Common mistake: Swapping domain and range

Swapping domain and range — domain is always the input (x), range is the output (y)

Common mistake: Forgetting that a hyperbola y = 1/(x−h) + k has asymptotes at x = h and y = k, not x = 0 and y = 0

Forgetting that a hyperbola y = 1/(x−h) + k has asymptotes at x = h and y = k, not x = 0 and y = 0

Common mistake: Applying transformations in the wrong order

Applying transformations in the wrong order — dilations before translations when reading from the rule

Common mistake: Assuming every relation is a function

Assuming every relation is a function — always check with the vertical line test

VCE Units 1–4 · Maths Methods

VCE Maths Methods — Functions and Graphs

Functions and graphs underpin almost every other topic in Mathematical Methods. A solid understanding of domain, range and transformations is essential before tackling calculus. This topic covers the main function types you need to know for SACs and the final exam, along with how transformations shift, reflect and scale their graphs.

Key Concepts & Formulas

A function assigns exactly one output to each input — use the vertical line test to check a graph is a function
Domain is the set of valid inputs (x-values); range is the set of outputs (y-values)
Key function types: linear y = mx + c, quadratic y = ax² + bx + c, cubic, hyperbola y = 1/x, truncus y = 1/x², square root y = √x, absolute value y = |x|
Asymptotes: hyperbola has vertical asymptote x = 0 and horizontal asymptote y = 0 (shifted by translations)
Transformations — dilation by factor a from x-axis: y = af(x); dilation by factor b from y-axis: y = f(x/b)
Reflection in x-axis: y = −f(x); reflection in y-axis: y = f(−x)
Translation: y = f(x − h) + k shifts right by h and up by k
Inverse function f⁻¹(x): reflect the graph in the line y = x; swap x and y to find the rule
For f⁻¹ to exist as a function, f must be one-to-one (restrict domain if necessary)

Practice Questions

5 questions

Attempt each question before reading the hint. These are styled to match VCE exam format.

Q1.State the domain and range of f(x) = √(x − 3) + 1.

2 marks

Q2.The graph of y = x² is transformed to y = −2(x + 1)² + 3. Describe the sequence of transformations applied.

3 marks

Q3.Find the rule and domain of the inverse function of f(x) = 2x − 5, x ∈ ℝ.

2 marks

Q4.Sketch the graph of y = 1/(x − 2) − 1, showing all asymptotes and axis intercepts.

3 marks

Show hint

Find where the graph cuts each axis by substituting x = 0 and y = 0 separately.

Q5.For what values of x is the function f(x) = √(9 − x²) defined? State the range.

2 marks

Common Mistakes to Avoid

These are the errors that VCE students most frequently make in Functions and Graphs — and that examiners are specifically watching for.

Swapping domain and range — domain is always the input (x), range is the output (y)
Forgetting that a hyperbola y = 1/(x−h) + k has asymptotes at x = h and y = k, not x = 0 and y = 0
Applying transformations in the wrong order — dilations before translations when reading from the rule
Assuming every relation is a function — always check with the vertical line test

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