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Math · Geometry & Patterns
Coordinate Plane: Ordered Pairs, Quadrants, Reflections & Distance
Extend input–output tables into the full coordinate plane by reading and plotting ordered pairs, identifying quadrants, measuring horizontal and vertical distance, and reflecting points across axes.
The idea
An ordered pair is written (x, y). Start at the origin: x tells you how far to move left or right, then y tells you how far to move down or up.
The x- and y-axes divide the plane into four quadrants. The signs follow a pattern: I (+,+), II (−,+), III (−,−), IV (+,−). Points on an axis are not inside any quadrant.
Coordinate geometry also makes transformations precise. Reflecting across the y-axis changes the sign of x; reflecting across the x-axis changes the sign of y.
Worked examples
Example 1 · Plot an ordered pair
Plot (−3, 4).
- Start at (0,0).
- Move 3 units left because x is −3.
- Move 4 units up because y is 4.
The point is in Quadrant II.
Example 2 · Read a quadrant
Which quadrant contains (5, −2)?
- x is positive, so move right.
- y is negative, so move down.
- Right and down is Quadrant IV.
Quadrant IV.
Example 3 · Horizontal distance
Find the distance between (−2,3) and (6,3).
- The y-values match, so the segment is horizontal.
- Find the difference in x-values.
- |6−(−2)|=8.
8 units.
Example 4 · Reflect across the y-axis
Reflect (4, −1) across the y-axis.
- A y-axis reflection keeps y unchanged.
- Change x from 4 to −4.
- Write the new ordered pair.
(−4, −1).
Example 5 · Connect a rule to a point
For y=2x+1, find the point when x=3.
- Substitute x=3.
- y=2(3)+1=7.
- Write input first, output second.
(3, 7).
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Coordinates are always written x first, then y.
Why this fails: If x=0 or y=0, the point lies on an axis, not inside a quadrant.
Why this fails: Distance is non-negative, so use the absolute difference for horizontal or vertical segments.
Why this fails: A y-axis reflection changes x only; an x-axis reflection changes y only.
Why this fails: Both x and y signs determine the quadrant.
Foundation
Foundation practice
Read and plot ordered pairs, axes and quadrants.
1. Point A is at (3, 5). What is its x-coordinate?
2. Point B is at (−2, 4). What is its y-coordinate?
3. Where does (0, 6) lie?
4. Where does (−5, 0) lie?
5. Which quadrant contains (4, 2)?
6. Which quadrant contains (−4, 2)?
7. Which quadrant contains (−3, −7)?
8. Which quadrant contains (6, −1)?
9. What are the coordinates of the origin?
10. To plot (2, −4), which way do you move from the origin?
Core
Core practice
Compare coordinates and calculate horizontal or vertical distance.
11. What is the horizontal distance between (1, 3) and (6, 3)?
12. What is the vertical distance between (−2, 5) and (−2, −1)?
13. Which point is farther right: (−1, 4) or (3, 4)?
14. Which point is higher: (2, −3) or (2, 5)?
15. A point is on the y-axis at height −7. What is the point?
16. A point is on the x-axis 8 units left of the origin. What is the point?
17. Three corners are (1,1), (1,4), (5,1). What is the missing corner of the axis-aligned rectangle?
18. Points (1,2), (2,4), (3,6) follow y=2x. What point comes next?
19. A function table gives x=−3 and y=7. Write the ordered pair.
20. Does (4,9) satisfy y=2x+1?
Reasoning
Reasoning practice
Reflect points and reason from sign changes.
21. Reflect (3, 5) across the y-axis.
22. Reflect (−4, 2) across the x-axis.
23. Reflect (2, −6) across both axes.
24. A point reflects across the y-axis to (5, −2). What was the original point?
25. How far apart are (−4, 1) and (3, 1)?
26. How far apart are (2, −5) and (2, 4)?
27. A point has x<0 and y>0. Which quadrant?
28. A point has x>0 and y<0. Which quadrant?
29. For y=x−3, what point matches x=−2?
30. For y=−x, what is the reflection pattern between (3,−3) and (−3,3)?
Stretch
Stretch practice
Combine plotting, rules, distance and transformations.
31. Start at (−2,3). Move 5 units right and 4 down. Where do you land?
32. Start at (4,−1). Move 7 left and 6 up.
33. An axis-aligned rectangle has corners (0,0), (0,3), (5,0), (5,3). What is its perimeter?
34. Using the same rectangle, what is its area?
35. For y=2x−1, find the point when x=−2 and name its quadrant.
36. Reflect (−3,4) across the y-axis, then across the x-axis.
37. Point A=(−6,2). Point B is 9 units to the right on the same horizontal line. Find B.
38. Point C=(4,−7). Point D is 10 units above C. Find D.
39. A student says (−5,2) is in Quadrant IV because one coordinate is negative. What is the error?
40. Why does reflecting across the y-axis preserve horizontal distance from the y-axis?
Answer key
- 1. 3
The first coordinate is x. - 2. 4
The second coordinate is y. - 3. On the y-axis.
An x-coordinate of 0 places the point on the y-axis. - 4. On the x-axis.
A y-coordinate of 0 places the point on the x-axis. - 5. Quadrant I.
Both coordinates are positive. - 6. Quadrant II.
x is negative and y is positive. - 7. Quadrant III.
Both coordinates are negative. - 8. Quadrant IV.
x is positive and y is negative. - 9. (0, 0)
Both axes meet at the origin. - 10. 2 right and 4 down.
Positive x moves right; negative y moves down. - 11. 5 units.
The y-values match; |6−1|=5. - 12. 6 units.
The x-values match; |5−(−1)|=6. - 13. (3, 4)
Greater x means farther right. - 14. (2, 5)
Greater y means higher. - 15. (0, −7)
Points on the y-axis have x=0. - 16. (−8, 0)
Left is negative x and the y-coordinate is 0. - 17. (5, 4)
Match x=5 with y=4. - 18. (4, 8)
Increase x by 1 and y by 2. - 19. (−3, 7)
Input x comes first, output y second. - 20. Yes.
2×4+1=9. - 21. (−3, 5)
Across the y-axis, x changes sign. - 22. (−4, −2)
Across the x-axis, y changes sign. - 23. (−2, 6)
Both coordinates change sign. - 24. (−5, −2)
Reverse the x-sign change. - 25. 7 units.
|3−(−4)|=7. - 26. 9 units.
|4−(−5)|=9. - 27. Quadrant II.
Negative x and positive y define Quadrant II. - 28. Quadrant IV.
Positive x and negative y define Quadrant IV. - 29. (−2, −5)
−2−3=−5. - 30. They are reflections across both axes, or a 180° turn about the origin.
Both coordinates change sign. - 31. (3, −1)
Add 5 to x and subtract 4 from y. - 32. (−3, 5)
4−7=−3 and −1+6=5. - 33. 16 units.
Width 5 and height 3, so 2(5+3)=16. - 34. 15 square units.
5×3=15. - 35. (−2, −5), Quadrant III.
y=−5; both coordinates are negative. - 36. (3, −4)
First (3,4), then (3,−4). - 37. (3, 2)
−6+9=3; y stays 2. - 38. (4, 3)
−7+10=3; x stays 4. - 39. Quadrant IV requires x>0 and y<0; (−5,2) is in Quadrant II.
The sign position matters, not just the number of negatives. - 40. The x-coordinate changes sign but keeps the same absolute value.
Distance from the y-axis is |x|.