Lines are nothing more than an infinite set
of points arrayed in a straight formation, but there are a number
of ways to analyze them. We look at some of the main properties,
formulas, and rules of lines.
The slope of a line is a measurement of how
steeply the line climbs or falls as it moves from left to right.
More technically, it is a line’s vertical change divided by its horizontal
change, informally known as “the rise over run.” Given two points
on a line, call them (x1, y1)
and (x2, y2), the
slope of that line can be calculated using the following formula:
The variable most often used to represent slope is m.
So, for example, the slope of a line that contains the
points (–2, –4) and (6, 1) is m = (1 – (–4)) ⁄
(6 – (–2)) = 5⁄8.
Positive and Negative Slopes
You can easily determine whether the slope of a line is
positive or negative just by looking at the line. If a line slopes
uphill as you trace it from left to right, the slope is positive.
If a line slopes downhill as you trace it from left to right, the
slope is negative. You can get a sense of the magnitude
of the slope of a line by looking at the line’s steepness. The steeper
the line, the greater the slope will be; the flatter the line, the
smaller the slope will be. Note that an extremely positive slope is larger then
a moderately positive slope while an extremely negative slope is smaller then
a moderately negative slope.
Look at the lines in the figure below and try to determine
whether the slope of each line is negative or positive and which
has the greatest slope:
Lines a and b have positive
slopes, and lines c and d have
negative slopes. In terms of slope magnitude, line a > b
> c > d.
For the Math IC, there are a few slopes you should recognize
by sight. If you can recognize one of these lines and identify its
slope without having to do any calculations, you will save yourself
a lot of time.
- A line that is horizontal has a slope of
zero. Since there is no “rise,” y2 – y1 =
0, and thus m = y2–y1
- A line that is vertical has an undefined slope. In this
case, there is no “run,” and x2 – x1 =
0. Thus, m = y2–y1
0, and any fraction with 0
in its denominator is, by definition, undefined.
- A line that makes a 45º angle with a horizontal line has
a slope of 1 or –1. This makes sense because the rise equals the
run, and y2 – y1 = x2 – x1,
or y2 – y1 =
–(x2 – x1).
the four lines pictured below, one has a slope of 0, one has a slope
of 1, another has a slope of –1, and another has undefined slope.
Decide which is which.
Line a has slope 0 because it is horizontal.
Line b has slope –1 because it slopes
downward at 45º as you move from left to right. Line c has
slope 1 because it slopes upward at 45º as you move from left to
right. Line d has undefined slope
because it is vertical.
Parallel and Perpendicular Lines
Parallel lines are lines that don’t intersect.
In coordinate geometry, they can also be described as lines with
the same slope.
Perpendicular lines are lines that intersect
at a right angle. In coordinate geometry, perpendicular lines have
opposite, reciprocal slopes. That is, a line with slope m is
perpendicular to a line with a slope of –1⁄m.
In the figure, lines q and r both
have a slope of 2, so they are parallel. Line s is
perpendicular to both lines q and r,
and thus has a slope of –1⁄2.
Equation of a Line
A line in coordinate geometry can be described by an equation
containing the variables x and y.
For the Math IC, you need to understand thoroughly two forms of
the equation of a line: the slope-intercept form and the point-slope
The slope-intercept form of the equation of a line is:
where m is the slope of the line, and b is
the y-intercept of the line. Both are constants.
The y-intercept of a line is the y-coordinate
of the point where the line intersects the y-axis.
Likewise, the x-intercept of a line is the x-coordinate
of the point where the line intersects the x-axis.
Therefore, if given the slope-intercept form of the equation of
a line, you can find both intercepts.
For example, in order to find the y-intercept,
simply set x = 0 and solve for the value of y.
For the x-intercept, set y = 0
and solve for x.
To sketch a line given in slope-intercept form, first
plot the y-intercept, and then use the slope of
the line to plot another point. Connect the two points to form your
line. In the figure below, the line y = –2x +
3 is graphed.
Since the slope is equal to –2, the line descends 2 units
for every 1 unit it moves in the positive x direction.
The y-intercept is at 3, so the line crosses the y-axis
at (0, 3). For practice, solve for the x-intercept.
The point-slope form of the equation of a line is:
where m is the slope of the line, and
is a point on the line.
The point-slope form and slope-intercept form are alternative
ways of expressing the same equation. In fact, the slope-intercept
form is the point-slope form taken at the y-intercept,
or the point (0, y1):
Since y1 = b (the y-intercept
is simply the y-coordinate of the point at which x =
0), the two forms are equal.
The slope-intercept form of the line equation is the more
common of the two, but the point-slope form is useful when your
only information is the slope and a point (hence the name point-slope form).
Example Problems with Slope
The Math IC test often asks questions that require you
to understand the slope-intercept form and the point-slope form,
and to be able to convert between the two.
Here are some practice questions:
is the slope-intercept equation of the line that contains the point
(3, 4) and is perpendicular to the line y =
To answer this question, you first need to find the slope
of the line whose equation you are trying to determine. Fortunately,
the question gives you the slope of a perpendicular line, and we
know that the slope of a line is the opposite reciprocal of the
slope of the line to which it is perpendicular. Thus, the slope
is –1⁄ (1⁄3) =
–3. If the line contains the point (3, 4), its point-slope equation
is y – 4 = –3(x – 3). To convert
this to slope-intercept form, use algebra:
Here’s another question:
is the slope-intercept form of the equation of the line that contains
the points (5, 3) and (–1, 8)?
Start by finding the slope of the line. You can calculate
the slope with the two points you’re given: m = 8–3⁄–1–5 =
–5⁄6. To put the
equation of this line in slope-intercept form, the only additional
information we need is the y-intercept. To find
it, use the x- and y-coordinates of
a point that you know is on the line and plug them into the equation y =
–5⁄6 x + b,
and solve for b. Using the point (5,
The slope-intercept form of the equation of this line
is y = –5⁄6 x + 43⁄6.