Let's start with the function
notation for the
basic quadratic: f(x)
= x^{2}.
A function transformation takes whatever is the basic function f(x) and then "transforms" it or "translates" it, which
is a fancy way of saying that you change the formula a bit and thereby
move the graph around.

For instance, the graph
for x^{2} + 3 looks like
this:

This is three units higher
than the basic quadratic, f(x)
= x^{2}.
That is, x^{2} + 3 is f(x)
+ 3. We added a "3"
outside the basic squaring function f(x)
= x^{2} and thereby went from the basic quadratic x^{2} to the transformed function x^{2} + 3.

This is always true: To move
a function up, you add outside the function: f(x)
+ b is f(x) moved up b units. Moving the function down
works the same way; f(x)
– b is f(x) moved down b units.

On the other hand, (x +
3)^{2} looks
like this:

In this graph, f(x) has been moved over three units
to the left: f(x + 3) = (x + 3)^{2} is f(x) shifted three units to the left.

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This is always true: To shift
a function left, add inside the function's argument: f(x + b) gives f(x) shifted b units to the left. Shifting
to the right works the same way; f(x – b) is f(x) shifted b units to the right.

Warning: The common temptation
is to think that f(x + 3) moves f(x) to the right by three, because "+3"
is to the right. But the left-right shifting is backwards from what you
might have expected. Adding moves you left; subtracting moves you right.
If you lose track, think about the point on the graph where x = 0. For f(x + 3), what does x now
need to be for 0 to be plugged into f ? In this case, x needs
to be –3,
so the argument is –3
+ 3 = 0, so I need to
shift left by three. This process will tell you where the x-values,
and thus the graph, have shifted. At least, that's how I was able to keep
track of things....

The last easy transformation
is –f(x).

Look at the graph of –x^{2}:

This is just f(x) flipped upside down. Any points
on the x-axis
stay on the x-axis;
it's the points off the axis that switch sides. This is always true: –f(x) is just f(x) flipped upside down.

For this next transformation,
I'll switch to g(x)
= x^{3}:

If I put –x in for x,
I get (–x)^{3} = –x^{3}:

This transformation rotated
the original graph around the y-axis.
Any points on the y-axis
stay on the y-axis;
it's the points off the axis that switch sides. This is always true: g(–x) is the mirror image of g(x),
as reflected in the y-axis.