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# Compose Transformation : Explanation With Graphs

### Key Concepts

• Understanding combination of transformation
• Understanding combination of transformation in a coordinate plane
• Describing combination of transformation in a coordinate plane

## Compose Transformations

#### What is transformation?

Transformation is a change.

What is a rigid transformation?

Transformation in which there is no change of size or shape after transformation is rigid transformation.

Give examples for rigid transformation or isometric transformation.

Translation, reflection, rotation are examples of rigid transformation.

What is translation?

Translation is a transformation where the image or point slides across the plane.

What is reflection?

Reflection is a transformation where the image flips across the line and that line is called the line of reflection.

What is rotation?

Rotation is a transformation that occurs around a fixed point and that point is called the center of rotation.

## Combination of Transformation

A composition of transformations is a combination of two or more transformations, each performed on the previous image.

Transformations can be combined by doing one transformation and then another.

When an object is transformed under two successive transformations, i.e., the initial image is re-transformed to obtain a final image, it is the called combination of transformation

### Understanding the combination of transformation in a coordinate plane

Translate triangle ALT if A (-5, -1), L (-3, -2), T (-3, 2) by the rule (x, y)  (x+5, y-2), then reflect the image over the y-axis.

In this case, the image is triangle A’L’T’, and A’(0, -3), L’(2, -4), T’(2, 0) are the coordinate points after translation, and then this image is reflected over the y-axis to form triangle A’’L”T” and

A”(0, -3), L”(-2, -4)T”(-2, 0) are the coordinate points after reflection.

### Describing combination of transformation in a coordinate plane

Here, line AB is rotated counterclockwise 90° so as to form line A’B’, and it is reflected about

the y- axis to form A’B’.

### Describe a sequence of transformations that maps △XYZ to △X′Y′Z′.

This image undergoes reflection followed by the translation.

—>A reflection across the y-axis followed by the translation rule ( x, y     (x+3, y-6) or

3 units right and 6 units down.

## Exercise

• Plot the points A(0,0), B(8,1), C(5,5) and undergo the transformation reflection with respect to the x-axis and rotate clockwise 180°.
• Triangle ABC where the vertices of ΔABC are A (−1,−3), B(−4,−1), and C(−6,−4) undergoes a composition of transformations described as:
• A translation 10 units to the right, then
• A reflection in the x-axis.
• What are the vertices of the triangle after both transformations are applied?
• Triangle XYZ has coordinates X(1,2), Y(−3,6) and Z(4,5). The triangle undergoes a translation of 2 units to the right and 1 unit down to form triangle X’ Y ‘Z ‘. Triangle X’ Y’ Z’ is then reflected about the y-axis to form triangle X”Y”Z”.  Determine the vertices for triangle X”Y”Z”.
• A point X has coordinates (1, -8). The point is reflected across the y-axis to form X’. X’ is translated over 4 to the right and up 6 to form X”. What are the coordinates of X’ and X”?
• A point A has coordinates (–2, –3). The point is translated over 3 to the left and up 5 to form A’. A’ is reflected across the x-axis to form A”. What are the coordinates of A’ and A”?
• A point P has coordinates (-5, -6). The point is reflected across the line y = −x to form P’. P’ is rotated about the origin 90◦CW to form P”. What are the coordinates of P’ and P”?
• Line JT has coordinates J(–3, 5) and T(2,3). The segment is rotated about the origin 180◦ to form J‘T’. J’ T’ is translated over 6 to the right and down 3 to form J”T”. What are the coordinates of J” T” and J’’T?
• Line SK has coordinates S(−1, 8) and K(1,2). The segment is translated over 3 to the right and up 3 to form S’ K’. S’ K’ is rotated about the origin 90◦CCW to form S”K”. What are the coordinates of S’ K’ and S”K”?
• A point K has coordinates (1, –4). The point is reflected across the line y = x to form K’. K’ is rotated about the origin 270◦CW to form K”. What are the coordinates of K’ and K”?
• Pre-image: D(9,–3), E(6,–7), F(3,–3), G(5,–1) Transformations: Reflection at x=2 followed by rotation 180°

### What we have learnt:

• Combination of transformation.
• Combination of transformation in a coordinate plane.
• Describe combination of transformation in a coordinate plane

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