These questions ask you to locate or move a point, name its quadrant, recognise a point on an axis, reflect it, or find a horizontal or vertical distance. The reliable method is to read x first and y second.
Turn each coordinate into a movement
Start at the origin. Read x as the horizontal move: positive goes right and negative goes left. Then read y as the vertical move: positive goes up and negative goes down. For example, (-2, 7) means 2 left and 7 up, so it is in Quadrant II. The quadrant numbers begin in the upper-right region and continue anticlockwise.
| Coordinate signs | Location |
|---|---|
| x positive, y positive | Quadrant I: right and up. |
| x negative, y positive | Quadrant II: left and up. |
| x negative, y negative | Quadrant III: left and down. |
| x positive, y negative | Quadrant IV: right and down. |
| x is zero or y is zero | The point is on an axis, not in a quadrant. |
A reflection changes the coordinate perpendicular to the mirror axis. Across the x-axis, the height reverses, so y changes sign and x stays fixed; (6, 3) becomes (6, -3). Across the y-axis, the sideways position reverses, so x changes sign and y stays fixed; (-4, -2) becomes (4, -2). Negating both coordinates crosses both axes and reaches the diagonally opposite quadrant.
For axis-parallel distance, first find the coordinate that stays equal. If y is equal, compare the x-values; if x is equal, compare the y-values. Values on opposite sides of zero contribute both distances to the axis. Thus points with x-values -2 and 5 at the same height are seven units apart. Values on the same negative side use the gap between magnitudes: y-values -10 and -6 are 4 units apart.
Coordinate traps to avoid
- Reading the signs in reverse order swaps horizontal and vertical movement and can send a point to the opposite sign pattern. Say x then y before naming a quadrant.
- Using only the first coordinate cannot locate a quadrant. The second coordinate decides whether the point is above, below or on the horizontal axis.
- Swapping the coordinate values changes the destination even when both signs are preserved. Move the x amount horizontally before using y vertically.
- Assigning a quadrant to a point with a zero coordinate ignores that the point lies on a boundary. Test for zero before consulting the four sign patterns.
- Changing the coordinate parallel to a reflection axis mirrors across the wrong line. Across the x-axis change y; across the y-axis change x.
- Calling an adjacent quadrant diagonally opposite changes only one sign. A diagonal move across the origin changes both signs.
Try it
The points (-5, 2) and (6, 2) are plotted on a four-quadrant grid. Enter the distance between them in units.
The points (3, -9) and (3, -2) are plotted on a four-quadrant grid. Enter the distance between them in units.