How Do Function Transformations Work?

[WannabeFeynman]

Hello all, I need some help to clear my doubts.

Why does a horizontal translation (f(x + c)) move to the left if c is positive?

Can someone graphically explain what effect a stretch and compression (vertical and horizontal) has on the original parent function?

Similar to the first question, why does f(ax) actually stretch by a factor of 1/a instead of a?

Thanks, I might have more questions later.

 

[.Scott]

Is you graph y1=f(x) and y2=f(x+C) and then compare them, whatever "y1" you see at any "x1" will be seen to the left (assuming C>0) at x2="x1-C" for y2 because y2 = f(x2+C) = f(x1-C+C) = f(x1) = y1.
Similarly:
f(Ax) will horizontally thin the graph by a factor of "A" (or stretch it by 1/A)
Af(x) will vertically stretch the graph by a factor of "A".

 

[Seydlitz]

Take a look at this simple example, suppose $f(x)=x$, a simple straight line through the origin. If $x=0$, then $f(x)=0$ as well. But if you take $f(x+a)$, the origin will be at $-a$. If you draw the line, the origin will move to the left because the origin is now in negative part of the x-axis.

 

[Mark44]

Take a look at this simple example, suppose $f(x)=x$, a simple straight line through the origin. If $x=0$, then $f(x)=0$ as well. But if you take $f(x+a)$, the origin will be at $-a$. If you draw the line, the origin will move to the left because the origin is now in negative part of the x-axis.

The origin doesn't move around, but the x-intercept does.