Transformations

Or: Is this the kind of problem where you can have more than one answer?

There are a select few topics in algebra that can get me tied up in knots. One of those is function transformations--you know, horizontal & vertical shifts, vertical stretches & compressions, and horizontal & vertical reflections--specifically, those that involve a multiple transformations.

Here's the task I got hung up on the other day:

The graph of a function f is shown. Sketch the graph of y = 2f(x+1) - 3.

On these types of tasks, it's not the transformations themselves that get me. There are three transformations at work here, and I can describe them easily enough:

1. Horizontal shift one unit left (because of the x+1).
2. Vertical shift three units down (because of the -3).
3. Vertical stretch by a factor of 2 (basically, double the y-values). 

The part that gets me hung up, at least when I have not done these kinds of problems in a while, is... in which order should I apply the transformations?

Because it makes a big difference! To illustrate, let's trace where the point (-2,1) ends up if we shift-then-double vs. double-then-shift:

• Shift-then-double: (-2,1) --left1--> (-3,1) --down3--> (-3,-2) --double y--> (-3,-4).
• Double-then-shift: (-2,1) --double y--> (-2,2) --left1--> (-3,2) --down3--> (-3,-1).

See? (-3,4) and (-3,-1).. we end up in two different spots.

This is not "the kind of problem you can have two different answers to" (Cathy Humphreys; clipped from one of the videos in this book).

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