VCE Methods Transformations: A Step-by-Step Guide to Mapping Notation

Transformations appear in almost every VCE Methods exam, often worth easy marks for students who are systematic and lost marks for those who guess. This guide gives you a method that works every time, without technology.

The six transformations

Starting from y = f(x):

Transformation	New rule	Point (x, y) moves to
Dilation by factor a from the x-axis	y = a f(x)	(x, ay)
Dilation by factor b from the y-axis	y = f(x/b)	(bx, y)
Reflection in the x-axis	y = −f(x)	(x, −y)
Reflection in the y-axis	y = f(−x)	(−x, y)
Translation h units right	y = f(x − h)	(x + h, y)
Translation k units up	y = f(x) + k	(x, y + k)

Changes inside the brackets act on x and feel backwards: x − 2 moves the graph right. Changes outside act on y and behave as expected.

Applying transformations in order

The graph of y = x² is dilated by factor 3 from the x-axis, reflected in the y-axis, then translated 2 units right and 1 unit down.

• Dilation: y = 3x²
• Reflection in the y-axis: y = 3(−x)² = 3x², unchanged because the parabola is symmetric about the y-axis
• Translation: y = 3(x − 2)² − 1

Working backwards with mapping notation

The harder exam question asks you to find the transformations. The reliable method: take a point (x, y) on the original graph and its image (X, Y) on the new graph, then match.

Find a mapping from y = √x to y = 2√(3 − x) + 1.

• The image satisfies Y = 2√(3 − X) + 1.
• Match the inside of the root: x = 3 − X, so X = 3 − x.
• Match the outside: Y = 2y + 1.
• Mapping: (x, y) → (3 − x, 2y + 1)

In words: reflect in the y-axis, translate 3 units right, dilate by factor 2 from the x-axis, then translate 1 unit up.

A second example

Find a mapping from y = x³ to y = −(2x − 4)³ + 1.

• Rewrite as y = −(2(x − 2))³ + 1.
• Inside: x = 2X − 4, so X = x/2 + 2.
• Outside: Y = −y + 1.
• Mapping: (x, y) → (x/2 + 2, −y + 1): dilate by factor ½ from the y-axis, reflect in the x-axis, translate 2 right and 1 up.

Mistakes that cost marks

• Translating the wrong way because of the sign inside the bracket
• Forgetting to factor out the coefficient of x before reading a horizontal translation (2x − 4 is a translation of 2, not 4)
• Applying transformations out of the order given
• Describing a dilation without saying "from the x-axis" or "from the y-axis"

Practice

1. Find the image of (2, 5) under (x, y) → (−x + 1, 3y − 2). Answer: (−1, 13)
2. The graph of y = 1/x is dilated by factor 2 from the y-axis, then translated 3 units up. Answer: y = 2/x + 3

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