1. Scaling and Shifting 1-D FunctionsWilliam T. Rhodes

11Jan08

We will frequently work with scaled and shifted versions of various functions. Consider the function

and three shifted-scaled versions given by

If you understand the procedures that take you from to , , and , you are prepared to handle most any scaling-shifting operation.The functions and are sketched in Fig. 1 so as to show their relationship to one another. Note that the total width of the function has been increased by a factor of 2 and what began at the origin for , i.e., the peak of the triangular section of the function, ends up 1 unit to the right. The associated procedure can be applied to the more general function , where the scaling-shifting operations consist of (a) stretching the function by the factor a, shifting the resulting scaled function in the positive-t direction by the amount b, and then scaling the amplitude by the factor c. Thus to obtain , you first stretch by a factor of 2, then shift the resulting stretched function right by 1 unit, and finally scale the amplitude by the factor 3. Practice with the function .

tt

( )f t

1 2

1

w

t

132

tf

1 1 52w

3

Fig. 1 Shows and its scaled-shifted version . The arrows suggest the operations (scale, shift, amplify) associated with the three respective parameters 1, 2, and 3.

I will refer a scaled-shifted version of written in the form

as being in standard scaled-shifted form. In most cases the value of the scaling parameter a will be positive (if not, see the note below on reflecting through the origin). Values of result in a stretching, i.e., the function gets longer along the t-axis. If , the scaling operation results in a compressed, or squeezed version of .

The function is not in standard scaled-shifted form, but it can be rewritten to be so:

.

Following the procedure taken for , we can easily sketch , as shown in the Fig. 2(a). Note that the + sign before the 2 in the standard-form numerator means that what lies at the origin for the starting function must now be moved 2 units to the left, i.e., minus 2 units in the positive-t direction. As a check on the reasoning, set t equal to 2 and note that you must be looking at .

t

23

tf

52

1

4 tt

23

tf

52

1

4t

22

tf

2 2

1

4 tt

22

tf

2 2

1

4

( )a ( )b

Fig. 2 Shows two scaled and shifted versions of , and . Note that the minus sign in the latter leads to a reflection of the starting function.

Dealing with the minus sign in the scaled-shifted function requires only slightly more effort. My approach is to first define the reflected version of , illustrated in Fig. 2(b) and given by

,and then write in the form

.

Knowing the shape of (you should sketch this function for practice), you can then easily sketch following the earlier procedures, the result being shown in Fig. 2(c).

The following comments relate to what we have discussed:

1. Any stretching or compression of a function is performed about the origin. Think in terms of a rubber sheet containing a picture of which you then stretch by, e.g., a factor of 2.

2. When working with standard-form scaled-shifted versions of , it is important that you perform the scaling and shifting operations in the correct order, for the two operations do not commute (i.e., you cannot in general reverse the order in which they are performed and obtain the same result). Verify this fact for yourself by taking , shifting it right by 1 unit, and then stretching the result by a factor of 2 ( remembering to stretch about the origin, not about any other point on the function). If you perform the operations as stated, you will not get the result obtained above for .