Transformation

Sub Topics
Transformation can be defined as the creation of duplicate object retaining the shape and certain properties of the original object. When we see two pictures our immediate tendency is to compare both. In cases where we find some similarities we further tend to study the types of similarities. That is, how the first picture has been 'transformed' into become like the second picture. In case of mathematical functions, the graphs which are the visual representations of the overall behavior of the function, such studies are interesting and important. The study is broadly known as 'transformations' in mathematics. Let us take a closer look at the different types of transformations.

Transformation can be classified as following
• Dilation
• Contraction
• Compression
• Translation
• Rotation
• Reflection

Dilation

Persons who have undergone an eye checkup must be aware of this term. In a eye check up, for a clear examinations the doctor 'dilates' your pupil so that the anatomy of the eye inside can clearly be seen. Thus, the meaning of 'dilation' means an expansion. In geometry when a shape is made larger it is said to have been dilated or stretched.  This is one of the transformations to explain how function changes by visualizing its graph. Look at the following diagram.

The triangle PQR, when carefully noticed is an enlargement of triangle ABC by two times. Hence the sizes are different but still both the shapes look similar. Further it can be seen that the ordered pair of any point in triangle ABC is doubled, you get the ordered pair of the corresponding point in triangle PQR.

These facts give rise to the definition of dilation as the stretch of a shape by a definite ratio and the ordered pairs of the corresponding points are also stretched by the same ratio. When dilated, the shapes are not congruent but they are similar.

Contraction

Contraction is just opposite to dilation. That is instead of the original shape being expanded, it undergoes a reduction in size. Referring to the same diagram of the previous section we can say that the triangle ABC is a contraction of the triangle PQR. Thus a contraction is a reduction of a shape by a definite ratio and the ordered pairs of the corresponding points are also reduced by the same ratio. In this case also the shapes are not congruent but they are similar.

Compression

In case of contraction and dilation we had noticed that the shapes get reduced or stretched respectively by a certain ratio. There are similar type of transformations called compression and expansion. The only difference in these cases is the the reference point of the shape remains the same. Imagine a ball is inflated and expanded to a larger size. For example, the mouth where the air is pumped in remains at the same position. Similarly when the shape is shrinks the transformation is called a compression. These concepts are more applicable with functions and hence are better known as transformation of functions.The following example diagrams will give a clearer idea.

The graphs in the above diagram shows how the function f(x) = x2 (red) becomes compressed when the function changes as f(x) = 3x2 (blue) and expanded when the change is f(x) = 0.5x2 (green). These are called vertical compression and vertical expansion respectively. Horizontal Compression and Expansion

Horizontal Compression and Expansion

Like we had seen a expansion or compression in the vertical direction, there could be an expansion and compression in horizontal direction. The graphs in the following diagram show the horizontal compression and expansion.

We notice vertical changes when the entire function is multiplied by a constant factor. If the factor is > 1, the change is an expansion and if it is < 1, then a compression takes place. Similarly horizontal changes take place when the variable is changed by any constant ratio. But in this case the pattern of change is reverse. When the ratio is greater than 1, the change is a compression and for x < 1, it is an expansion.

When a change is made from a given function by changing it by a constant ratio or changing its variable by a constant ratio, the given function is called the 'parent function' and the changed functions are known as transformed functions. Thus compression and expansion come under transformation of functions. It may be noted that in case of dilation or contraction, the shapes change in size but they remain similar. But in case of expansion or compression, there is no congruency or similarity. But there are more types of transformations where the changed shapes are congruent to the parent shapes. Let us study about them in subsequent sections.

Translation

Translation means conveying the message from one language to another language. In geometry, translation of a function is shifting that from one point of reference to another point of reference. The overall operation of the function is not affected. It is more perceived graphically when the shape is shifted from one position to another position without change in size. Thus the congruency is maintained. The following diagram explains what a translation is.

In the above diagram is we see a triangle is shifted to right by 4 units. This means the variable in the function is 'added' by 4 units. Same way a shape can be 'translated' to any position, by shifting vertically up or down and or by horizontally moving left or right. The x-coordinate undergo a change when the shape is shifted horizontally (towards right when a constant is added or towards left when there is a subtraction). Similarly the y-coordinate undergo a change when the shape is shifted vertically (towards up when a constant is added or towards down when there is a subtraction).

Hence, when a function f(x) translates to f(x) + c, it moves up if 'c' is positive  and moves down if 'c' is negative. Similarly, when a function f(x) translates to f(x + c), it moves right if 'c' is positive and moves left if 'c' is negative.

Reflection

Reflection normally refers to the processes of producing an image for a given object across a reference line, usually a mirror. In such cases the image and the object are equidistant from the reference line but in opposite directions. In geometry, a reflection means transferring a point to an equidistant place in the opposite direction from a reference axis. If the reference axis is x-axis then the reflection is known as reflection across x-axis and if it is across y-axis then the reflection is called reflection across y-axis. In these cases, the object and image are symmetric across x-axis and y-axis respectively.

In the above diagram, the figure (i) shows a triangle in quadrant 1 of the coordinate axes. The reflection across y-axis is shown in figure (ii) and the reflection across x-axis is shown in figure (iv). Figure (iii) is the reflection across y-axis and again a reflection across x-axis. It is also same as reflection across x-axis first and then reflection across y-axis.
Thus, in general, an image in quadrant (ii) is a reflection across y-axis and an image in quadrant iv is the reflection across x-axis. An image in quadrant (iii) is successive reflections either first across x-axis and then across y-axis or vice-verse. It  may be seen that the ordered pairs of any point gets changed with the type of reflection.

Suppose (x, y) is the ordered pair of a point, it changes with the reflection as follows:

Reflection across y-axis (-x, y)
Reflection across x-axis (x, -y)
Reflection across both x-axis and y-axis (-x, -y)

The double reflection across x-axis and y-axis is also the same as or can be defined as the reflection along the line y = -x.

Alternatively, we can identify the type of reflection by observing the change in sign using the same concept. The reflection of a function can also be defined as follows:

g(x) = f(-x), as the reflection of f(x) across y-axis.
g(x) = - f(x) as the reflection of f(x) across x-axis.

Having studied all types of transformations we can form a combined formula for the transformation of a function f(x) as, a* f[(bx + c) + d], where,

1) 'a' denotes a vertical expansion or compression. If l a l > 1, it is a vertical expansion, if l a l < 1, then it is a vertical compression. If a < 0, then it is a reflection across x-axis.

2) 'b' denotes a horizontal expansion or compression. If l b l < 1, it is a horizontal expansion, if l b l > 1, then it is a horizontal compression. If b < 0, then the reflection is across y-axis.

3) 'c' denotes a horizontal translation. If c > 0, the translation is a horizontal shift towards right and if c < 0, then the translation is a horizontal shift towards left.

4) 'd' denotes a vertical translation. If d > 0, the translation is a vertical shift upwards and if d < 0, then the translation is a vertical shift downwards.

Rotation

In all cases of transformations studied so far, we had seen that the ordered pairs change in a straight line. But it is also possible that a translation can occur by rotation of points by rotating the ordered pairs. In this section let us consider rotation of a point (the concept can be extended to an object) only around origin. A rotation is generally measured in terms of degrees of angle and the counter clock wise direction is considered as positive. The common rotations are by angles 90o, 180o and 270o. Let us study how does the ordered pair of a point changes when rotated by 90o counter clock wise. This will help us to find an algorithm in general for rotation transformations.

Let P is a point in quadrant (i) which has ordered pair as (x, y). Suppose the point is given a rotation of 90o around origin. The x-coordinate of P is 'x' and when it is rotated by 90o, ox becomes as oy', that is the x-coordinate of P now becomes as the y-coordinate of the transformed point. Similarly the y-coordinate of P is 'y' and when it is rotated by 90o, oy becomes as ox', that is the y-coordinate of P now becomes as the x-coordinate (in the negative direction) of the transformed point. The transformed point P' is located with the help of the transformed coordinates. Thus, for the transformation of any point (x, y) by 90o, the ordered pair changes as (-y, x). By the same method it can be shown that the ordered pair changes as (-x, -y) for 180o rotation and as (y, -x) for 270o rotation.

The translation of a full shape under rotation can be formed by first drawing the reflections of the vertices or the key points of the original shape and joining them as in the original shape.