Sunday, April 1, 2007

Graphing - Slopes

When you have a line on a graph, you will probably need to know its SLOPE. A slope on a graph means essentially the same thing as if you were talking about the slope of a driveway or the slope of a ski hill... it is a measure of how steep the driveway/hill/line is.

The minimum amount of information you need to find the slope of a line is the location of two points on the line. These could be endpoints for a line segment, or just points on a line that goes on forever. Since it's a single, straight line, ANY two different points that are on the line can be used and will give you the exact same slope as any other two points on the same line. Makes sense right? The slope is a property of the line, and all the points on the line make up the line.

The slope formula is easy to remember. The complicated way of saying it is 'the slope is the difference in height of two points on a line, divided by the difference in width of the same two points.' The much easier way is 'slope equals RISE OVER RUN.'

slope = rise / run

The rise (think rising = height) is the difference of the y-coordinates of two points on a line. The run (think about running on the street = horizontal) is the difference of the x-coordinates of the same two points.

Usually, slope is represented by the letter 'm.' So then, the slope equation can be written like:

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

This appears to be a little more complex than the first one, but it means the same thing. The 1 and 2 are just names for the y's and x's. (They could be anything... A and B, or whatever.) The (y2-y1) means 'the difference between two y-coordinates, and the (x2-x1) means 'the difference between two x-coordinates.' IMPORTANT: Make sure that the point you use for y2 is the same point you use for x2, and likewise for y1 and x1. (Otherwise, you'll get the wrong slope.)

Another IMPORTANT thing to recognize: if the graph rises to the right, it is said to have a POSITIVE SLOPE. If it is falling towards the right, it has a NEGATIVE SLOPE. That means your slope value will have a positive or negative sign with its number. You can check that when you're done. (It's easy to make sign errors. It happens to everyone.)

This figure should hopefully clear things up.

Let's look at the black line first. Let's call the top point 'point 1' and the bottom point 'point 2'. So:

rise = (y2-y1) = (7-1) = 6
run = (x2-x1) = (7-2) = 5 **Notice that I didn't use (2-7)!

slope = rise/run = 6/5 or 1.2

That's it! Let's look at the red one now. It's a little trickier because of the negative signs, but you do the exact same thing. Point 2 on the left, point 1 on the right:

rise = (y2-y1) = (2-(-5)) = (2+5) = 7
run = (x2-x1) = ((-6)-(-4)) = ((-6)+4) = (-2)

slope = rise/run = 7/(-2) = (-7/2) or (-3.5)

Notice how the first graph had slope of (positive) 1.2 and was going up to the right, and the second one had slope (-3.5) and was falling to the right.

As long as you keep y2 and x2 coming from the same set of coordinates (and y1, x1), and you keep track of your signs, you'll get the right answer for your slope! And with the slope, you can do more complex things, like find an EQUATION that describes the line.


  1. I have been reading through your blog trying to figure out a graphing problem. I have been trying to use geometry and trigonometry to solve the problem, keep getting stuck and can't seem to put all the pieces together to find the solution. Am hoping maybe you or one of your readers can help point me in the right direction.

    Essentially, I have polygonal chain (aka polyline) defined by a series of points. I am trying to find the points necessary to create parallel polygonal chains above and below the original set of points. I created a graphic representing the problem here:

    Figuring out the slope is easy enough. The complex part of the problem is that the length of the parallel line segments changes as the polylines "bend". I can't figure out how to calculate how much I need to change each parallel segment's length.

    Any ideas or pointers on how I can go about solving this type of problem?

  2. Hi eric_c,

    I will admit that I've never dealt with problems like this one. Can I assume that the original (black line) formula is a piecemeal formula, which states that for the domain 0 - 6, y = 0; for the domain 6 - 10, y = 1/2x - 3, etc. You are right, the slopes will be the same for the similar segments of each line, but the equation describing that segment will have a different intercept, and so a different overall equation with a different domain.

    However, (considering I have not encountered a problem like this one,) I bet the solution is not that simple, and you are looking for a way to describe how the top line maintains a 2 unit distance above the base line, and the bottome line maintains a 2 unit distance below it, independent of specific domains or equations. Does it have anything to do with solving for the perpendicular slope through each segment and calculating a 2 unit distance in either direction? Just an initial thought...

    I'll continue to work on it, to see if I can be of more help. Sorry I'm not more of an immediate help, but thanks for posting your question! Maybe others will be able to comment as well.

  3. When subtracting Y2 from Y1 I noticed that you did not use the exact numbers that correspond to these y' even mentioned "**Notice that I didn't use (2-7)!" Can you please explain WHEN to know to subtract a particular Y from the other Y... Thanks

  4. Hi Nancy,

    To do this, you need to know the x and y values of 2 points. For these 2 points, you can say that point 1 is at (x1, y1), and point 2 is at (x2, y2). It doesn't matter which is point 1 or point 2. What DOES matter is that you keep the x1 with the y1, and the x2 with the y2.

    So, if we have a point at (1,2), and a point at (4,3), the slope of the line between them can be calculated as (2-3) / (1-4) = (-1/-3) = 1/3. If you reverse the points, then the calculation becomes (3-2) / (4-1) = 1/3. It's the same thing. However, if you only reverse the y's, or the x's, then you will get the answer wrong. You have to pay attention to the ordered pairs. Hopefully that helps you. Send another message if you need more help. :)

  5. THANK YOU It is so helpfull :D hahaha thanks


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