Transformation Geometry.

Transformation geometry involves making images or copies of an object.

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Grade 9 syllabus introduces you to transformation geometry which involves translation, reflection, dilation and rotation. These notes or concept, you will use until grade 12. At that time, you will only be asked about these processes in the exams and they will never be covered again in class. My child, know them by heart.

Now that you are in grade 9, transformation geometry will be about creating an image of a shape by performing the above procesess. As you get to higher grades, you will apply these processes to functions/graphs.

The general rule or notation for transformation geometry is that a new point is called an image, symbolised with an apostrophe (') next to the name of a point, i.e,:

$${A \rightarrow A'}$$

1. Translation

When you translate objects/functions, you are moving them to a new point. Translations move objects either horizontally of vertically. Translation is also known as shifting.

The general rule for translation is:

$${(x;y) \rightarrow (x+a;y+b)}$$

Where \(a\) and \(b\) are parameters that for horizontal and vertical shifting, respectively.

Horizontal translation

Horizontal translation moves objects leftwards or rightwards. The rule:

$${(x;y) \rightarrow (x+a;y)}$$

Where \(a\) is positive when an object is moved to the right and negative when moved to te right.

Example: Move point A(2;3)

• 6 units left.

  The new point will be:
  

  \begin{gather*} (2;3) \rightarrow (2-6;3)\\ (2;3) \rightarrow (-4;3) \end{gather*}

  The new point is A'(-4;3)

• 3 units to the right.

  The new point will be:
  

  \begin{gather*} (2;3) \rightarrow (2+3;3)\\ (2;3) \rightarrow (5;3) \end{gather*}

  The new point is A'(5;3)

The sketch below shows the results of the above example. In a graph, the results can be obtained simply by moving according to number stated for the translaton. For example, a shift of 6 units to the left means that you should count six numbers to the left of your point.

Vertical translation

For the vertical translation, the function will either move up or down.

$${(x;y) \rightarrow (x;y+b)}$$

Where \(b\) is positive when an object is moved up and negative when moved down.

Example: Move point A(2;3)

• 6 units down.

  The new point will be:
  

  \begin{gather*} (2;3) \rightarrow (2;3-6)\\ (2;3) \rightarrow (2;-3) \end{gather*}

  The new point is A'(2;-3)

• 2 units up.

  The new point will be:
  

  \begin{gather*} (2;3) \rightarrow (2;3+2)\\ (2;3) \rightarrow (2;5) \end{gather*}

  The new point is A'(2;5)

2. Reflection

In this transformation, an object will be reflected across a line, creating an image. In reflection transformations, each point in an object appears at an equal distance on the opposite side of the line of reflection.

Reflection on the x-axis

When reflecting in the x-axis, the x-values remain constant while the y-values change the sign.

The transformation rule is:

$${(x;y) \rightarrow (x;-y)}$$

Example: Reflect A(2;3) on the x-axis

$${(2;3) \rightarrow (2;-3)}$$

The new point is A'(2;-3)

As we can see from this result, the new point is the opposite side of the y-axis.

Reflection in the y-axis

When reflecting in the y-axis, the y-values remain constant while the x-values change the sign.

The transformation rule is:

$${(x;y) \rightarrow (-x;y)}$$

Reflection in the line y = x

Reflection in the line y = x, simply requires you to interchange the values. What was the x-value will now become the y-value and what was the y-value will now become the x-value.

The transformation rule is:

$${(x;y) \rightarrow (y;x)}$$

3. Dilation

Dilation is about changing the size of the object either by enlarging or shrinking by a factor.

The transformation rule is:

$${(x;y) \rightarrow (kx;ky)}$$

Where \(k\) is a dilation factor.

1. Enlargement: \(k\) is a whole number.
2. Shrinking: \(k\) is a decimal number (a fraction).

4. Rotation

We will explain rotation by summarising the methods or rules in the table below:

If you are using a small screen, please rotate your screen to have a better view of the table below!!

Process	Rule	Example
Rotation 90° clockwise	(x;y) → (y;-x)	(2;3) → (3;-2)
Rotation 90° anticlockwise	(x;y) → (-y;x)	(2;3) → (-3;2)
Rotation 180°	(x;y) → (-x;-y)	(2;3) → (-2;-3)
Rotation 360°	(x;y) → (x;y)	(2;3) → (2;3)

From the table above, we can summarise that:

• 90° rotation: x and y-values interchange.
• 180° rotation: x and y-values remain the same but have opposite signs.
• 360° rotation: x and y-values remain the same.