Determining Transformations Revision Notes for VCE SSCE Mathematical Methods

Determining Transformations

Introduction

When you have two functions and need to find the transformation that maps one to the other, you can work backwards from the transformed graph. This is called determining transformations. Instead of applying a transformation to see what happens to a graph, you're starting with both the original and final graphs and figuring out what transformation connects them.

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The key difference with this approach is that you already know what the final graph looks like, and your task is to work backwards to identify the transformation that was applied. This is the reverse of the usual process where you start with a function and apply a transformation to see the result.

The reverse method

The reverse method involves working backwards through the transformation steps. Here's how it works:

Start with the original function $y = f(x)$ and the image function $y' = g(x')$
Rearrange the image equation to match the form of the original function
Identify how $x$ relates to $x'$ and how $y$ relates to $y'$
Express the transformation in terms of these relationships
Describe the transformation using geometric terms (dilation, translation, reflection)

Key principle

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You can often determine transformations by inspection if you recognise the form of the image function. However, the systematic method ensures you don't miss any steps and is especially valuable for complex transformations or exam situations where you need to show clear working.

Worked example: Quadratic and square root transformations

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Worked Example: Transforming $y = x^2$

Find the transformation that takes $y = x^2$ to $y = 2(x - 2)^2 + 3$

Step 1: Start by writing the image equation:

y' = 2(x' - 2)^2 + 3

Step 2: Rearrange to match the original form:

\frac{y' - 3}{2} = (x' - 2)^2

Step 3: Choose to write:

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

Step 4: Solve for $x'$ and $y'$ :

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

Step 5: Express as a sequence of transformations:

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

The transformation is:

A dilation of factor 2 from the x-axis
Followed by a translation of 2 units right and 3 units up

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Worked Example: Transforming $y = \sqrt{x}$

Find the transformation that takes $y = \sqrt{x}$ to $y = \sqrt{5x - 2}$

Step 1: Start with:

y = \sqrt{x} \text{ and } y' = \sqrt{5x' - 2}

Step 2: Choose to write:

y = y' \text{ and } x = 5x' - 2

Step 3: Solve for $x'$ :

x' = \frac{x + 2}{5} = \frac{x}{5} + \frac{2}{5}

And we have $y' = y$

The transformation is:

A dilation of factor $\frac{1}{5}$ from the y-axis
Followed by a translation of $\frac{2}{5}$ units in the positive direction of the x-axis

Worked example: Rational and quadratic transformations

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Worked Example: Transforming to $y = \frac{1}{x^2}$

Find the transformation that takes $y = \frac{3}{(x-1)^2} + 6$ to $y = \frac{1}{x^2}$

Step 1: Write the equations:

\frac{y - 6}{3} = \frac{1}{(x-1)^2} \text{ and } y' = \frac{1}{(x')^2}

Step 2: Choose to write:

y' = \frac{y - 6}{3} \text{ and } x' = x - 1

Step 3: Express as a sequence:

(x, y) \to (x - 1, y - 6) \to \left(x - 1, \frac{y - 6}{3}\right)

The transformation is:

A translation of 1 unit left and 6 units down
Followed by a dilation of factor $\frac{1}{3}$ from the x-axis

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Worked Example: Transforming from $y = (5x - 1)^2 + 6$

Find the transformation that takes $y = (5x - 1)^2 + 6$ to $y = x^2$

Step 1: Write:

y - 6 = (5x - 1)^2 \text{ and } y' = (x')^2

Step 2: Choose to write:

y' = y - 6 \text{ and } x' = 5x - 1

One possible transformation is:

A dilation of factor 5 from the y-axis
Followed by a translation of 1 unit left and 6 units down

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Important note: There are infinitely many transformations that can map one graph to another. The transformations shown here are conventional choices that follow standard mathematical practices.

Key terminology

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Dilation from the x-axis: Changes the y-coordinates by a scale factor while $x$ -coordinates remain unchanged

Dilation from the y-axis: Changes the x-coordinates by a scale factor while $y$ -coordinates remain unchanged

Translation: Shifts the graph horizontally and/or vertically without changing its shape

Image: The result of applying a transformation to an original graph

Exam tips

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Rearrange systematically: Always rearrange the image equation to match the form of the original function

Show your working: Write out the relationships $x' = ...$ and $y' = ...$ clearly

Use sequence notation: Express transformations as $(x, y) \to ... \to ...$ to show the order

Remember the order matters: The sequence of transformations affects the final result

Check your answer: Verify by applying your transformation to see if it produces the correct image

Remember!

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Key Points to Remember:

The reverse method works backwards from the transformed equation to find the transformation
Rearrange the image equation to match the original function's form
Identify relationships between $(x, y)$ and $(x', y')$ to determine the transformation
Express transformations as a sequence showing the order of operations
There are infinitely many possible transformations between two graphs, but conventional choices are preferred
Dilations from the x-axis affect y-coordinates; dilations from the y-axis affect x-coordinates

Determining Transformations (VCE SSCE Mathematical Methods): Revision Notes