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In my book, the definition of the slope of the straight line is:

The slope is a measure of the direction of the line.

1) When the line has no slope, it tells that it is vertical or moving vertically along the $y$ and $y'$ axis.

2) When the slope value is equal to $0$ it tells that the line is moving horizontally along $x$ and $x'$ axis.

3) When the slope value is positive, it tells that the line is rising to the right.

4) When the slope value is negative, it tells that the line goes downward to the right.

5) A large positive slope value tells that the line goes along the $y-axis$ and is rising steeply to the right, and a small positive slope value tells that the line goes along the $x-axis$ and is rising slowly to the right.

Well, I'm not sure about the fifth one. Do a large positive slope value and a small positive slope value judge the direction of the straight line or if its rising steeply or slowly, does that judge the direction of the line?

Samama Fahim
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  • If you're familiar with the measurement of slopes of roads in mountaineous regions of the world (measured in %, like 7%, which is pretty steep for a road) the slope of a line in the $xy$-plane is exactly the same thing, only we don't usually use %, so 7% would mean a slope of $0.07$. PS: This should tell you that roads are usually very flat when viewed from the side. – Arthur Apr 30 '13 at 20:54

2 Answers2

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Do a large positive slope value and a small positive slope value judge the direction of the straight line or if its rising steeply or slowly, does that judge the direction of the line?

Both:

You can use slope to determine how steep a line is; if a line is steep, it's slope will be larger than a line that is less steep, and the steeper the line, the larger its slope.

So they are mutually correlated: steepness increases as slope increases (directly and positively proportional): (each gives information about the other...

But steep, as a description itself, is relative to some orientation. Usually we mean that steepness is a measure of the absolute value of the slope: the larger the magnitude of the slope, the closer a line with that slope is to the y-axis.

The sign of the slope tells us in what in what direction the line is tilted, if it is "tilted" whether y is increasing from "left to right" (positive), whether $y$ is decreasing from left to right, or neither(0 = horizontal, or slope is not defined = vertical.)

amWhy
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  • And the less the slope value, does that mean that the line with that slope is closer to the x-axis? – Samama Fahim Apr 30 '13 at 20:57
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    It is rising, but more slowly, when as the slope decreases, so it lies closer to the x-axis than to the y-axis. Compare, e.g., $y = \dfrac{1}{2}$ with $y = 2$. We know that at $y = x$, the angle between the line and the x axis is equal to the measure of the angle between the line and the x-axis, rising at a $45^\circ$ inclination. So $y = \dfrac 12 \implies$ measure of the angle between the line and the x-axis, is smaller than $45\circ$, so closer to the x axis than the y. in contrast $y = 2x$ is steeper than $y = x$, and so the angle between it and the x-axis is greater than $45^\circ$ – amWhy Apr 30 '13 at 21:02
  • Does that make sense? If you see a picture of two lines graphed, you can compare their steepness, and know which has a slope of greater magnitude. If you don't have a graph, but you have the equations for each of two lines, you will know which line, if graphed, would be steeper than the other. – amWhy Apr 30 '13 at 21:07
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    You need both the magnitude of the slope, and its sign, to know the direction and steepness of a line. – amWhy Apr 30 '13 at 21:09
  • That means am I right in saying that the slope value whether a large positive value or small also determines the orientation of the line with that slope? – Samama Fahim Apr 30 '13 at 21:12
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    Yes, absolutely... – amWhy Apr 30 '13 at 21:13
  • nitrous2 has a different opinion, and I'm confused. Whether a line is rising steeply or slowly, it is rising to the right. In both the cases, the direction is the same. – Samama Fahim Apr 30 '13 at 21:39
  • If you mean by direction the answer to whether it is rising or falling to the right, then sign is all you need. But orientation (how steeply/quickly a line rises or falls) also depends on the magnitude of its slope. For example, the minute hand of a clock is almost 90 degrees with respect to the x-axis, when 1/2 minute has elapsed from the top of the hour, but almost 0 degrees when 14.5 minutes have elapsed from the top of the hour. – amWhy Apr 30 '13 at 21:50
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    @amWhy: such detailed and nice descriptions! +1 – Amzoti May 01 '13 at 00:26
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Take a few of these examples.

a) $y= x+1$

b) $y= 10x+1$

c) $y= -10x+1$

In line a, we find that the slope is x which is equal to $\dfrac 11$ which means rise 1, run 1. The graph is positive due to the slope being positive. This also means it is going to the right.

In line b, we find that the slope is $10x$ which is equal to $\dfrac {10}1$ which means rise 10, run 1. The graph is positive due to the slope being positive. This also means it is going to the right.

In line c, we find the slope is -10x which is equal to $\dfrac {-10}1$ which means drop -10, run 1. The graph is negative and goes to the left.

All of these graphs have the same y intercept, the only differences is the slope.

Examples B & C are very similar, the only difference is that C has a negative slope and goes to the left and B has a positive slope that goes to the right.

Another thing to point out is notice the steepness of example a vs. b and c.

This might be a handy tool to help you visualize these examples: https://www.desmos.com/calculator

nitrous2
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  • So if a line is rising steeply to the right or rising slowly to the right doesn't define its orientation? – Samama Fahim Apr 30 '13 at 21:05
  • A line of A) $y = 25x + 1$ and a line of b) $y = 1x + 1$ are both examples of positive lines and positive slopes. A is a steep line and B) is a line with a steady consistent increase. Both lines are going in the same direction which is positive. – nitrous2 Apr 30 '13 at 21:18