Topic 3: Functions and Sketching Graphs

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

How do the zeros of a numerator and denominator dictate the shape of a rational function's graph?

Sketch graphs of rational functions, identifying intercepts, vertical asymptotes, and horizontal or oblique asymptotes.

Sketching rational functions by locating intercepts and vertical asymptotes from the denominator, and finding horizontal or oblique asymptotes from the degrees of numerator and denominator.

9 min read →

How does taking the absolute value reshape a graph, and how do we solve equations and inequalities involving it?

Sketch graphs involving the modulus (absolute value) function and solve equations and inequalities containing modulus expressions.

How the modulus reflects negative parts of a graph above the axis, the difference between |f(x)| and f(|x|), and solving modulus equations and inequalities by cases or squaring.

8 min read →

When can a function be reversed, and how do composition and inversion interact with domain and range?

Form composite functions and find inverse functions, attending to domain and range and the condition for an inverse to exist.

Building composite functions and tracking their domains, the one-to-one condition for an inverse to exist, finding inverse functions algebraically, and the reflection relationship in y equals x.

8 min read →