exc4libur and

Paulli1982 :

I have learned that the sign of a slope just tells us which direction it goes, not whether it is steeper or flatter.

I have learned that the slope of a line tells us the rate of change of y relative to corresponding change in x.

So it is hard for me to understand the reasons you answered the way you did.

As far as I can tell, you rest your case on the number line alone. Is that correct?

As I said, I learned that the slope of a line tells us the

rate of change of y relative to x.

That seems to imply that whether its slope is positive or negative, a line that is close to vertical has a very steep slope.

It also seems to imply that whether its slope is positive or negative, a line that is close to horizontal has a very gradual slope.

If the slope is 4, y increases rapidly compared to x.

If the slope is -4, y decreases rapidly compared to x.

Given those two mathematical relationships, it would seem that the magnitude of the slope, not its sign, determines steepness.

True, the sign tells you whether it goes "right and up", or "right and down."

But I cannot see how a positive or negative sign tells us how quickly y increases or decreases in relation to x.

Stated again: The slope of a linear function is a

rate of change.

A horizontal line, with a slope of zero, does not change. At all. Ever.

Think of a skateboarder.

First she is on a flat sidewalk. Zero slope.

Then she climbs up the left side of a hill facing that sidewalk.

She skateboards down the right side.

Are you willing to argue that she deals with a greater slope while on a flat sidewalk than while going down the hill?

If so, would you please explain why?

Or suppose there is a train.

It goes up a perfectly symmetrical mountain, which looks like this: /\

The train makes it to the top. At the top, it loses its brakes.

Now it is a runaway train. It is hurtling down the other side of the mountain.

Because, and only because, it hurtles downward and to the left, (and has a slope of, say -5), would you say its slope while hurtling without brakes is LESS than the slope on flat ground?

If so, would you please explain why?

I respect disagreement. That said,

Paulli1982 I do not think there is any need for words such as "obvious."

If it were so obvious, I would not be taking a different position.

Please explain to me how and why a horizontal line with no change (y is always the same) has a greater slope than a line that looks like this: \

Please explain how to account for the rate of change in your scenario.

Please explain how the sign of the number denoting of a rate of change in a linear function tells me how steep my hike will be, both up the mountain and down?

I think magnitude of change, or absolute value of the number that describes rate of change, determines greater or smaller slope, steeper or less steep slope.

I think a slope of -4 is greater than a slope of \(\frac{1}{2}\). Why? It has a greater rate of change.

Greater rate of change? Steeper slope.

Smaller rate of change? Flatter slope.

I am very curious to know

how you are measuring steepness. And

why. Maybe I am missing something.

If there is some way to order slopes about which I do not know, I would appreciate knowing.

Thanks in advance!

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"