# HORIZONTAL EXPANSIONS AND COMPRESSIONS

## How to Do Horizontal Expansions or Compressions in a Function

Let y = f(x) be a function.

In the above function, if we want to do horizontal expansion or compression by a factor of "k", at every where of the function, "x" co-ordinate has to be multiplied by the factor "k". Then, we get the new function

y  =  f(kx)

The graph of y = f(kx) can be obtained by expanding or compressing the graph of  y = f(x) horizontally by the factor "k".

It can be done by using the rule given below.

## Horizontal Expansion - Example

Question :

Perform the following transformation to the function        y = √x.

"a horizontal expansion by a factor 0.5"

And also write the formula that gives the requested transformation and draw the graph of both the given function and the transformed function

Step 1 :

Since we do horizontal expansion by the factor "0.5", we have to replace  "x" by "0.5x" in the given function y = √x.

Step 2 :

So, the formula that gives the requested transformation is

y =  √0.5x

Step 3 :

The graph y =  √0.5x  can be obtained by expanding the graph of the function y = √x horizontally by the factor 0.5.

(x , y ) -----------------> ( 0.5x , y )

Step 4 :

The graph of the original function (given function)

Step 5 :

The graph of the function in which horizontal expansion made by the factor "0.5".

How to sketch the graph of the function which is  horizontally expanded or compressed ?

Let "y = f(x)" be the given function and (x , y) by any point on the graph of the function y  =  f(x).

If we want to perform horizontal expansion in the graph of the function  y = f(x) by the factor "0.5", we have to write the point (x , y) as (0.5x ,  y).

That is, "x" co-ordinate of each and every point to be multiplied by the factor 0.5.

Therefore, any point on the horizontally expanded graph will be in the form of   (0.5x , y)

So, each and every point to be changed according to (0.5x, y) and plot them on the graph.

After having plotted the points, if we connect all the points, we will get the horizontally expanded graph.

The same procedure to be followed for horizontal compression.

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