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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* = 2*f*(*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:

- Horizontal shift one unit left (because of the
*x*+1).
- Vertical shift three units down (because of the -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).