Mastering Mathematics Smartly
by Wee Wen Shih

Given the graph of y = f(x), it is often mind-boggling for the student to wonder how he/she should go about to sketch the graph of y = f(1 - 2x)?

First, let us take a closer look at the RHS-expression f(1 - 2x). We observe that there are three transformations to be performed, namely:

T1. Translation, parallel to the x-axis,
T2. Reflection about the y-axis,
T3. Scaling of factor 1/2, parallel to the x-axis.

Knowing the transformations is not enough, for we do not know the sequence of transformations to help us to sketch the final graph. Thus, it is natural for us to ask these questions:

Q1. Does it matter whether reflection comes before scaling OR scaling comes before reflection?
A1. You could use a simple example (such as y = x versus y = -2x) to convince yourself that order does not matter.

Q2. Does it matter whether reflection & scaling come before translation OR translation comes before reflection & scaling?
A2. Yes, and we wish to discuss further by considering two possibilities.

Case 1: Reflection & scaling before translation
We start with the graph of y = f(x). 

After a reflection about the y-axis (i.e. T2), we obtain the graph of y = f(-x). In short, we replace x with -x in the RHS-expression.

After a scaling of factor 1/2, parallel to the x-axis (i.e. T3), we obtain the graph of y = f(-2x). In short, we replace x with 2x in the RHS-expression.

In order for us to obtain the graph of y = f(1 - 2x), we will need a translation of 1/2 units to the right (i.e. T1), since we replace x with x - 1/2 in the RHS-expression, i.e. f(-2(x - 1/2)) = f(-2x + 1).

Case 2: Translation before reflection & scaling
Again, we start with the graph of y = f(x).

After a translation of 1 unit to the left (i.e. T1), we obtain the graph of y = (x + 1). In short, we replace x with x + 1 in the RHS-expression.

After a reflection about the y-axis (i.e. T2), we obtain the graph of y = f(-x + 1). In short, we replace x with -x in the RHS-expression.

After a scaling of factor 1/2, parallel to the x-axis (i.e. T3), we obtain the graph of y = f(-2x + 1). In short, we replace x with 2x in the RHS-expression.

In conclusion, we see that the action of translation differs in each case (1/2 units to the right in case 1 as opposed to 1 unit to the left in case 2), depending on the order of transformations one considers.

Hopefully, this discussion will provide you with the general approach of investigating the sequence of transformations to be taken to sketch the right graph in your tests and examinations!

Try this: Given the graph of y = f(x), what is the sequence of transformations necessary to sketch the graph of y = f(-|x/2|)?

The next difficulty a student typically faces is the manual sketching of the final graph. Take the example of y = f(1 - 2x) which is the result of three transformations, it is rather tedious to draw a version after each transformation ya?

We can cut down much work by means of a table. The first column indicates all the transformations and the subsequent columns indicate the important features of the y = f(x) graph (such as points and asymptotes) and how they change with each transformation. By looking at the last row, we can then construct the graph of y = f(1 - 2x) rather easily.

What if the graph of y = f(1 - 2x) is given and we are asked to sketch the graph of y = f(x)?

We need to know the sequence of transformations from y = f(x) to y = f(1 - 2x). Then we simply reverse the sequence and actions, i.e.

1. Scaling of factor 2 (note that it was 1/2 previously), parallel to the x-axis.

2. Reflection about the y-axis.

3. Translation of 1 unit to the right (note that it was left previously).

The table will again be very handy to help us sketch the graph of y = f(x) manually.