Reflections in Standard Lines | Transformations

🎯 Learning Objectives

reflect figures in the x-axis and y-axis;
reflect in lines x=a and y=b;
use equal perpendicular distances;
find a mirror line from corresponding points.

Introduction

This lesson develops Reflections in Standard Lines from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

Reflection Principle

The mirror line is the perpendicular bisector of the segment joining each point to its image.

x-axis

(x,y)→(x,−y)

y-axis

(x,y)→(−x,y)

x=a

Horizontal coordinates lie equally far from a.

y=b

Vertical coordinates lie equally far from b.

Worked Example

Reflect P(5,−2) in x=1.

P is 4 units right of x=1, so P′ is 4 units left: P′=(−3,−2) .

Practice Questions

Reflect (3,−5) in the x-axis.
Reflect (−4,2) in the y-axis.
Reflect (6,1) in x=2.
A(1,3) maps to A′(7,3). Find the mirror line.

✅ Check Your Work

Show answers and working

(3,5).
(4,2).
The point is 4 units right of x=2, so image is (−2,1).
The midpoint x-coordinate is (1+7)/2=4, so mirror line x=4.

🧠 Learn It Visually: Reflections in Standard Lines

A reflection is a flip across a mirror line. The mirror line is the perpendicular bisector of every segment joining a point to its image.

In the x-axis

(x,y) → (x,−y). The x-coordinate stays; the y-sign changes.

In the y-axis

(x,y) → (−x,y). The y-coordinate stays; the x-sign changes.

In x=a or y=b

Count equal perpendicular distances on opposite sides of the mirror line.

🎛️ See the Transformation Happen

Select an option and watch the object change. Compare orientation, size and position.

Choose a transformation.

Think Like a Mathematician: Which points remain fixed when a shape is reflected in the x-axis?

🧷 Remember

Points on the mirror line do not move. They are invariant points.

⚠️ Common Mistake

Measuring diagonally from the mirror line instead of perpendicularly.

🎯 Exam Tip

Name the exact mirror line, such as y=0, x=−2 or the x-axis.

Common Mistakes

Choosing a rule before identifying what the question describes.
Skipping working or changing notation part-way through a solution.
Accepting an answer without checking its size, sign or units.

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary

Understand the meaning of each idea before selecting a method.
Use definitions, properties and notation consistently.
Show working and check that the final result is sensible.