powered by Tutorvista.com

Sales Toll Free No: 1-855-666-7440

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

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.

The graphs in the above diagram shows how the function f(x) = x

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.

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.

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.

Let P is a point in quadrant (i) which has ordered pair as (x, y). Suppose the point is given a rotation of 90

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.