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Coordinate Geometry Basics: Understanding the Coordinate Plane

Capítulo 1

Estimated reading time: 4 minutes

+ Exercise

1) Ordered Pairs (x, y) and How to Move on the Plane

A coordinate plane is formed by two perpendicular number lines:

x-axis: the horizontal number line
y-axis: the vertical number line

They intersect at the origin, written as (0, 0). Every point on the plane can be named using an ordered pair (x, y).

What an ordered pair means

In (x, y):

x tells the horizontal movement from the origin.
y tells the vertical movement from the origin.

Use these visual conventions:

Positive x: move right; negative x: move left.
Positive y: move up; negative y: move down.

Step-by-step plotting rule (the “x then y” rule)

Start at the origin (0, 0).
Move horizontally to match x (right if positive, left if negative).
From there, move vertically to match y (up if positive, down if negative).

Example: Plot (-3, 2).

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Start at (0,0).
Move left 3 to reach x = −3.
Move up 2 to reach y = 2.

2) Guided Practice: Reading Coordinates and Naming Quadrants

The axes divide the plane into four regions called quadrants. Quadrants are numbered using Roman numerals, starting in the upper-right and moving counterclockwise.

Quadrant	x sign	y sign	Location
I	+	+	upper-right
II	−	+	upper-left
III	−	−	lower-left
IV	+	−	lower-right

Practice A: Identify coordinates of marked points

Use the grid below. Each intersection is 1 unit. Read each point by counting from the origin: first x (left/right), then y (up/down).

      y  5 |                • A(?,?)            • B(?,? )  5  y      4 |                                              4      3 |        • C(?,?)                               3      2 |                                              2      1 |                                              1  ------+---------------------------------------------- x        -5 -4 -3 -2 -1 0 1 2 3 4 5             0      -1 |                               • D(?,?)        -1      -2 |                 • E(?,?)                     -2      -3 |                                              -3

Step-by-step for point A:

From the origin, count horizontally to A to find x.
Then count vertically to A to find y.
Write the ordered pair (x, y).

Now you try: Find the coordinates of A, B, C, D, and E.

Answer key for Practice A

A is at (2, 5)
B is at (5, 5)
C is at (-3, 3)
D is at (4, -1)
E is at (-2, -2)

Practice B: Name the quadrant (or axis)

For each point, name its quadrant using the sign pattern table. If a point lies directly on an axis, it is not in any quadrant.

(2, 5) is in Quadrant ____
(-3, 3) is in Quadrant ____
(4, -1) is in Quadrant ____
(0, -4) is on the ____ axis
(-6, 0) is on the ____ axis

Answer key for Practice B

(2, 5): Quadrant I
(-3, 3): Quadrant II
(4, -1): Quadrant IV
(0, -4): y-axis
(-6, 0): x-axis

3) Error-Check: Avoid Common Coordinate Mistakes

Mistake 1: Reversing x and y

What happens: You plot (x, y) as if it were (y, x).

Example: The point (1, 4) means right 1, up 4. If you reverse it, you would plot right 4, up 1, which is a different location.

Quick fix: Say out loud: (x, y) = “across, then up/down.”

Mistake 2: Sign errors (mixing up left/right or up/down)

Common confusion: Thinking negative means “down” for both coordinates. Actually:

Negative x means left.
Negative y means down.

Check yourself with quadrants:

If x is negative and y is positive, the point must be in Quadrant II (upper-left).
If x is positive and y is negative, the point must be in Quadrant IV (lower-right).

Mistake 3: Misreading points on the axes

If a point has a zero coordinate, it lies on an axis:

(0, y) is on the y-axis.
(x, 0) is on the x-axis.

Important: Points on axes are not in any quadrant.

Error-check mini practice

Each statement below contains a possible mistake. Identify and correct it.

“Point (-2, 3) is right 2 and up 3.”
“Point (5, -1) is in Quadrant III.”
“Point (0, 4) is in Quadrant I.”
“To plot (2, -6), go down 2 and right 6.”

Corrections

(-2, 3) is left 2 and up 3.
(5, -1) is in Quadrant IV (positive x, negative y).
(0, 4) is on the y-axis, not in a quadrant.
(2, -6): go right 2, then down 6 (x then y).

4) Quick Self-Check: Directions ↔ Ordered Pairs

Translate between movement directions and ordered pairs. Remember: horizontal movement is x, vertical movement is y.

A. Directions → ordered pair

Write each as (x, y).

Right 3, up 2: (__, __)
Left 5, down 1: (__, __)
Left 4, up 6: (__, __)
Right 2, down 7: (__, __)
No horizontal move, down 3: (__, __)

Answers (A)

Right 3, up 2: (3, 2)
Left 5, down 1: (-5, -1)
Left 4, up 6: (-4, 6)
Right 2, down 7: (2, -7)
No horizontal move, down 3: (0, -3)

B. Ordered pair → directions

Write each ordered pair as “left/right __, up/down __.”

(-2, -4): __________
(6, 1): __________
(-3, 0): __________
(0, 5): __________

Answers (B)

(-2, -4): left 2, down 4
(6, 1): right 6, up 1
(-3, 0): left 3, no vertical move (on the x-axis)
(0, 5): no horizontal move, up 5 (on the y-axis)

Now answer the exercise about the content:

Using the “x then y” rule, which directions correctly plot the point (-4, 6) from the origin?

You are right! Congratulations, now go to the next page

You missed! Try again.

In an ordered pair (x, y), x is the horizontal move and y is the vertical move. A negative x means left, and a positive y means up, so (-4, 6) is left 4, then up 6.