Reflections in Axes Revision Notes for HSC SSCE Mathematics Advanced

Reflections in Axes

What you'll learn

By the end of this note, you will be able to:

Describe how replacing x with -x in $y = f(x)$ creates a reflection in the y-axis
Describe how replacing y with -y in $y = f(x)$ creates a reflection in the x-axis
Find the equation of a function after reflection in either axis
Sketch functions and their reflections, identifying key points and features

What is a reflection?

A reflection is a type of transformation that creates a mirror image of a shape or graph on the opposite side of a given line.

When we reflect graphs in coordinate geometry, we typically use either the x-axis or y-axis as the line of reflection.

infoNote

Think of a reflection like looking in a mirror - the image appears on the opposite side of the mirror line, at the same distance but in the reverse direction.

Reflection in the y-axis

Understanding y-axis reflections

A reflection in the y-axis flips a graph horizontally. This transformation maps each point $(x, y)$ to a new position at $(-x, y)$ .

Notice that:

The y-coordinate stays the same
The x-coordinate becomes its opposite (changes sign)
The distance from the y-axis remains equal but on the opposite side

chatImportant

Key Point: In a y-axis reflection, only the x-coordinates change - they become negative. The y-coordinates remain unchanged.

The transformation rule

To reflect a function $y = f(x)$ in the y-axis, we replace x with -x:

$y = f(-x)$

This gives us the equation of the reflected graph.

Simple example

Consider the linear function $y = x$ .

When we replace $x$ with $-x$ , we get $y = -x$ , which is the reflection of the original line in the y-axis.

The graph shows how $y = -x$ is the mirror image of $y = x$ across the y-axis.

lightbulbExample

Worked Example 1: Reflecting a cubic function

Question: For the function $y = x^3$ , find the equation of the graph after reflection in the y-axis, then sketch both graphs showing key points.

Part a: Finding the equation

Strategy: Substitute $x$ with $-x$ .

Working:

y = x^3

(Write the function)

= (-x)^3

(Substitute $x = -x$ )

= -x^3

(Simplify)

The reflected graph has equation y = -x³.

Part b: Sketching both graphs

To sketch the graphs accurately, we choose simple x-values such as $-1$ , $0$ , and $1$ , then substitute them into each equation.

For $y = x^3$ :

If $x = -1$ , then $y = (-1)^3 = -1$
If $x = 0$ , then $y = 0^3 = 0$
If $x = 1$ , then $y = 1^3 = 1$

This gives key points: $(-1, -1)$ , $(0, 0)$ and $(1, 1)$ .

For $y = -x^3$ :

If $x = -1$ , then $y = -(-1)^3 = -(-1) = 1$
If $x = 0$ , then $y = -(0)^3 = 0$
If $x = 1$ , then $y = -(1)^3 = -1$

This gives key points: $(-1, 1)$ , $(0, 0)$ and $(1, -1)$ .

The graph shows that $y = -x^3$ is the reflection of $y = x^3$ in the y-axis.

Key observation: When we reflect in the y-axis, points on opposite sides of the y-axis swap positions. For instance, the point $(1, 1)$ on $y = x^3$ corresponds to $(-1, 1)$ on the reflected graph.

Reflection in the x-axis

Understanding x-axis reflections

A reflection in the x-axis flips a graph vertically. This transformation maps each point $(x, y)$ to a new position at $(x, -y)$ .

Notice that:

The x-coordinate stays the same
The y-coordinate becomes its opposite (changes sign)
The distance from the x-axis remains equal but on the opposite side

chatImportant

Key Point: In an x-axis reflection, only the y-coordinates change - they become negative. The x-coordinates remain unchanged.

The transformation rule

To reflect a function $y = f(x)$ in the x-axis, we replace $y$ with $-y$ , which is equivalent to negating the entire function:

$y = -f(x)$

This gives us the equation of the reflected graph.

Simple example

Consider the quadratic function $y = x^2$ .

When we apply the reflection rule, we get $y = -x^2$ , which reflects the parabola downward across the x-axis.

lightbulbExample

Worked Example 2: Reflecting a linear function

Question: For the function $y = 2x + 1$ , find the equation of the graph after reflection in the x-axis, then sketch both graphs showing key points.

Part a: Finding the equation

Strategy: Substitute $y$ with $-y$ .

Working:

y = 2x + 1

(Write the function)

-y = 2x + 1

(Substitute $y = -y$ to apply reflection)

y = -(2x + 1)

(Multiply both sides by $-1$ )

= -2x - 1

(Simplify)

The reflected graph has equation y = -2x - 1.

Part b: Sketching both graphs

When sketching linear graphs, we only need two points since a straight line is completely determined by any two distinct points. A good choice is to use the y-intercept (when $x = 0$ ) and one other simple x-value, such as $x = 1$ .

For $y = 2x + 1$ :

If $x = 0$ , then $y = 2 \times 0 + 1 = 1$
If $x = 1$ , then $y = 2 \times 1 + 1 = 3$

This gives key points: $(0, 1)$ and $(1, 3)$ .

For $y = -2x - 1$ :

If $x = 0$ , then $y = -2 \times 0 - 1 = -1$
If $x = 1$ , then $y = -2 \times 1 - 1 = -2 - 1 = -3$

This gives key points: $(0, -1)$ and $(1, -3)$ .

The graph shows that $y = -2x - 1$ is the reflection of $y = 2x + 1$ in the x-axis.

Key observation: When we reflect in the x-axis, points above and below the x-axis swap positions. The y-coordinate of each point becomes its opposite, while the x-coordinate remains unchanged.

bookmarkSummary

Key Points to Remember:

Y-axis reflection: Replace x with -x in the function to get $y = f(-x)$ . This creates a horizontal flip, mapping each point $(x, y)$ to $(-x, y)$ .
X-axis reflection: Negate the entire function to get $y = -f(x)$ . This creates a vertical flip, mapping each point $(x, y)$ to $(x, -y)$ .
Finding key points: To sketch reflections accurately, substitute simple values (like $-1$ , $0$ , $1$ ) into both the original and reflected equations to find corresponding points.
Quick check: After reflection in the y-axis, the x-coordinates of points change sign. After reflection in the x-axis, the y-coordinates change sign.
Points on the axis: Points that lie on the axis of reflection remain unchanged. For example, any point on the y-axis stays in the same position when reflected in the y-axis.

Reflections in Axes (HSC SSCE Mathematics Advanced): Revision Notes