and reflections, and apply combined transformations in the correct order
A focused answer to the H2 Mathematics outcome on graph transformations. Translations, stretches and reflections, the effect of each parameter, the order of combined transformations, and the effect on asymptotes and key points.
9 min answer →