Grade 9 Transformation Geometry

Grade 9 transformations: translations, reflections in the axes and y = x, rotations and enlargements on the Cartesian plane, with coordinate rules and…

Transformation geometry describes how shapes move on the Cartesian plane. In Grade 9 you translate (slide), reflect (flip), rotate (turn) and enlarge or reduce shapes, and you describe each transformation with a coordinate rule.

Learning the rules, such as (x; y) → (x; −y) for a reflection in the x-axis, makes these questions quick. The same ideas return in Grade 10 and 11 functions, where graphs are shifted and reflected.

Key concepts

  • Translation: (x; y) → (x + a; y + b)
  • Reflection in the x-axis: (x; y) → (x; −y); in the y-axis: (x; y) → (−x; y)
  • Reflection in y = x: (x; y) → (y; x)
  • Enlargement by a factor k about the origin: (x; y) → (kx; ky)
  • Translations, reflections and rotations keep size and shape; enlargements change size

Worked examples

Point B(−4; 6) is enlarged by a scale factor of ½ about the origin. Find the image.

Multiply both coordinates by ½: (−4 × ½; 6 × ½) = (−2; 3).

Point C(3; −1) is reflected in the line y = x. Find the image.

Reflecting in y = x swaps the coordinates: C′(−1; 3).

Describe the transformation that maps P(2; 5) to P′(−2; 5).

The x-coordinate changes sign and y stays the same, so it is a reflection in the y-axis.

A square with sides of 3 cm is enlarged by a factor of 2. How does its area change?

The sides become 6 cm, so the area changes from 9 cm² to 36 cm². Area is multiplied by 2² = 4.

Triangle ABC has A(1; 2), B(4; 2) and C(1; 5). It is translated 3 units left and 1 unit down. Find the image of each vertex.

The rule is (x; y) → (x − 3; y − 1). So A′(−2; 1), B′(1; 1) and C′(−2; 4). The image is the same size and shape as the original.

Common mistakes

  • Changing the wrong coordinate's sign in a reflection.
  • Thinking an enlargement by 2 doubles the area (it multiplies it by 4).
  • Translating in the wrong direction (left is negative x, down is negative y).
  • Forgetting to label the image points with a dash (A′).

Exam tips

  • Write the rule before applying it to each point.
  • Plot the original and the image to check your answer looks right.
  • For enlargements, check the distances from the origin have been multiplied by the scale factor.

Frequently asked questions

Which transformations change the size of a shape?

Only enlargements and reductions. Translations, reflections and rotations keep the same size and shape, so the image is congruent to the original.

What is the rule for reflecting in the x-axis?

(x; y) → (x; −y): the x-coordinate stays the same and the y-coordinate changes sign.

What does a scale factor between 0 and 1 do?

It reduces the shape. For example, a factor of ½ halves every length.

Snap&Learn gives South African Matric learners instant, CAPS-aligned, step-by-step Mathematics solutions. Snap or upload a question and the AI shows every method mark the way the NSC memo awards them. Your first three AI solutions are free.