Graphing a Line Given a Point on It and Its Slope

Date: 03/07/2004 at 20:51:17
From: Sandra
Subject: Sandra, trying to hang on at 46!

I am completely confused about how to graph lines.  The question is
asking me to graph the line that goes through the point (0,2) with a
slope m = 1/4.  I am not sure what slope m = 1/4 means.

I'm 46, and this course is not my cup of tea.  I will be glad when I'm
finihed with it completely--it is so frustrating!

Date: 03/07/2004 at 21:49:48
From: Doctor Achilles
Subject: Re: Sandra, trying to hang on at 46!

Hi Sandra,

Thanks for writing to Dr. Math.

"M" is just a letter used for the slope of a line.  It's confusing, 
and no one I've ever met can tell me why "m" stands for slope.

I'm going to walk you through a lot of the basics of graphing here.  
Some of this may be familiar, but I hope it will be helpful.  I 
suggest you have some graph paper out while you read this and you try 
drawing lines on the paper that illustrate what I describe.

The slope of a line is a measurement of how steep it is.  If you're 
looking at a graph, imagine yourself walking along the line from the 
left of the graph to the right.  If the line is completely flat, then 
slope is 0.  If you are going uphill, then the slope is positive.  
The steeper it is, the bigger the slope is.  If you are going 
downhill, the slope is negative; again, the steeper it is the more 
negative the slope will be.

I hope all of that is clear; let me know if it's confusing.

Let's also talk about what coordinates mean.  Coordinates are a way 
to tell someone where you are on a piece of graph paper.  Start by
drawing two lines on your paper, one horizontal and one vertical.  The
point where those two lines meet is called the origin, which has
coordinates of (0,0).  All other points in the plane are referenced
from the origin.

Did you ever play the game Battleship?  The board had numbers along
the left side and letters across the top, and players would call their
shots by saying something like, 'B7.'  That meant to find the block on
the board where the B column and the 7 row intersected.

Coordinates work much the same way.  The horizontal line through the
origin is called the 'x-axis' and the vertical one is the 'y-axis.' 
When we give coordinates, they consist of two numbers which refer to
those axes.  In a coordinate or ordered pair, the x number always
comes first, followed by the y value.

If you move to the right of the origin one space, you will be at (1,0)
since you are at the point where x = 1 and y = 0.  If you go back to
the origin and then move to the left 3 spaces from the origin, you are
at (-3,0).  You can also move up and down along the "y-axis".  Up is
positive and down is negative.  So if you move up 2 from the origin,
you are at (0,2) since x = 0 and y = 2.  If you start at the origin
and move right 5, then move down 7 from that spot, you will be at
(5,-7).  Again, this is the point that matches up with +5 on the
x-axis and -7 on the y-axis, just like in Battleship.

Now let's talk about a few examples of slopes.  Slope is essentially a
ratio which describes how much x and y change in relation to each
other as you move from point to point on the line.  Slope is often
referred to as:

  rise              change in y values
  ----   which is   ------------------
  run               change in x values

If the slope of a line is 1, think of that as the fraction or ratio
1/1.  That means that for every one unit you move to the right (change
in x), you will move up one unit as well (change in y).  So a line
that passes through (0,0) and has a slope of m = 1 will start at (0,0)
and will go through (1,1) and (2,2) and so on.  It will also go
through (-1,-1) and (-2,-2).

If a line has a slope of 2, think of that as 2/1.  That means for
every one unit you move to the right, you move up 2 units.  So if the
line goes through the origin (0,0), it will also go through (1,2) and
(2,4) and (3,6) and (-1,-2) and (-2,-4) and so on.

If a line has a slope of 1/3, that means you move up 1 for every 3 
you move right.  So if the line goes through the origin (0,0), it 
will go through (3,1) and (6,2) and (-3,-1) and (-6,-2) and so on.

If a line has a slope of -1, that's -1/1, so that means you go DOWN 1
for every 1 you go right.  Let's say that we have a line that has a
slope of -1 which passes through (0,3).  To find the next point, we go
down 1 and right 1 to (1,2) and then to (2,1) and then to (3,0) and
then to (4,-1).  It will also continue in the other direction to
(-1,4) and (-2,5) and so on.

Again, remember that as you walk the line from left to right, a
negative slope will go downhill while a positive one will go uphill.

If a line has a slope of -3, that's -3/1, so that means you go down 3
for every 1 you go right.  Let's say this line goes through the origin
(0,0).  It will also go through (1,-3) and (2,-6) and back the other
way through (-1,3) and (-2,6).

If a line has a slope of -2/3, that means you go down 2 for every 3 
you go right.  Let's start at the origin (0,0).  It goes through
(3,-2) and (6,-4) and also the other way through (-3,2) and (-6,4).

Let's do one more.  A line has a slope of -3/4 and goes through the 
point (2,10).  The slope of -3/4 means we go down 3 for every 4 to 
the right.  So, starting at (2,10) the line will go through (6,7) and 
(10,4) and (14,1) and (18,-2) and (22,-5) and going back the other 
way through (-2,13) and (-6,16) and (-10,19) and so on.

Ok, so now let's talk about your question.

You have a line that has a slope of 1/4 and it goes through (0,2).  
To graph this, you can start at (0,2) and mark that as one of the
points on the line.  Then you move right 4 and up 1, which puts you at
(4,3).  Then you move right another 4 and up another 1, which puts you
at (8,4), and you keep going like that.  You can also go backwards to
(-4,1) and (-8,0) and (-12,-1), and so on.

Finally, remember that a line contains infinite points.  We have just
found a few of them, so once you have plotted a few points, draw a
line passing through those points and that will give you the graph of
the whole line.

Hope this helps.  If you have other questions about this or if you'd 
like to talk about graphs and slopes some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/