VERTICAL EXPANSIONS AND COMPRESSIONS

How to Do Vertical Expansions or Compressions in a Function

Let y = f(x) be a function.

In the above function, if we want to do vertical 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 

ky  =  f(x)

or

y  =  (1/k)f(x)  

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

It can be done by using the rule given below.

Vertical Expansion - Example

Question :

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

               "a vertical 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

Answer :

Step 1 :

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

Step 2 :

So, the formula that gives the requested transformation is    

0.5y  =  √x

or

y  =  2√x  

Step 3 :

The graph y =  2√x  can be obtained by expanding the graph of the function y = √x vertically by the factor 0.5. 

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

Step 4 :

The graph of the original function (given function)

Step 5 :

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

How to sketch the graph of the function which is  vertically 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 vertical expansion in the graph of the function  y = f(x) by the factor "0.5", we have to write the point (x , y) as (x , 0.5y).

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

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

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

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

The same procedure to be followed for vertical compression.

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