1. Connecting Algebra
with the
Coordinate Plane

2. The year is Lying on his back, French mathematician René Descartes, watches a fly crawl across the ceiling. Suddenly, an idea comes to him. He visualizes two number lines, intersecting at a 90° angle. He realizes that he can graph the fly's location on a piece of paper. Descartes called the main horizontal line the x-axis and the main vertical line the y-axis. He named the point where they intersect the origin.

3. Descartes represented the fly's location as an ordered pair of numbers.
The first number, the x-value, is the horizontal distance along the x-axis, measured from the origin.
The second number, the y-value, is the vertical distance along the y-axis, also measured from the origin.

4. Y-COORDINATE
The locations in the plane where the x and y values intersect are called coordinates.
ORIGIN
X-COORDINATE
In the image on the right, three points — (0, 0), (-2, 5), and (3, -6.5) — are represented as graphed coordinates.

5. The plane containing these points is called the Cartesian plane (in honor of Descartes), or the coordinate plane.
Together, the x-axis, the y-axis, the coordinate plane, and all the coordinates make up the Cartesian coordinate system.
CARTESIAN
COORDINATE
SYSTEM

6. The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers.
COORDINATE PLANE
The point (0, 0) on a coordinate plane, where the x-axis and the y-axis intersect.
ORIGIN

7. Y-COORDINATE
ORIGIN
X-COORDINATE
CARTESIAN
COORDINATE
SYSTEM

8. Points in Quadrant 2 have negative x but positive y coordinates.
Points in Quadrant 1 have positive x and positive y coordinates.
To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants.
Points in Quadrant 3 have negative x and negative y coordinates.
Points in Quadrant 4 have positive x but negative y coordinates.

10. Write the coordinates of each point.
B: (5, -3)
C: (-1, -5)

11. WRITE THE COORDINATES FOR EACH POINT
B: (4, 6)
C: (2, 1)
D: (-4, 3)
E: (-5, 0)
F: (-3, -2)
G: (0, -4)
H: (4, -4)

12. EVERY POINT THAT FITS ON A LINE IS CALLED “COLLINEAR”.
Every straight line can be represented by an equation: y = mx + b. The coordinates of every point on the line will solve the equation if you substitute them in the equation for x and y.
EVERY POINT THAT FITS ON A LINE IS CALLED “COLLINEAR”.
THESE POINTS ARE “COLLINEAR”.
y = 2x + 1
0 = 2(1) + 1== YES
3 = 2(1) + 1== YES
5 = 2(2) + 1== YES

13. All segments shown are parallel to either the x- or y-axis
All segments shown are parallel to either the x- or y-axis. Determine the ordered pair that represents each point.
J: ( , )
– 6 12
G: ( , )
– 6
B: ( , ) 1
N: ( , ) 12
F: ( , ) 8 1
E: ( , ) 8
– 2

14. Determine whether the following points are “collinear” to “G” and “H”.
POINTS G(1, – 1) AND H(3, 3) LIE ON THE LINE “y = 2x – 3”.
Determine whether the following points are “collinear” to “G” and “H”.
I (0, – 3)
y = 2x – 3  – 3 = 2(0) – 3
YES
J (2, 1)
 1 = 2(2) – 3
YES
K (– 3, –8)
 –8 = 2(–3) – 3 NO
L ( 5, 7)
 7 = 2(5) – 3
YES
 16 = 2(10) – 3 NO
M ( 10, 16)