Let's say we have a graph f(x). f(x) equals some other random equation like x^3 or something.

Now let's say g(x) is a transformation of f(x). Take a look at the following implications looking at different values of g(x)

Let's say g(x) = f(Ax)

if A = 1, then the graphs are equal obviously. 

if A = 2, then the graph is compressed horizontally
if A = 5, then the graph is compressed horizontally even further.

if A = 0.5, then the graph is stretched horizontally
if A = 0.1, then the graph is stretched horizontally even further

As you can see, as A increases, the graph is compressed horizontally. And as A decreases, the graph is stretched horizontally. Using that logic, the further A decreases, then the more the graph is stretched.

But that's not the case. Because once A dips below zero, it's like the graph reverses or something. As you go from 1 to 0.1 to 0.01 to 0.001...etc. the graph continues to stretch. But as you go from to -0.01 to -0.1 to -1 etc, the graph begins to compress, which completely changes the direction it was previously going in. This doesn't make sense to me. Shouldn't the effect of increasing or decreasing A be continuous even when passing zero?

I understand the rules and know how to use it I started thinking about the effects of a graph as you change A, and this just seemed strange. As with every other transformation (verticle stretch, vertical shift, horizontal shift), the change seems to correlate continuously with their value but not with horizontal stretches. Does anyone find this strange?

FUCK YOUR MATH NERD

This is sacred ps4 only time!

ah dammit. I'm probably going to have to bump this in several hours as it's already nearly off the front page.

I will try to bump it as well. This was literally the worst time to post this. :D:D:D:D

It's to do with the absolute value of A, not whether it is positive or negative. I think it is a property of homogeneous functions, as -f(Ax) = f(-Ax), so it has to have that kind of symmetry

It makes sense that the graph's behavior would change when transitioning from positive to negative, as well as the other special threshold in this case which is the number 1 (since we are multiplying). Take a number X, where (inf. > abs(x) > 1), here it means that it is raised to a positive power, and (1 > abs(x) > 0) means a negative power.

Scaling (stretching) depends on the absolute value of the scaling factor, and how it relates to the number one. Flipping depends on whether the factor is positive, or negative. Apply at the same time and you get all those changing intervals, notice the change at each threshold.

(1 - inf.) = compressed, non-flipped.
[1] = equal, non-flipped.
(0 - 1) = stretched , non-flipped.

[0] = no graph i.e. a single point.

(-1 - 0) = stretched , flipped.
[-1] = equal, flipped.
(-inf. - -1) = compressed, flipped.

Also, the same does happen with vertical stretching. However, in the case of shifting, we are not multiplying, so the number 1 is no longer a critical point of concern, only the sign of the factor.

I see, It's just strange that horizontal stretches are based on absolute value rather than positive or negative, while vertical stretches, vertical shifts, and horizontal shifts are completely different. 

Never heard of homogeneous functions. Should I have?

Isn't a horizontal stretch just a vertical squish though? I'm pretty tired as it's late here so I can't really picture it, but I'm not sure the vertical transformations behave differently. I'm thinking of g(x) = Af(x) meaning the same as (1/A)g(x) = f(x), again only with homogeneous.

Homogeneous just means that you can put the scalar out, i.e. f(Ax) = Af(x). So for y=2x, or y=x^2 (except that homogeneous of degree 2 as the scalar is squared).

Huh? The same doesn't happen with vertical stretching. The direction of the graph correlates with whether the positivity/negativity of A in Af(x).

In vertical stretching, as A decreases the graph is pulled downward. Even as A goes negative, the graph continues to go downward. The opposite is true when positive of course.

This is different from horizontal stretching. As A in f(Ax) decreases, the graph widens away from the Y-axis (assuming the graph is symetrical along the Y axis), but once it hits zero, it stops widening away from the Y-axis and begins to shrink toward the Y-axis

This is impossible since, like Osc89 said, a vertical stretch is just a horizontal compression.

...Define vertical stretching?

@bolded, what you're describing sounds like shifting...