What does slope mean?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Slope refers to the steepness of a line.

A line written in slope-intercept form is written as y=mx+b where m is the slope of the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls as you scan from left to right. A horizontal line has zero slope, while a vertical line has no slope (the slope is undefined.)

Graphically the slope is the ratio of the rise over the run -- the vertical change divided by the horizontal change.

We can compute the slope of the line if we are given any two points on the line -- the slope is the difference in the y-values divided by the difference in the x-values. For example, the slope of the line containing (1,1) and (3,5) has slope (5-1)/(3-1)=2. (Note that we can go the other direction -- (1-5)/(1-3)=2)

Slope is the constant or average rate of change. For example, if a car is moving at 60mph, the distance function is a line with slope 60.

In calculus we find the slope of the line tangent to a curve at a point using teh derivative of the function.

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studyfaq1 | eNotes Newbie

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The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line is, or how much y increases as x increases. The slope is constant (the same) anywhere on the line.

One way to think of the slope of a line is by imagining a roof or a ski slope. Both roofs and ski slopes can be very steep or quite flat. In fact, both ski slopes and roofs, like lines, can be perfectly flat (horizontal). You would never find a ski slope or a roof that was perfectly vertical, but a line might be.

We can usually visually tell which ski slope is steeper than another. Clearly, the three ski slopes get gradually steeper.

You could write the relationship like this:

Slope = (Change in height)/(Change in width)

Or:

Slope = rise/run

If y represents the vertical direction on a graph, and xrepresents the horizontal direction, then, this formula becomes:

Slope = (Change in y)/(Change in x)

Or:

In this equation, m represents the slope. The small triangles are read 'delta' and they are Greek letters that mean 'change.'

For the first ski slope example, the skier travels four units vertically and ten units horizontally. So, the first slope is m = 4/10.

The second ski slope involves a seven unit change vertically and ten units horizontally. So, the slope is m = 7/10. The second slope is steeper than the first because 7/10 is greater 4/10.

The third ski slope involves a seven unit change vertically and six units horizontally. So, the slope is m = 7/6. The third slope is the steepest of all.

Slope on the Cartesian Plane

The Cartesian plane is a two-dimensional mathematical graph. When graphing on it, a line may not start at zero as in the ski slope examples. In fact, a line goes on forever at both ends. The slope of a line, however, is exactly the same everywhere on the line. So, you can choose any starting and ending point on the line to help you find its slope. It is also possible that you might be given a line segment, which is a section of a line that has a beginning and an end. Or, you might be given two points and you are expected to draw (or imagine) the line segment between them. In all these situations, finding the slope works the same way.

Just like with the ski slope, the goal is to find the change in height and the change in width. For the line segment in the image, you can simply count the squares on the grid.

Graph of a line segment

The difference in height between the two points is three units (three squares). The difference in width between the two points is two units (two squares). So, the slope of the line segment (the slope between the two points) is m = 3/2.

In mathematics class, you may memorize a formula to help you get the slope. The formula looks like this:

 
 
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kandukurimaths | Student, Graduate | (Level 1) Honors

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straight line have one good property that rate of change is constant.means

ax+by+c=0 is general straight line equation. let x is independent variable and dependent variable and two arbitrary points (x1,y1),(x2,y2) then change in y coordinates by change in x coordinates is constant

                     slope=(y2-y1)/(x2-x1)

example y=2x 

if we draw graph y=2x

lets take two points on line (1,2) and (2,4)

slope=(4-2)/(2-1)=2

from this example we need to observe two points

point one:

          if you write straight line in the form of y=mx than m is slope

second point:

          if see in graph we can form right angle triangle with given two points

and intersection of y=y1 and x=x2 that is (2,2)

using trigonometry function  tan(theta)=(y2-y1)/(x2-x1)

any straight line equation can write in the form of y=mx+c

m is slope of line 

if we know two points on line than we can calculate slope using (y2-y1)/(x2-x1)

if we know line making angle with x-axis is theta than we can calculate slope=tan(theta)

In example intercept zero but it wont effect the slope of line

thank you 

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