slope which is the amount of change in y relative to a change in x. Slope is calculated by taking the ratio of the change in y between two points
relative to the change in x for those same points:
(Slope is usually represented by the letter m
for various reasons).
Here are some properties of the slope of a line:
- A slope of 0 means that y doesn't change, i.e., the line is horizontal
(parallel to the x axis):
- A slope greater than 0 represents a line the is moving upwards from left to right
- A slope less than 0 represents a line that is moving downwards from left to right
- A slope of 1 means x and y change at the same rate; i.e., the line rises at a 45° angle from left-to-right
- A slope of -1 means x and y change at the same rate, but here the line descends at a 45° angle from left-to-right
- A vertical line has no slope - the x value is not moving, and thus the y component increases 'infinitely' with respect to x
- In terms of the slope equation, the denominator is 0 (since both x values are the same) which would result in a
division-by-zero, thus the slope is undefined
- A slope between 0 and 1 (or between 0 and -1) is shallow, the line makes more horizontal progress than vertical progress
- A slope greater than 1 (or less than -1) is steep, the line makes more vertical progress than horizontal progress
Line Pairs
Our last investigation of the Cartesian plane will be pairs of lines. These are simply represented as two lines, each line represented — as above — using two endpoints; e.,g (0, 0), (1, 1) and
(-1, -1), (-1, 0):
Parallel, and Perpendicular
- Two lines are parallel if they have the same slope
- If they both rise at the same rate they will never intersect
- On the other hand if the slopes of the two lines are negative reciprocals of each other
(i.e., one has slope
m
and the other -1/m
, then the lines are perpendicular to each other.
Point of Intersection of the Two Lines
While the point of intersection of two lines can be calculated with a fairly straightforward formula, deriving it is just a bit involved, so we will not deal with it.