Transformations

1) Hook — A Fun Real-Life Example

Imagine you are watching a thrilling cricket match at the iconic Wankhede Stadium in Mumbai. The batsman hits the ball, and it bounces off the pitch, changes direction, and zooms towards the boundary. This change in the ball’s path is like a transformation in mathematics — the ball's position changes but its shape and size remain the same. Just like the ball moves, shapes in math can move, flip, slide, or turn on a plane. Let’s explore these magical movements called transformations!

2) Core Concepts — What Are Transformations?

In mathematics, a transformation changes the position or orientation of a shape on a plane. The shape may move, flip, or turn, but its size and shape stay the same (except for some special cases). There are four main types of transformations:

Translation in Detail

Translation moves every point of a shape the same distance in the same direction. If a point P(x, y) moves a units right and b units up, the new point P'(x', y') is:

Example: If a point is at (3, 2) and translated 4 units right and 3 units up, the new point is (7, 5).

Reflection in Detail

Reflection flips a shape over a mirror line (axis). Common reflection lines are:

• x-axis: (x, y) → (x, -y)
• y-axis: (x, y) → (-x, y)
• y = x: (x, y) → (y, x)

Example: Reflecting point (5, 3) over the y-axis gives (-5, 3).

Rotation in Detail

Rotation turns a shape around a fixed point (usually the origin) by 90°, 180°, or 270° counterclockwise:

Example: Rotate (3, 4) by 90° → (-4, 3).

Dilation (Scaling) in Detail

Dilation changes the size of a shape by a scale factor k, keeping the shape similar. Coordinates multiply by k:

Example: Dilate (2, 3) by scale factor 2 → (4, 6).

3) Key Formulas / Rules

4) Did You Know?

In Bollywood movies, especially in dance sequences, the dancers often perform moves that are like rotations and reflections on stage! The famous “mirror dance” is a perfect example of reflection, where dancers mimic each other's movements as if looking into a mirror. Maths is everywhere — even in your favourite movies!

5) Exam Tips

• Label points carefully: Always write the coordinates before and after transformation clearly.
• Check direction of rotation: Rotation is usually counterclockwise unless told otherwise.
• Use grid paper: Drawing shapes on graph paper helps visualize transformations accurately.
• Don’t confuse reflection lines: Remember the rules for reflecting over x-axis and y-axis.
• Practice coordinate rules: Most exam questions require applying formulas to find new coordinates.
• Board exam pattern: Questions may ask you to identify type of transformation, write coordinates after transformation, or draw the transformed shape.