Combinations of Transformations Revision Notes for VCE SSCE Mathematical Methods

Combinations of Transformations

When working with function graphs, you'll often need to apply more than one transformation sequence. This section shows you how to combine dilations, reflections, and translations in sequence to create new graphs.

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Combining transformations allows you to create complex graphs from simple ones. Instead of memorizing dozens of different function shapes, you can learn how to transform a few basic functions to get the graphs you need.

Why transformation order matters

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The order in which you apply transformations is crucial. Different sequences produce different results. Think of it like getting dressed - putting on socks then shoes gives a different result than shoes then socks!

When combining transformations, you build up the mapping step by step. Each transformation takes the output from the previous one.

Building transformation sequences

Let's look at how to combine transformations systematically.

Example: Dilation followed by reflection

Start with a dilation of factor $2$ from the $x$ -axis, followed by a reflection in the $x$ -axis.

We write the mapping as:

(x, y) \to (x, 2y) \to (x, -2y)

First we apply the dilation, then the reflection.

For instance, the point $(1, 1)$ transforms like this:

(1, 1) \to (1, 2) \to (1, -2)

Example: Dilation followed by translation

Now consider a dilation of factor $2$ from the $x$ -axis, followed by a translation of $2$ units right and $3$ units down.

The mapping becomes:

(x, y) \to (x, 2y) \to (x + 2, 2y - 3)

For the point $(1, 1)$ :

(1, 1) \to (1, 2) \to (3, -1)

Finding equations after combined transformations

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When you need to find the equation of a transformed graph, follow this systematic method. The key is to work with primed coordinates $(x', y')$ for the image, then work backwards to express the original coordinates in terms of the new ones.

Step 1: Write out the complete mapping by applying each transformation in order

Step 2: Let $(x', y')$ be the coordinates of the image point. Express $x'$ and $y'$ in terms of $x$ and $y$

Step 3: Rearrange these expressions to write $x$ and $y$ in terms of $x'$ and $y'$

Step 4: Substitute these expressions into the original equation

Step 5: Simplify and rewrite using $x$ and $y$ (dropping the primes)

Let's see this method in action with some examples.

Worked example: Square root function with dilation and reflection

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Worked Example: Square Root Function with Dilation and Reflection

Find the equation of the image of $y = \sqrt{x}$ under a dilation of factor $2$ from the $x$ -axis followed by a reflection in the $x$ -axis.

Solution

The mapping is:

(x, y) \to (x, 2y) \to (x, -2y)

If $(x, y)$ maps to $(x', y')$ , then:

x' = x \text{ and } y' = -2y

Rearranging these:

x = x' \text{ and } y = \frac{y'}{-2}

Substitute into the original equation $y = \sqrt{x}$ :

\frac{y'}{-2} = \sqrt{x'}

Therefore:

y' = -2\sqrt{x'}

The equation of the image is y = -2√x

Worked example: Square root function with dilation and translation

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Worked Example: Square Root Function with Dilation and Translation

Find the equation of the image of $y = \sqrt{x}$ under a dilation of factor $2$ from the $x$ -axis followed by a translation of $2$ units right and $3$ units down.

Solution

The mapping is:

(x, y) \to (x, 2y) \to (x + 2, 2y - 3)

If $(x, y)$ maps to $(x', y')$ , then:

x' = x + 2 \text{ and } y' = 2y - 3

Rearranging:

x = x' - 2 \text{ and } y = \frac{y' + 3}{2}

Substitute into $y = \sqrt{x}$ :

\frac{y' + 3}{2} = \sqrt{x' - 2}

Therefore:

y' = 2\sqrt{x' - 2} - 3

The equation of the image is y = 2√(x - 2) - 3

Sketching sequential transformations

When sketching the result of multiple transformations, it helps to draw each stage separately. This lets you track how the graph changes at each step.

Worked example: Sketching step by step

Consider a parabola transformed by:

Translation of $1$ unit right and $2$ units up
Dilation of factor $2$ from the $y$ -axis
Reflection in the $x$ -axis

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At each stage, key points move according to the transformation rule. By tracking these points, you can sketch the final image accurately. The diagram above shows how the parabola changes at each transformation stage.

Worked example: Complex transformation sequence

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Worked Example: Complex Transformation Sequence

The graph of $y = \sqrt{x}$ is transformed by:

Translation $6$ units left
Reflection in the $y$ -axis
Dilation by factor $2$ from the $x$ -axis

Find the equation of the final image.

Solution

Apply each transformation in sequence:

Translation $6$ units left:

(x, y) \to (x - 6, y)

Reflection in the $y$ -axis:

(x - 6, y) \to (-(x - 6), y)

Dilation factor $2$ from $x$ -axis:

(-(x - 6), y) \to (-(x - 6), 2y)

The complete mapping is:

(x, y) \to (-(x - 6), 2y)

Let $(x', y')$ be the image coordinates. Then:

x' = -(x - 6) \text{ and } y' = 2y

Rearranging:

x = -x' + 6 \text{ and } y = \frac{y'}{2}

Substitute into $y = \sqrt{x}$ :

\frac{y'}{2} = \sqrt{-x' + 6}

Therefore:

y' = 2\sqrt{6 - x'}

The rule of the transformed function is y = 2√(6 - x)

Worked example: Parabola transformation sequence

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Worked Example: Parabola Transformation Sequence

For the graph of $y = x^2$ , sketch the image and state its rule after these transformations:

Translation $1$ unit right and $2$ units up
Dilation factor $2$ from the $y$ -axis
Reflection in the $x$ -axis

Solution

The complete mapping is:

(x, y) \to (x + 1, y + 2) \to (2(x + 1), y + 2) \to (2(x + 1), -(y + 2))

Let $(x', y')$ be the image of $(x, y)$ . Then:

x' = 2(x + 1) \text{ and } y' = -(y + 2)

Rearranging:

x = \frac{1}{2}(x' - 2) \text{ and } y = -y' - 2

Substitute into $y = x^2$ :

-y' - 2 = \left(\frac{1}{2}(x' - 2)\right)^2

-y' - 2 = \frac{1}{4}(x' - 2)^2

y' = -\frac{1}{4}(x' - 2)^2 - 2

The rule of the image is y = -¼(x - 2)² - 2

Using technology to verify transformations

Graphing calculators can help you verify transformation equations and visualize the results.

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Using the TI-Nspire:

Define the original function, for example f(x) = x²
Enter the transformation in function notation: -f(½(x-2)) - 2
The calculator will simplify this to standard form
You can also graph both functions to see the transformation visually

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Using the Casio ClassPad:

Define f(x) = x²
Enter the transformed function: -f(½(x-2)) - 2
Use the simplify function to expand the expression
Graph both functions to compare them

Remember!

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Key Points to Remember:

Order matters: Apply transformations in the exact sequence given. Different orders produce different results.
Build step by step: Write out the complete mapping by applying each transformation one at a time.
Use primed coordinates: Let $(x', y')$ represent the final image, then work backwards to express the original coordinates in terms of the new ones.
Substitute carefully: After finding $x$ and $y$ in terms of $x'$ and $y'$ , substitute these into the original equation and simplify.
Verify with technology: Use your calculator to check your transformed equation and visualize the graph.

Combinations of Transformations (VCE SSCE Mathematical Methods): Revision Notes