Combinations of Transformations Revision Notes for HSC SSCE Mathematics Advanced

Combinations of Transformations

Introduction

When working with graphs, we can apply multiple transformations one after the other. In this topic, we'll explore how different transformations interact and whether the order in which we apply them affects the final result.

infoNote

For this section, we'll treat reflections in the axes as dilations with factor $-1$ . A rotation of $180°$ about the origin is equivalent to two reflections. This simplifies our work to focus on four main types of transformations: two translations (horizontal and vertical) and two dilations (horizontal and vertical).

Understanding commuting transformations

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Two transformations are said to commute if they produce the same final result regardless of the order in which they are applied. This is an important concept because it tells us when we can rearrange the order of transformations without changing the outcome.

Summary of transformations

Before exploring combinations, let's review the four basic transformations and how they're applied:

Each transformation can be expressed either by variable replacement or by function rule notation. Understanding both methods will help you work flexibly with transformations.

When transformations commute

Two translations always commute

When applying two translations (shifts) to a graph, the order doesn't matter. You'll always get the same result.

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Worked Example: Translations with parabola $y = x^2$

Consider shifting the parabola right 3 units and down 1 unit.

Method 1: Shift right first, then down

Shifting right 3: $y = (x - 3)^2$
Then shifting down 1: $y + 1 = (x - 3)^2$ , which gives $y = (x - 3)^2 - 1$

Method 2: Shift down first, then right

Shifting down 1: $y + 1 = x^2$ , which gives $y = x^2 - 1$
Then shifting right 3: $y = (x - 3)^2 - 1$

Both methods produce the same equation: $y = (x - 3)^2 - 1$

This demonstrates that horizontal and vertical translations can be applied in either order.

Two dilations always commute

Similarly, when applying two dilations (stretches) to a graph, the order of application doesn't affect the final result.

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Worked Example: Dilations with circle $x^2 + y^2 = 1$

Consider stretching the circle vertically by factor 2 and horizontally by factor 3.

Method 1: Vertical stretch first, then horizontal

Stretching vertically by factor 2: $x^2 + \left(\frac{y}{2}\right)^2 = 1$ , which simplifies to $x^2 + \frac{y^2}{4} = 1$
Then stretching horizontally by factor 3: $\left(\frac{x}{3}\right)^2 + \frac{y^2}{4} = 1$ , which gives $\frac{x^2}{9} + \frac{y^2}{4} = 1$

Method 2: Horizontal stretch first, then vertical

Stretching horizontally by factor 3: $\frac{x^2}{9} + y^2 = 1$
Then stretching vertically by factor 2: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

Both methods produce the same equation: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

Cross-direction transformations commute

A horizontal dilation and a vertical translation will always commute. Similarly, a vertical dilation and a horizontal translation will commute.

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Worked Example: Cross-direction transformations with circle $(x - 2)^2 + (y - 1)^2 = 1$

Apply a reflection in the $y$ -axis (horizontal dilation with factor $-1$ ), then shift up 2 units.

Method 1: Reflect first, then shift up

Reflecting in the $y$ -axis: $(-x - 2)^2 + (y - 1)^2 = 1$ , which simplifies to $(x + 2)^2 + (y - 1)^2 = 1$
Then shifting up 2: $(x + 2)^2 + (y - 2 - 1)^2 = 1$ , giving $(x + 2)^2 + (y - 3)^2 = 1$

Method 2: Shift up first, then reflect

Shifting up 2: $(x - 2)^2 + (y - 3)^2 = 1$
Then reflecting in the $y$ -axis: $(-x - 2)^2 + (y - 3)^2 = 1$ , giving $(x + 2)^2 + (y - 3)^2 = 1$

Both methods produce: $(x + 2)^2 + (y - 3)^2 = 1$

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Summary of commuting transformations:

Any two translations commute
Any two dilations commute (including reflections)
A translation and a dilation commute if one is vertical and the other horizontal

When transformations don't commute

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The critical case where order matters is when both transformations act in the same direction. Specifically:

A vertical translation and a vertical dilation do not commute
A horizontal translation and a horizontal dilation do not commute

Worked example: Horizontal transformations

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Worked Example: Non-commuting horizontal transformations

Problem: Apply a reflection in the $y$ -axis (horizontal dilation with factor $-1$ ) and a translation left 2 units to the circle $(x + 3)^2 + y^2 = 1$ .

Part a: Reflect first, then translate left

Applying the reflection:

$(-x + 3)^2 + y^2 = 1$
$(x - 3)^2 + y^2 = 1$

Then applying the translation left 2:

$(x + 2 - 3)^2 + y^2 = 1$
$(x - 1)^2 + y^2 = 1$

Part b: Translate left first, then reflect

Applying the translation left 2:

$(x + 2 + 3)^2 + y^2 = 1$
$(x + 5)^2 + y^2 = 1$

Then applying the reflection:

$(-x + 5)^2 + y^2 = 1$
$(x - 5)^2 + y^2 = 1$

The final equations are different: $(x - 1)^2 + y^2 = 1$ versus $(x - 5)^2 + y^2 = 1$

Worked example: Vertical transformations

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Worked Example: Non-commuting vertical transformations

Problem: Apply a vertical dilation with factor $\frac{1}{2}$ and a translation down 3 units to the parabola $y = x^2 + 4$ .

Part a: Dilate first, then translate down

Applying the dilation:

$y = \frac{1}{2}(x^2 + 4)$
$y = \frac{1}{2}x^2 + 2$

Then applying the translation down 3:

$y = \frac{1}{2}x^2 + 2 - 3$
$y = \frac{1}{2}x^2 - 1$

Part b: Translate down first, then dilate

Applying the translation down 3:

$y = x^2 + 4 - 3$
$y = x^2 + 1$

Then applying the dilation:

$y = \frac{1}{2}(x^2 + 1)$
$y = \frac{1}{2}x^2 + \frac{1}{2}$

The final equations are different: $y = \frac{1}{2}x^2 - 1$ versus $y = \frac{1}{2}x^2 + \frac{1}{2}$

chatImportant

Key rule for non-commuting transformations:

When a translation and a dilation both act in the same direction (both horizontal or both vertical), the order of application matters and produces different results.

Universal formula for combinations

For functions, there's a general formula that combines all four transformation types:

$y = kf(a(x + b)) + c$

This formula applies the following transformations in sequence:

Stretch horizontally with factor $\frac{1}{a}$
Shift left $b$ units
Stretch vertically with factor $k$
Shift up $c$ units

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The vertical transformations (steps 3 and 4) can be done before the horizontal transformations (steps 1 and 2) because they commute.

Understanding the universal formula

Starting with $y = f(x)$ :

Stretching horizontally with factor $\frac{1}{a}$ gives: $y = f(ax)$
Shifting left $b$ gives: $y = f(a(x + b))$
Stretching vertically with factor $k$ gives: $y = kf(a(x + b))$
Shifting up $c$ gives: $y = kf(a(x + b)) + c$

Alternative analysis:

You can also work backwards from the formula by progressive rewriting:

Starting with $y = kf(a(x + b)) + c$ :

Subtract $c$ : $y - c = kf(a(x + b))$
Divide by $k$ : $\frac{y - c}{k} = f(a(x + b))$
Rewrite: $\frac{y - c}{k} = f\left(\frac{x + b}{1/a}\right)$
Express as shifts: $\frac{y - c}{k} = f\left(\frac{x - (-b)}{1/a}\right)$

infoNote

This formula is particularly useful for trigonometric functions and computer programs, but most problems are more easily solved using the standard transformation methods.

Remember!

bookmarkSummary

Key Points to Remember:

Commuting transformations: Any two translations commute, any two dilations commute, and a horizontal transformation commutes with a vertical transformation
Non-commuting transformations: A translation and a dilation in the same direction (both horizontal or both vertical) do not commute—order matters!
Universal formula: $y = kf(a(x + b)) + c$ combines all four transformations, but is trickier to use than applying transformations step by step
When order matters: Always check whether your transformations are in the same direction—if they are both horizontal or both vertical, the order of application will affect your final result
Practical approach: For most problems, apply transformations one at a time using the replacement method or function rule notation rather than trying to use the universal formula

Combinations of Transformations (HSC SSCE Mathematics Advanced): Revision Notes