Using
Slope and y-Intercept to Graph Lines (page
1 of 2)

Given two points (x_{1}, y_{1}) and (x_{2}, y_{2}),
the formula for the slope of the straight line going through these two points is:

...where the subscripts
merely indicate that you have a "first" point (whose coordinates
are subscripted with a "1")
and a "second" point (whose coordinates are subscripted with
a "2");
that is, the subscripts indicate nothing more than the fact that you have
two points to work with. Note that the point you pick as the "first"
one is irrelevant; if you pick the other point to be "first",
then you get the same value for the slope:

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(If you're not sure that
the two formulas above give exactly the same values, no matter the pair
of points plugged into them, then pick some points and try them out. See
what you get.)

The formula for slope is
sometimes referred to as "rise over run", because the fraction
consists of the "rise" (the change in y,
going up or down) divided by the "run" (the change in x,
going from left to the right). If you've ever done roofing, built a staircase,
graded landscaping, or installed gutters or outflow piping, you've probably
encountered this "rise over run" concept. The point is that
slope tells you how much y is changing for every so much that x is changing.

Pictures can be helpful,
so let's look at the line y = ( ^{2}/_{3} )x – 4;
we'll compute the slope, and draw the line.

To find points from the
line equation, we have to pick values for one of the variables, and then
compute the corresponding value of the other variable. If, say, x = –3,
then y = ( ^{2}/_{3} )(–3) – 4 = –2 – 4 = –6,
so the point (–3,
–6) is on the line.
If x = 0, then y = ( ^{2}/_{3} )(0) – 4 = 0 – 4 = –4,
so the point (0,
–4) is on the line.
Now that we have two points on the line, we can find the slope of that
line from the slope formula:

In stair-stepping
up from the first point to the second point, our "path"
can be viewed as forming a right triangle:

The distance between
the y-values
of the two points (that is, the height of the triangle) is the "y_{2} – y_{1}"
part of the slope formula. The distance between the x-values
(that is, the length of the triangle) is the "x_{2} – x_{1}"
part of the slope formula. Note that the slope is ^{2}/_{3},
or "two over three". To go from the first point to the
second, we went "two up and three over". This relationship
between the slope of a line and pairs of points on that line is
always true.

To
get to the
"next" point, we can go up another two (to y = –2), and over
to the right another three (to x = 3):

With these three
points, we can graph the line y = ( ^{2}/_{3} )x – 4.

(If you're not sure of
that last point, then put 3 in for x,
and verify that you get –2 for y.)

Let's try another line
equation: y = –2x + 3. We've
learned that the number on x is the slope, so m
= –2 for this line.
If, say, x = 0, then y = –2(0) + 3 = 0 + 3 = 3.
Then the point (0,
3) is on the line.
With this information, we can find more points on the line. First, though,
you might want to convert the slope value to fractional form, so you can
more easily do the "up and over" thing. Any number is a fraction
if you put it over "1",
so, in this case, it is more useful to say that the slope is m = ^{–2}/_{1}.
That means that we will be going "down two and over one" for
each new point.

We'll start at the
point we found above, and then go down two and over one to get to
the next point:

Go down another two,
and over another one, to get to the "next" point on y = –2x + 3:

Then the point (2,
–1) is also on this
line; with three points, we can graph the line.