My thoughts (I am not an expert, but I wanted to take a crack at it)


What does a line with a positive slope less than one look like? It's going to be increasing, but much flatter than > 1. We're moving 1 "y-space" for every z "x-spaces", if the slope is 1/z.
What's the slope of a very steep line? As the line gets steeper, we eventually end up with an undefined slope. We have a huge change in Y with no change in X. A straight vertical line has an undefined slope, since the change in x = 0, and we'd be dividing by 0.
What's the slope of a line with negative x- and y-intercepts?

What happens if you increase "b"? The Y-intercept will increase
What happens if we Decrease "b"? The Y intercept will decrease

I assume for the last two that the equation is setup as y = mx + b. B is then the Y-intercept, since when x = 0, y = b.

The x-intercept can be derived by calculating x = (y-b)/m. If Y = 0, then x = -b/m. So the x intercept is a function of b AND m.

for example, if b was 3 and m was -1/3, then the x intercept would increase if we increased b to 4.

-3/(-1/3) = 9 /// -4/(-1/3) = 12

Please feel free to correct me if I'm wrong.

Thanks for the reply.Yeah I agree with you on everything actually. I just wanted to add that when you increase/decrease b, the x-intercept also changes it's value. Since b stands for Y-intercept,it's obvious that Y-intercept increases when b increases and Y-intercept decreases when b decreases. However,there's more than that!You can check for yourself and you'll find out that as b increases at the same time the X-intercept will decrease, and as b decreases the X-intercept will increase.

P.S. I thought that the X-intercept can be derived by setting y=0, I didn't know about x=(y-b)/m.

You are correct that x the intercept does change - we're just not sure whether it increases or decreases.

You are also right that the x-intercept is when y = 0. It's just that we still have 2 variables when y = 0; b and m.

if y = mx + b, we can rearrange to isolate x -> x = (y-b)/m. If we set y = 0, then x = -b/m.

When x is zero, it's easier, because it takes out the m variable and we're left with y = b.

The easy way to identify >1 versus <1 slopes is to picture a slope of 1 for a line that cuts through the origin. That line will form an angle of 45 degrees with each axis. Looking at any other line going through the origin, if the line in quadrant one is below our original line, the slope is less than one. If the line in quadrant one is above our original line, the slope is greater than one.