California Standards:
Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
Measurement and Geometry 3.2
What it means for you:
You’ll see how to use a grid system to plot numbered points.
Key words:
• coordinate
•x-axis
•y-axis
• quadrant
You Plot Coordinates on a Coordinate Plane
The coordinate plane is a two-dimensional (flat) area where points and lines can be graphed.
Check it out:
Coordinates are sometimes called ordered pairs.
This means that the order of the numbers matters — (1, 2) is different from (2, 1).
When you draw a graph you draw it on a coordinate plane.
This is a flat grid that has a horizontal axis and a vertical axis.
You can describe where any point on the plane is using a pair of numbers called coordinates.
Plotting Points
Coordinates Describe Points on the Plane
The plane is formed by the intersection of a vertical number line, or y-axis, and a horizontal number line, or x-axis. They cross where they are both equal to 0 — a point called the origin.
The x and y coordinates of a point describe where on the plane it lies.
The coordinates are written as (x, y).
When you plot points on the coordinate plane you plot them in relation to the origin, which has coordinates of (0, 0).
1 2 3 4
–1 –2 –3 –4 y
0 x
1 2 3 4
–1 –2 –3 –4
( , 2 –3 ) ( , 2 –3 )
x-coordinate
x-coordinate y-coordinatey-coordinate
• The x-coordinate tells you how many spaces along the x-axis to go.
Negative values mean you go left.
Positive values mean you go right.
• The y-coordinate tells you how many spaces up or down the y-axis to go.
Positive values mean you go up.
Negative values mean you go down.
So a point with the coordinates (2, –3) will be two units to the right of the origin, and three units below.
Section 3.2
Section 3.2 — The Coordinate Plane
Lesson
3.2.1
151 When you are reading the coordinates of a point on a graph you can use the same idea.
What are the coordinates of point A?
Solution
Start at (0, 0). To get to point A on the graph you need to move 2 units to the left. So the x-value of your coordinate is –2.
Then you need to go 2 units straight up. So the y-value is 2.
The coordinates of the point A are (–2, 2).
Example 2
1 2 3 4
–1 –2 y
0 x
1 2 3 4 –1
–2 –3 –4
A
Plot the point with the coordinates (3, 4).
Solution
Step 1: start at the origin, (0, 0).
Example 1
Check it out:
A positive x-coordinate tells you to move right along the x-axis, while a negative x-coordinate tells you to move left along the x-axis.
A positive y-coordinate tells you to move up the y-axis, while a negative y-coordinate tells you to move down the y-axis.
Step 3: now move straight up 4 units and plot the point.
Step 2: move right along the x-axis 3 units.
1 2 3 4
–1 –2 y
0 x
1 2 3 4
–1 –3 –2
–4
Section 3.2 — The Coordinate Plane
Guided Practice
Plot and label each of the coordinate pairs in Exercises 1–6 on a coordinate plane.
1. (1, 4) 3. (–1, –2) 5. (0, 3)
2. (2, –3) 4. (–4, 2) 6. (–4, 0)
152
Check it out:
A point that is on either the x-axis or the y-axis is not in any of the quadrants.
The Coordinate Plane is Divided into Four Quadrants The x-axis and y-axis divide the coordinate plane into four sections.
Each of these sections is called a quadrant. The quadrants are represented by Roman numerals, and are labeled counterclockwise.
x 1
2 3 4
–1 –2 –3 –4 y
0
1 2 3 4 –1
–2 –3 –4
II
(–, +)
I
(+, +)
III
(–, –)
IV
(+, –)
The signs of the x and y values are different in each quadrant.
For instance, in quadrant I both the x and y values are positive.
But in quadrant II the x value is negative and the y value is positive.
You can tell which quadrant a point will fall in by looking at the signs of the x and y coordinates.
Which quadrant is the point (1, –4) in?
Solution
The x-value is 1. This is positive, so the point must be in quadrant I or IV.
The y-value is –4. This is negative, so the point must be in quadrant IV.
The point (1, –4) is in quadrant IV.
Example 3
Which quadrant is the point (–3, –6) in?
Solution
Both coordinates are negative, so the point (–3, –6) is in quadrant III.
Example 4
Section 3.2 — The Coordinate Plane
153 Coordinates allow you to describe where points are plotted — they’re written as pairs of numbers, such as (1, –5). The first number tells you the horizontal or x-coordinate. The second tells you the vertical or y-coordinate. Plotting points on coordinate planes is a big part of drawing graphs, and will be used a lot in the rest of this Chapter and in the next Chapter.
Independent Practice
Round Up Round Up
In Exercises 1–6 say which quadrant the point is in.
1. (1, 1) 3. (–1, 2) 5. (3, –2)
7. Do the coordinate pairs (–3, 4) and (4, –3) correspond to the same point on the plane?
Plot each of the points in Exercises 8–13 on a coordinate plane.
8. (0, 0) 10. (–2, –3) 12. (1, –2)
14. Sophie and Jorge are playing a game. Sophie marks out a coordinate plane on the beach, and buries some objects at different points. Jorge has to use the map below to find the objects.
2. (–1, –1) 4. (–2, 1) 6. (–61, 141)
9. (3, 2) 11. (–3, 4) 13. (0, 4)
a) What are the coordinates of the keys?
b) What are the coordinates of the baseball cap?
c) What are the coordinates of the apple?
d) How many units are there between the keys and the flashlight?
e) How many units are there between the baseball mitt and the keys?
Guided Practice
In Exercises 7–14 say which quadrant the point lies in.
7. (1, 3) 9. (7, –2) 11. (–2, –2) 13. (–1.5, 2.5)
8. (–2, –4) 10. (–3, 6) 12. (2, 2) 14. (1, –1)
1 2
–1 –2 –3 –4 y
0
1 2 3 –1
–3 –2
x
Section 3.2 — The Coordinate Plane Now try these:
Lesson 3.2.1 additional questions — p445
154