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VCE Mathematical Methods (Year 12) · Functions, Relations and Graphs · 19m read

Functions, Domain and Transformations

Ever wonder why some functions only work for certain numbers? That's where domain comes in—it's simply the set of all inputs your function can actually handle. Understanding domain, range (the outputs you get), and how to shift, flip, and stretch graphs onto each other is like learning the alphabet before writing essays. Master these fundamentals and you'll breeze through Methods questions, saving yourself precious time whenever functions appear.

Learning objective: State the domain and range of a function and describe any sequence of dilations, reflections and translations that maps one graph onto another.

By the end of this lesson you can:

  • I can write domain and range using interval notation.
  • I can apply a composition of dilations, reflections and translations in the correct order.
  • I can find the image of a point after a transformation.

Transformations are photo edits on a graph. Stretch, flip and drag the parent graph — the picture is the same, just edited.

What are functions and how do transformations work

A function f maps each x in its domain to exactly one y in the range. The "maximal domain" is the largest set of real x-values for which the rule is defined — you exclude anything that makes the denominator zero or forces the square root of a negative number.

Transformations come in three families. Dilations stretch or compress the graph either from an axis. Reflections flip the graph across an axis. Translations shift the graph horizontally or vertically. The order they are applied matters: "DRT" (Dilate, Reflect, Translate) is the safe default order, matching how transformations stack inside a general equation y = a·f(b(x − h)) + k.

Transformation cheat sheet

  • Dilation factor a (y-axis): y = a·f(x) stretches vertically by factor a.
  • Dilation factor 1/b (x-axis): y = f(b·x) compresses horizontally by factor b.
  • Reflection: y = −f(x) reflects in x-axis; y = f(−x) reflects in y-axis.
  • Translation: y = f(x − h) + k shifts right h, up k.
  • Domain rules: Exclude roots of denominators and negatives under even roots.

Quick reference

| y = a f(n(x − h)) + k | Effect |
|---|---|
| a | Dilation from x-axis (reflect if a < 0) |
| n | Dilation from y-axis by 1/n |
| h | Translate right h |
| k | Translate up k |
| Domain/range | Track through transformations |

Worked example (Medium)

Describe a sequence of transformations that maps y = x² onto y = −2(x − 3)² + 5, and state the image of (0, 0).

  1. Identify a, h, k. Match to y = a(x − h)² + k. a = −2, h = 3, k = 5
  2. Dilate. Dilation by factor 2 from the x-axis. (x, y) → (x, 2y)
  3. Reflect. Reflect in the x-axis (because a is negative). (x, y) → (x, −y)
  4. Translate. Shift right 3, up 5. (x, y) → (x + 3, y + 5)
  5. Apply to (0, 0). Track the point through the sequence. (0,0) → (0,0) → (0,0) → (3, 5)

Answer: Dilate ×2 from the x-axis, reflect in the x-axis, translate +3 right and +5 up. The image of (0,0) is (3, 5).

Worked example (Hard)

The graph of $y = f(x)$ is transformed by the rule $T(x, y) = (x/3 - 2,; -2y + 1)$. Express the transformed function in the form $y = a,f(b(x-h)) + k$ and identify $a, b, h, k$.

  1. Read the rule. Image $(x', y') = (x/3 - 2,; -2y + 1)$.
  2. Invert to express $x, y$ in terms of $x', y'$. Solve $x' = x/3 - 2$ for $x$ and $y' = -2y + 1$ for $y$. x = 3(x' + 2),; y = (1 - y')/2
  3. Substitute into $y = f(x)$. Replace. (1 - y')/2 = f(3(x' + 2))
  4. Solve for $y'$. Rearrange. y' = 1 - 2 f(3(x' + 2)) = -2 f(3(x' + 2)) + 1
  5. Match form. Compare with $y = a f(b(x-h)) + k$. a = -2,; b = 3,; h = -2,; k = 1

Answer: Transformed function $y = -2 f(3(x + 2)) + 1$, so $a = -2$, $b = 3$, $h = -2$, $k = 1$.

Common mistakes

  • Applying translations before dilations or reflections The order DRT (Dilate, Reflect, Translate) matches the general form $y = a \cdot f(b(x - h)) + k$. Reversing the order gives incorrect image coordinates because translations shift the reference point for subsequent transformations. Fix: Always apply transformations in DRT order. If the question gives a different order, rewrite the equation step by step to confirm each mapping.
  • Writing the horizontal dilation factor as $b$ instead of $\frac{1}{b}$ In $y = f(bx)$, the graph is compressed by factor $b$, meaning the dilation factor from the $y$-axis is $\frac{1}{b}$, not $b$. Students often confuse the coefficient with the scale factor. Fix: State: "dilation of factor $\frac{1}{b}$ from the $y$-axis". For example, $y = f(2x)$ is a dilation of factor $\frac{1}{2}$ from the $y$-axis.
  • Confusing domain restrictions for square roots vs. logarithms Square roots require the radicand $\geq 0$ (allowing equality), while logarithms require the argument $> 0$ (strict inequality). Mixing up the boundary loses a mark. Fix: Use $\geq 0$ for even roots and $> 0$ for logarithms. Write the domain in interval notation with square bracket for $\geq$ and round bracket for $>$.

Matrix representation of transformations and commutativity

In Specialist Mathematics, every affine transformation $y = a \cdot f(b(x - h)) + k$ can be encoded as a matrix equation. Setting $(x', y') = (bx + h,\, ay + k)$ leads to the mapping $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} b & 0 \\ 0 & a \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h \\ k \end{pmatrix}$. The $2 \times 2$ matrix handles dilations and reflections simultaneously (a negative entry on the diagonal encodes a reflection), while the column vector handles translations.

An important consequence is that dilations about different axes commute with each other — a vertical dilation followed by a horizontal dilation gives the same graph regardless of order. However, a dilation and a translation in the same direction do not commute: dilating then translating differs from translating then dilating. This is precisely why DRT order works — it matches the algebraic nesting of the general equation and avoids the trap of composing non-commutative operations in the wrong sequence.

Practice questions

  1. State the maximal domain of f(x) = √(2x − 6).

    1. [3, ∞)
    2. (−∞, 3]
    3. ℝ
    4. (3, ∞)

    (2 marks)

  2. The graph of y = f(x) is transformed to y = 3f(x + 2) − 1. Which is NOT part of this transformation?

    1. Reflection in the x-axis
    2. Dilation of factor 3 from the x-axis
    3. Translation 2 units left
    4. Translation 1 unit down

    (3 marks)

  3. The point (2, 5) lies on y = f(x). Find its image on y = −f(x − 1) + 4.

    (3 marks)

  4. State the maximal domain of $g(x) = \dfrac{1}{\sqrt{9 - x^2}}$.

    (3 marks)

Transformation Rules

  • General form: y = a \cdot f(b(x-h)) + k (DRT order)
  • Vertical dilation: (x, y) \to (x, ay) (Factor $a$ from $x$-axis)
  • Horizontal dilation: (x, y) \to (x/b, y) (Factor $1/b$ from $y$-axis)
  • Translation: (x, y) \to (x+h, y+k) (Right $h$, up $k$)

Summary

  • Domain: all valid input $x$ values. Range: all output $y$ values.
  • Maximal domain: exclude values making denominator 0 or radicand negative.
  • Transformations: $y = af(b(x-h))+k$. Apply in DRT order: Dilate, Reflect, Translate.
  • Vertical dilation factor $a$ from $x$-axis; horizontal dilation factor $1/b$ from $y$-axis.
  • Image of point: apply each transformation step by step to the coordinates.

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