###
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 functionOn 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:fis shown. Sketch the graph ofy= 2f(x+1) - 3.

- 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).

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).

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).

Of course, after thinking it through, it becomes clear that we must mirror the order in which the function 2

*f*(

*x+*1) - 3 is evaluated

*.*If we try to do the transformation in the wrong order, we get stuck in conversations like:

The only sequence of transformations that can work is the one that traces the operations involved from input to output. That is, we must:Me:I'm going to subtract 3 first.

You:Subtract 3 from what?

Me:From 2f(x+1).

You:But you we haven't calculated that yet.

Me:Oh.

- Evaluate (
*x*+1), - Apply
*f*, - Double the result, and finally
- Subtract 3.

*f*(

*x+*1) - 3

*for any particular value of*

*x*, we must first calculate

**. The horizontal shift happens first.**

*f*(*x*+1)f(x+1) |

Once we have

**, we double it:**

*f*(*x*+1)**2**. So the vertical stretch comes second.

*f*(*x+*1)2f(x+1) |

And once we have

**2**, we can perform the final transformation:

*f*(*x*+1)**subtract 3**.

2f(x+1) - 3 |

Until then... "Mathbots, roll out!"

## No comments:

## Post a Comment