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Transformations

- Calculus
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Imagine you are lying in your bed and you see a fly enter your room and sit on the ceiling of your room. From time to time it moves from one place to another. How do you keep track of the locations of the fly?

Imagine another scenario, you are on a roller coaster and you go around in twists and turns. Are the actions taken by the fly on the ceiling and you on the roller coaster the same or are they different? How do we track the exact motions in these scenarios?

In this article, we will learn some **fundamental movements in two-dimensional space**. These are transformations and we will learn their definition, and types of transformations, and see examples.

**Transformations** are movements in space of an object.

We say a transformation is **rigid** if the object does not change its size or shape during the transformation. If the object changes its size during a transformation, then we call it a **non-rigid** transformation.

A rigid transformation does not change the size or shape of the object transformed. Examples of rigid transformations include:

**Translation**- moving of the shape, left, right, up or/and down;**Reflection**- reflecting a shape with respect to a line, the line could also be the x-axis or y-axis;**Rotation**- rotating a shape around a point, clockwise or anti-clockwise.

A non-rigid transformation can change the size or shape, or both size and shape, of the object. An example of a non-rigid transformation is **d****ilation **- blowing up or shrinking an object.

Translation can be thought of as the process of moving an object around in a graph sheet. For knowing the movement of an object we look at how its edge points are transformed.

The translation of a point (x, y) to a new point (x', y') can be understood from its **change** in the x and y coordinates. Under this transformation, the point has moved along the x-direction and along the y-direction.

Moreover,

a

**positive value in the x-direction**indicates the movement to the**right**and a**negative value**indicates movement to the**left**;

a

**positive value in the y-direction**indicates the movement**upwards**and the**negative value**indicates movement**downwards**.

If a is positive then you move right and if a is negative you move left.

If b is positive you move up and if b is negative you move down.

For example, translating the object by (2, −3) means that the x-coordinate of every point in an object will increase by two, and the y-coordinate of every point in an object will decrease by three. Successfully, the object will move two units to the right and three units downward.

Translate the given triangle ABC by (–7, –4).

**Solution**

Translate (–7, –4) means "move the given triangle to 7 units left and 4 units downwards". We can move the triangle if we move its edge points A(4, 6), B(1, 2), and C(5, 2).

Applying the translation to point A by moving 7 units left and 4 units down we have A'(–3, 2).

Similarly, we get on applying the translation to B and C the points B'(–6, –2) and C'(–2, –2). By joining A', B' and C' we have the translated triangle.

Translate the given hexagon ABCDEF by (–7, 7).

**Solution**

Translation by the vector (–7, 7) means we move the hexagon 7 units to the left and 7 units upwards.

To do this we apply the transformation to the edge points and join the translated points to obtain the hexagon A'B'C'D'E'F'.

A reflection can be thought of as seeing something through a mirror. So it is always with respect to a given line where the mirror is placed. The distance between the object and its image from the mirror is the same. Similar to translation to reflect an object you reflect the edge points of the object.

**Reflecting** **an object about a line **, means to move every point in the object at an equal distance to the other side of the line.

For example to reflect the point (1, 0) about the y-axis we first see the distance the point is from the y-axis. In this case, the point (1, 0) is 1 unit from the y-axis and so it will be 1 unit on the other side of the y-axis and so at (–1, 0).

Reflect shape A about the line . Label the resulting shape with the letter B.

**Solution**

To obtain the reflection we first draw the line of reflection . Then we move each corner of the shape the same distance from the line of reflection on the ‘other side'.

For example, the bottom left corner of A is the point (3, 1), which is 2 units from the line . On reflection, it will be 2 units on the other side of the line. So its reflection point is (–1, 1).

Notice there is no change in the y-coordinate of the point and its reflection. This is because the line of reflection is parallel to the y-axis. We do the same for all the edge points to obtain the reflected image.

Reflect shape A about the line (x-axis). Label the resulting shape with the letter A’.

**Solution**

Rotations are transformations where the object is rotated through some angles. Examples of rotations include the minute needle of a clock, merry-go-around, and so on.

In all cases of rotation, there will be a centre point which is not affected by the transformation. In the clock the point where the needle is fixed in the middle does not move at all. In other words, the needle rotates around the clock about this point.

**Rotating an object d ^{o }about a point (a, b)** is to rotate every point of the object such that the line joining the points in the object and the point (a, b) rotates at an angle d

If d is positive then it is clockwise, otherwise, it is anticlockwise. In both transformations, the size and shape of the figure stay exactly the same.

The general rule of transformation of rotation about the origin (0, 0) is as follows.

Type of Rotation | Original Points | Switched Points (Anticlockwise Rotation) | Switched Points (Clockwise Rotation) |

To rotate 90 ^{0} | (x, y) | (−y, x) | (y, −x) |

To rotate 180 ^{0} | (x, y) | (−x, −y) | (−x, −y) |

To rotate 270 ^{0} | (x, y) | (y, −x) | (−y, x) |

Rotate the given shape ABC, 90^{º} clockwise and anticlockwise about the origin.

Original Points(x, y) | Switched Points(anticlockwise rotation)(−y, x) | Switched Points (clockwise rotation)(y, −x) |

(−3, 5) | (−5, −3) | (5, 3) |

(−6, 2) | (−2, −6) | (2, 6) |

(−3, 2) | (−2, −3) | (2, 3) |

**Solution**

Rotate the given shape to 270^{º} clockwise and anticlockwise about the origin.

Original Points (x, y) | Switched Points(counter-clockwise rotation)(y, −x) | Switched Points(clockwise rotation)(−y, x) |

(−4, 6) | (6, 4) | (−6, −4) |

(−6, 4) | (4, 6) | (−4, −6) |

(−2, 4) | (4, 2) | (−4, −2) |

(−3, 1) | (1, 3) | (−1, −3) |

Dilation is a transformation, which is used to resize the object to be larger or smaller. This transformation produces an image that is the same as the original in shape, but there is a difference in the size of the object.

- If a dilation creates a larger image, then it is known as
**enlargement**(a stretch).

- If a dilation creates a smaller image, then it is known as
**reduction**(a shrink).

A description of a dilation includes the** ****scale factor** (or ratio) and the **centre of the dilation**.

**Dilating an object by a scale factor k and about the centre of dilation (a, b)** means to move every point in the object by the scale times the distance of the point from the centre of dilation.

- If the scale factor is greater than 1, the image is an enlargement (a stretch).

- If the scale factor is between 0 and 1, the image is a reduction (a shrink).

For the scale factor , and origin being the centre of dilation we have the rule

.

That is, the original point (x, y) is changed as (3x, 3y). In this case, the dilation image will be stretched.

For, we get

.

In this case, the dilation image will be shrunk.

Dilate the given shape A by a factor of 2 with the origin as the centre of dilation.

**Solution**

The edges of shape A have the coordinates (1, 1), (1, 3), (3, 0), and (3, 3).

Now the coordinates of the given shape are multiplied by 2. They are (2,2), (2,6), (6,0), (6,6).

The Original Shape A and the Enlarged Shape B are represented in the following diagram.

Dilate the given shape A by a factor of 0.5 with the origin as the fixed point.**Solution**

The edges of shape A have the coordinates (6, 6), (6, 2), and (4, 2).Now the coordinates of the given shape are multiplied by 0.5. We then get the new coordinates (3, 3), (3, 1), (2, 1).The Original Shape A and the Shrinked Shape B are represented in the following diagram.

The edges of shape A have the coordinates (6, 6), (6, 2), and (4, 2).Now the coordinates of the given shape are multiplied by 0.5. We then get the new coordinates (3, 3), (3, 1), (2, 1).The Original Shape A and the Shrinked Shape B are represented in the following diagram.

When an object undergoes more than one transformation sequentially we call it a composite transformation. For example, a triangle undergoing translation first followed by dilation. There can be two or more transformations done one after another.

Rotate the given shape A to 90^{º} counter-clockwise direction about the origin, then reflect the resultant shape about the line x = 0, and finally translate the resultant shape into (–1, 7).

Original Points (x, y) | Switched Points (counter-clockwise rotation) (−y, x) |

(–6, 2) | (–2, –6) |

(–3, 2) | (–2, –3) |

(–2, 5) | (–5, –2) |

(–4, 5) | (–5, –4) |

- Transformations are movements of objects in space.
**Rigid Transformations**do not change the size or shape of the object after transformation.- Examples of rigid transformations include translation, reflection and rotation.

- Examples of rigid transformations include translation, reflection and rotation.
**Non-Rigid Transformations**can change the size or shape, or both, of the object.- Dilation is an example of a non-rigid transformation.

**Translation**(sometimes called ‘movement’) is the process of moving something around.- To translate an object by a vector means to move every point in the object a units in the x-direction and b units in the y-direction.
- If a is positive then you move right and if a is negative you move left.
- If b is positive you move up and if b is negative you move down.

**Reflection**occurs when each point in the shape is reflected about a line of reflection.- After reflection, the image is at the same distance from the line as the pre-image but on the other side of the line.

**Rotation**rotates each point in the shape at a certain degree with respect to a point.- The shape rotates counter-clockwise when the degrees are positive;
- And rotates clockwise when the degrees are negative.

**Dilation**is a transformation which is used to resize an object, making it larger or smaller. A description of a dilation includes the scale factor (or ratio) and the centre of the dilation.

♦ If the scale factor is**greater than 1**, the image is an enlargement (a stretch).

♦ If the scale factor is**between 0 and 1**, the image is a reduction (a shrink).

Transformations are movements in space of an object.

A translation is an example of a transformation.

The five transformations are translation, reflection, rotation, dilation, and shear.

More about Transformations

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