Coordinate System (Parallel/Perpendicular/Intercepts)


Coordinate System

The coordinate system is a grid system, similar to a map. Its lines are formed by two axes that are drawn perpendicular to each other.  The horizontal axis is called the x-axis, and the vertical axis is called the y-axis.  The two intersecting axes form four quadrants, numbered I through IV.

 

The point of intersection (0, 0) is called the origin.  In an ordered pair, the x-coordinate is always listed first and the y-coordinate second.

Intercept

The x-intercept is the point at which a graph crosses the x-axis.  The y-intercept is the point at which the graph crosses the y-axis. To find the x-intercept of a line, substitute 0 for y into the equation and solve for x. To find the y-intercept of a line, substitute 0 for x into the equation and solve for y. You can use the intercepts to graph the line. 

Example 1  

The y-intercept is 4

The x-intercept is 6

  • The graph of 2x + 3y = 12 crosses the x-axis at (6, 0), so its x-intercept is 6.
  • The graph of 2x + 3y = 12 crosses the y-axis at (0, 4), so its y-intercept is 4.

Example 2 

Find the intercepts of the graph

y = x − 5


Solution

To find the x-intercept, let y = 0 and solve for x:

y = x − 5
0 = x − 5
5 = x
10 = x
To find the y-intercept, let x = 0 and solve for y:
y = x − 5
y = (0) − 5
y = 0 − 5
y = −5

The x-intercept is 10, and the y-intercept is −5.

The graph of the equation contains the points (10, 0) and (0, −5).  You could plot these two points and connect them to determine the graph of the line.


Compare the Slopes of Lines

Two lines in a plane are parallel if they do not intersect. Two lines in a plane are perpendicular if they intersect to form a right angle. The slope can be used to determine whether two different lines are parallel or perpendicular.

Consider two different non-vertical lines 1 and 2. Line 1 has a slope m1, and line 2 has a slope m2. The lines are parallel if and only if they have the same slope:


m1 = m2

 

The lines are perpendicular if their slopes are negative reciprocals of each other.  Therefore, perpendicular lines have the following relationships between their slopes: 


   or  




Practice

1.  Determine the quadrant for the following points.

a.  (4, −1)   b. (−2, −8)  c. (−9, 3)


2.  Find the x- and y-intercepts for the graph of the equation 6x − 2y = 24.


3.  Given the line y = x + 4, find the following.

a.  the slope of a line parallel to its graph

b.  the slope of a line perpendicular to its graph.















Answers

1.  a. IV  b. III  c. II

2.  x-intercept: 4,  y-intercept: −12

3.  a.   b.