Question

the equation g(x) = a(x-h) +k
h is a horizontal transltion (left or right)
k is a vertical translation (up or down)
.. but i don't understand what a is..
it supposed to be a "vertical stretch or compression" but what does that mean? and how can i apply that to the graph?

Answer 1

The only way to really understand this is to visualize it:

I suggest downloading geogebra (a graphing program, free, works on all operating systems) and graphing your problem. You can toggle all the values a, h, and k.

http://www.geogebra.org/cms/en/installers

Answer 2

So once you understand one function, and once you understand how translations, stretches, and compressions (transformations in general) work,
then you can understand an entire class of functions.

For example:
Let’s say you know how to draw
f(x) = x^2.

It’s a parabola.

The question is:
How is that related to g(x) = (x – 1)^2?
How is that related to h(x) = (1/2)x^2?
How is that related to p(x) = 2x^2 + 4x + 2?

It turns out that all of these “quadratic polynomials” take on
essentially the same shape, they’re all parabolas.

So you don’t have to learn how to draw each function separately.
You just need to know the shape of the original function f
and g, h, and p are just translations and stretches of f.

That’s how we can apply knowledge of transformations to graphs.

I think the reason why you’re having trouble understanding “stretch”
is because the function you gave is a line.
So when you “stretch it vertically,”
it just appears like you’re rotating the line more than anything else.

y = 2x is a vertical stretch of y = x.
But you don’t see that. You see it as a rotation
(because of our experience with levers rotating.)

You only see why it’s called a stretch when you
try f(x) = x^2 and h(x) = 10x^2. Then try g(x) = (1/4) x^2.
h(x) is a “stretch”, and g(x) is a “compression”
and in general, |a| > 1 gives a stretch while |a| < 1 is called a compression.

Negative values for a, such as -1, flip the graph.
Like f(x) = -x^2.