What is Transformation?

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Before getting into the concept, let us do an activity. Take a sheet of paper and draw a picture using straight lines. For example, if you are drawing a hut, you will have to use straight lines to make its walls, roof, windows, and doors. Now take a pack of match sticks from the matchbox. Place one stick on each line of the hut. What do you observe from this? Initially, the match sticks were kept horizontally in a box. But you might have kept it vertically over the walls. For the roofs, you might have placed the sticks in a slanting position. This is how we change the position of the stick from one to another. You can shift the sticks to the left or the right as you wish. Similarly, you can move them upwards or downwards.

Similarly, when you draw a line in a graph sheet, you can move the position of the line by adding some units, subtracting some units, multiplying some units, or dividing some units from the original function. Let us see the technical definition given below.

How to do Transformation on a Graph?

The transformation is done by shifting the functions to the right or left or by stretching or compressing the functions and so on. Some of the ways in which the transformation is done are given below.

What is Translation of Graph?

The translation is to move an object in such a way that every single point of the object moves in the same direction and for the same distance. The function’s domain remains the same.

There are two types of translation. They are given below.

Vertical Translations

Suppose that you are holding a ruler in your hand. Stretch your hands fully and hold the ruler horizontally parallel to your body. The ruler will be parallel to your shoulders. Now if someone asks you to shift it a little lower than the actual position in such a way that the ruler is still parallel and horizontal, you may bring the ruler down to your shoulder level. This is called vertical translation because the graph is translating along the vertical axis. That is, the movement is along the y-axis. This is also called a vertical shift.

Let us see how it appears in a graph. Take a graph sheet and draw a line parallel to the x-axis at a distance of k units upwards from the x-axis. The function is defined as f(x) = k for any value of x. You can translate the line by adding or subtracting ‘a’ units to the function to get a new function. This would give f(x) = k + a or f(x) = k – a. This will make the line either shift ‘a’ units upwards or ‘a’ units downwards along the y-axis.

The coordinates of any point on the graph are of the form (x, f(x) + a) or (x, f(x) – a) where ‘a’ is a constant.

Practice Problem

Consider the function$f(x)=x^2$. Graph the function and translate it along the vertical axis. (This is a linear function).

Solution

For the given function$f(x)=x^2$, the vertical translation will be given by the functions $g(x)=f(x)+a$ or$h(x)=f(x)-a$, where $a$ is any constant.

Consider$a=2$. Then,$g(x)=x^2+2$ and $h(x)=x^2-2$.

The red, blue and green curve represents the graph of$f(x)$, $g(x)$ and $h(x)$ respectively.

Horizontal Translations

Take a graph sheet and draw a line parallel to the y-axis at a distance of k units to the right of the y-axis. You can translate the line by adding or subtracting ‘c’ units. This will make the line either shift ‘c’ units to the right or ‘c’ units to the left along the x-axis.

This is called horizontal translation because the graph is translating along the horizontal axis. That is, the movement is along the x-axis. This is also called a horizontal shift.

The coordinates of any point on the graph is of the form (x + c, f(x)) or (x - c, f(x)) where c is a constant.

Practice Problem

Consider the function$f(x)=x^2$. Graph the function and translate it along the horizontal axis.

Solution

For the given function$f(x)$, the horizontal translation will be given by the functions $g(x)=f(x+a)$ or$h(x)=f(x-a)$, where $a$ is any constant.

Consider$a=2$. Then, $g(x)={(x+2)}^2$and $h(x)={(x-2)}^2$.

The red, blue and green curve represents the graph of$f(x)$, $g(x)$ and $h(x)$ respectively.

Dilation

Dilation means to either stretch or contract a graph in such a way that the direction and shape of the graph are not changed. But the size of the graph is altered. The stretching can be vertical or horizontal. It is done by multiplying or diving the coordinates such that it changes the location and structure of the graph. If the function is defined as f(x) = kg(x) for any value of x and some function g(x), the stretching is vertical.

Reflection of Graph

Have you seen your reflection in the mirror? Try carrying a board with some words written on it. If you stand in front of the mirror holding the board in your hand, what do you observe? The words in the board appear to be backward and retrace the path from where it travelled.

Reflection means the mirror image of an object. The reflection of a graph is done by multiplying by -1. We can reflect about the y-axis by multiplying x by -1. This gives –x. To reflect about the x-axis, we multiply the function f(x) by -1. This gives –f(x). By adding a minus sign, the positive values become negative values.

When y = f(-x), the graph reflects about y-axis. Whereas for y = -f(x), the graph reflects about x-axis.

Practice Problem

Find out if the following is a reflection, scaling or if the graph shifts horizontally or vertically:

1. y = f(x + 2)
2. y = f(x) + 2
3. y = 7f(x)
4. y = f(x/4)

Solution

1) For a function$f(x)$, the horizontal translation will be given by the functions $g(x)=f(x+a)$ or$h(x)=f(x-a)$, where $a$ is any constant. This implies that the given function shifts 2 units horizontally.

2) For a function$f(x)=x$, the vertical translation will be given by the functions $g(x)=f(x)+a$ or$h(x)=f(x)-a$, where $a$ is any constant. This implies that the given function shifts 2 units vertically.

3) If a function$f(x)$ is multiplied by a constant value$k$, then the function is said to be scaled by a factor of $k$vertically. This implies that the given function represents scaling by a factor of$7$vertically.

4) If the $x$ value in a function$f(x)$ is multiplied by a constant value$k$, then the function is said to be scaled by a factor of $k$horizontally. This implies that the given function represents scaling by a factor of $\frac{1}{4}$ horizontally.

Try to do some transformations to the constant function (f(x) = c), linear function (f(x) = x), quadratic function, absolute value function, square root function and cubic function. This activity is to enhance your leaning.

Real-Life Examples of Transformation

• The clock or watch has a minute and hour hand which keeps changing its position.
• Keeping a box on the floor and moving it in a straight line.
• While playing chess, the pieces are translated up and down.
• The rays of the sun reflected by the water.

Formulas

The formulas for different transformations of a function f(x) are as follows:

• Vertical translation: y=f(x)+c
• Horizontal translation: y=f(x+c)
• Vertical dilation: y=kf(x)
• Horizontal dilation: y=f(kx)
• Vertical reflection: y=-f(x)
• Horizontal reflection: y=f(-x)

Here c is any real number and k is any positive real number.

Context and Applications

It is used in the K-12 curriculum, undergraduate and post-graduate mathematics, and is asked in entrance examinations.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for      

• Bachelors Mathematics
• Masters Mathematics