HORIZONTAL STRETCH AND COMPRESSION

How to Do Horizontal Stretch in a Function

Let f(x) be a function.

Let g(x) be a function which represents f(x) after an horizontal stretch by a factor of k.

where, k > 1.

In the function f(x), to do horizontal stretch by a factor of k, at every where of the function, x co-ordinate has to be multiplied by k.

The graph of g(x) can be obtained by stretching the graph of f(x) horizontally by the factor k.

Note :

Point on the original curve : (x, y)

Point on the curve after it is horizontally stretched by the factor of k :

(kx, y)

How to Do Horizontal Compression in a Function

Let f(x) be a function.

Let g(x) be a function which represents f(x) after a horizontal compression by a factor of k.

where k > 1.

In the function f(x), to do horizontal compression by a factor of k, at every where of the function, x co-ordinate has to be multiplied by 1/k.

The graph of g(x) can be obtained by compressing the graph of f(x) horizontally by the factor k.

Note :

Point on the original curve : (x, y)

Point on the curve after it is vertically compressed by the factor of k :

((1/k)x, y)

Example 1 :

Perform a horizontal stretch by a factor 2 to the function

f(x) = (x - 1)2

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 :

Let g(x) be a function which represents f(x) after the horizontal stretch by a factor of 2.

Since we do horizontal stretch by the factor 2, we have to replace x by (1/2)x in f(x) to get g(x).

Step 2 :

So, the formula that gives the requested transformation is

g(x) = f[(1/2)x]

g(x) = [(1/2)x - 1]2

Step 3 :

The graph g(x) = [(1/2)x - 1]2 can be obtained by stretching the graph of the function f(x) = (x - 1)2 vertically by the factor 2.

(x, y) -----> (2x, y)

Step 4 :

Table of values :

Step 5 :

Graphs of f(x) and g(x) :

Example 2 :

Perform an horizontal compression by a factor 2 to the function

f(x) = (x - 1)2

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 :

Let g(x) be a function which represents f(x) after the vertical compression by a factor of 2.

Since we do vertical compression by the factor 2, we have to replace x by 2x in f(x) to get g(x).

Step 2 :

So, the formula that gives the requested transformation is

g(x) = f(2x)

g(x) = (2x - 1)2

Step 3 :

The graph g(x) = (2x - 1)2 can be obtained by compressing the graph of the function f(x) = (x - 1)vertically by the factor 2.

(x, y) -----> ((1/2)x, y)

Step 4 :

Table of values :

Step 5 :

Graphs of f(x) and g(x) :

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