Math Help - Is Vertical expansion the same as horizontal compression?

A simple Math 12 question:

Consider a) y=x and b) y=5x.

So, b) is vertically expanded by a factor of 5 compared to a) right?
Can I also say that it is horizontally compressed by a factor of 5, and vertically compressed by a factor of 1/5?

Thanks.

Yes and No.

For simpler forms, it is easy enough to see that you CAN design a vertical expansion that looks like any given horizontal compression. This does not mean they are the same thing, only that the result is the same.

Odd Example

$1 = 50¢+50¢
$1 = 25¢+25¢+25¢+25¢
$1 = 25¢+25¢+25¢+10¢+10¢+5¢

It's still a dollar, you just get there different ways.

More-to-the-point example

Start at (0,0).
Walk Up 1 - You are at (0,1)
Walk Right 1 - You are at (1,1)

Remember that and do this.

Start at (0,0).
Walk Right 1 - You are at (1,0)
Walk Up 1 - You are at (1,1)

You ended up in the same place, but took a different path to get there.

It may not seem all that important, and often it isn't. As long as things are nice and continuous and well-behaved it is likely no one cares. When you start adding singularities, it requires greater care. Ask Han Solo! "Traveling through hyperspace ain't like dusting crops, boy. Without precise calculations we could fly right through a star or bounce too close to a supernova, and that'd end your trip real quick, wouldn't it?"