Reflections (VCE SSCE Mathematical Methods): Revision Notes

Reflections

Introduction

Reflections are transformations that flip a graph across a line called the mirror line. In this note, we focus on reflections in the $x$-axis and $y$-axis only.

When we reflect a graph, each point on the original graph is mapped to a new position on the opposite side of the mirror line. The distance from the point to the mirror line remains the same, but the point appears on the other side.

Let's explore what happens to a general point $(x, y)$ when we reflect it in each axis.

Reflection in the x-axis

When we reflect a graph in the $x$-axis, the $x$-axis acts as the mirror line. This means that each point flips vertically across the horizontal axis.

The transformation rule

A reflection in the $x$-axis follows this mapping rule:

$$(x, y) \to (x, -y)$$

This can also be written as:

$$x' = x \text{ and } y' = -y$$

What this means: The x-coordinate stays exactly the same, but the y-coordinate changes sign (positive becomes negative, and vice versa).

For example, the point $(1, 1)$ maps to $(1, -1)$ when reflected in the $x$-axis.

Effect on functions

For a function with equation $y = f(x)$, there are two equivalent ways to describe a reflection in the $x$-axis:

• Apply the transformation rule $(x, y) \to (x, -y)$ to every point on the graph
• Replace $y$ with $-y$ in the equation to get $y = -f(x)$

Both methods produce the same result.

Notice how the reflected graph is upside down compared to the original. Every point that was above the $x$-axis is now below it, and at the same distance from the axis.

Reflection in the y-axis

When we reflect a graph in the $y$-axis, the $y$-axis acts as the mirror line. This means that each point flips horizontally across the vertical axis.

The transformation rule

A reflection in the $y$-axis follows this mapping rule:

$$(x, y) \to (-x, y)$$

This can also be written as:

$$x' = -x \text{ and } y' = y$$

What this means: The y-coordinate stays exactly the same, but the x-coordinate changes sign (positive becomes negative, and vice versa).

For example, the point $(1, 1)$ maps to $(-1, 1)$ when reflected in the $y$-axis.

Effect on functions

For a function with equation $y = f(x)$, there are two equivalent ways to describe a reflection in the $y$-axis:

• Apply the transformation rule $(x, y) \to (-x, y)$ to every point on the graph
• Replace $x$ with $-x$ in the equation to get $y = f(-x)$

Both methods produce the same result.

Notice how the reflected graph appears on the left side of the $y$-axis. Every point that was to the right of the $y$-axis is now to the left, at the same distance from the axis.