Given the following
graph off(x),
graph the transformation–f(x+ 1) – 3

This transformation formula
has just about everything: there's a left-shift of one (the "+1"
inside), a move-down by three (the "–3"
outside), and a flip-upside-down (the "minus" sign out front).
And, worse yet, I have no formula for f(x),
so I can't cheat; I have to do the transformation.

The way the original graph
is drawn, there are a few points that I can use to keep track of things.
If I move those points successfully, then I can draw the rest of the
graph at the end.

What I'm going to show
you below is not what you would hand in. You would not show all this
work in your homework, and certainly not on a test. But this displays
the thinking that should be going through your head as you transform
each point you've chosen.

I'll do the
points from left to right, starting with the point (–3,
–2).

The first point I'll
work with is the point (–3,
–2).

First, I shift the
point left by one unit, to (–4,
–2).

Then I flip the point
over the x-axis,
up to (–4, 2).

Then I move the point
down three units, to (–4,
–1).

Now I'll look at the
second point marked on the graph, (–2,
–4).

I move the point back
to the left by one unit, to (–3,
–4).

Flipping across the
x-axis
moves the point to (–3,
4).

Moving the point three
units down takes it to (–3,
1).

Flipping across the
x-axis
takes the point to (1,
3).

Moving down three
units takes the point to (1,
0), on the axis.

The last point that
I need to move is (4,
2).

Shifting the point
one unit to the left takes it to (3,
2).

Flipping across the
x-axis
takes the point to (3,
–2).

Moving the point three
units down takes it to (3,
–5).

Now that I've moved
all the points, I can graph the transformation.

If you're not sure of the
order in which to do the transformation's steps, then stick to working
from the inside out (from the argument, to the function, to anything done
outside the function), like I did above. Pick a point to move, and trace
out the movements with your pencil tip, drawing in the point once you
reach the final location. Once you've moved all the points, draw in the
transformation.

For the above transformation,
I could have factored the minus sign out front of the expression and viewed
the transformation as being –[f(x
+ 1) + 3]. Then the point
movements would have been "left one, up three, and then flip across
the x-axis".
The end result would have been the same.

Stapel, Elizabeth.
"Function Transformations / Translations: Moving the Points."
Purplemath. Available from http://www.purplemath.com/modules/fcntrans3.htm.
Accessed