Let’s start with a very simple thing.

Let's see how functions work.

Well, here is the x axis, full of numbers.

x axis

The function takes certain numbers from these, and assigns another number to them.

For example, it assigns their square.

This function is written like this:

Most of the time we will use this third type of notation.

Those lucky x values for which the function assigns something, are called the Domain, denoted by .

For x2, this is the entire x axis.

But here comes, for instance, ,

which is not defined for negative x values.

Therefore, the domain:

The part of the y axis that is assigned to the x values is called the Image, or the Range.

The Range is denoted by

And now, let’s get back to function x2.

The graph of function x2 is a parabola. The vertex of this parabola is in the origin.

But if we replace x with

then, well, the vertex moves.

The vertex of the parabola is always at the point where this is zero.

Right now it is at .

And then we have this, for instance.

If we add three outside the squaring function,

the graph will move up by 3 along the y axis.

This is called internal transformation,

and this one is external.

If there is such a thing as:

then, due to the internal transformation, it is shifted along the x axis,

and due to the external transformation, it is also shifted along the y axis.

Let’s see what happens if we write 2x here.

Well, then we see a bit of stretching along the y axis,

but this is not too exciting.

What is much more interesting is that the shifting has changed as well.

And now let’s see what this would look like.

We already know .

We have to move this along the x axis, let’s see...

by 3.

And by 2 along the y axis.

If there is such a thing as:

then, well, due to the 3x it will stretch somewhat,

and then the usual stuff.

If we write a negative sign in front of , then we will reflect the graph about the x axis.

If we insert the negative sign inside the radical, then we will reflect the graph about the y axis.

And if we feel like it, we can reflect the function about both axes.

It gets really exciting if we combine it with the shifting business.

Let's see, for instance, what this function would look like.

There will be some shifting along the x axis,

then along the y axis as well,

and finally, due to the negative sign, a reflection.

But if the negative sign is on the outside, then it is an entirely different case.

This is splendid; let's see what else could possibly come.

The art of “completing the square”.