Math101 · Independent Ontario learning library
Math101learn.math101.caCoordinate Plane
The coordinate plane locates points with ordered pairs and turns numerical relationships into visible geometry and graphs.
The coordinate plane uses a horizontal $x$-axis and vertical $y$-axis to locate every point with an ordered pair $(x,y)$.
Axes and origin
The axes meet at the origin $(0,0)$. Positive $x$-values lie to the right and negative $x$-values to the left. Positive $y$-values lie above the origin and negative $y$-values below it.
The scale must be read before plotting. One grid square might represent $1$, $5$, $0.5$, or another interval.
Ordered pairs
Coordinates are ordered: move horizontally using $x$, then vertically using $y$. The points $(3,-2)$ and $(-2,3)$ are different.
A point on the $x$-axis has $y=0$. A point on the $y$-axis has $x=0$.
The four quadrants
The axes divide the plane into four regions numbered counterclockwise:
| Quadrant | Sign of $(x,y)$ |
|---|---|
| I | $(+,+)$ |
| II | $(-,+)$ |
| III | $(-,-)$ |
| IV | $(+,-)$ |
Points on an axis are not in any quadrant.
Horizontal and vertical change
From $A(x_1,y_1)$ to $B(x_2,y_2)$, horizontal change is $x_2-x_1$ and vertical change is $y_2-y_1$. These differences lead to slope and the distance formula.
For $A(-1,2)$ and $B(4,5)$, the horizontal change is $5$ and vertical change is $3$.
Distance and midpoint
The Pythagorean theorem gives distance:
The midpoint averages the coordinates:
For $(-1,2)$ and $(4,5)$, the midpoint is $(1.5,3.5)$ and the distance is $\sqrt{34}$.
Graphing relationships
A graph turns pairs of related values into a visual pattern. If $y=2x+1$, substitute chosen $x$-values to create points such as $(0,1)$, $(1,3)$, and $(2,5)$. Their alignment reveals a linear relationship.
Not every context should connect points with a continuous line. A graph of ticket count may use separate points because fractional tickets are impossible.
Transformations
Coordinates describe geometric movement precisely. Adding $3$ to every $x$-coordinate translates a figure right $3$ units. Negating every $x$-coordinate reflects it across the $y$-axis. These rules connect coordinate geometry with function transformations.
Common mistakes
Reversing coordinates. Read and plot $x$ before $y$.
Ignoring the scale. Check labelled ticks instead of assuming one square equals one unit.
Misnumbering quadrants. Begin in the upper right and move counterclockwise.
Dropping negative signs in differences. Use brackets when substituting negative coordinates into formulas.
Quick self-check
- What does each grid interval represent?
- Did I move horizontally before vertically?
- Do the coordinate signs match the point’s region?
- Does the context justify connecting plotted points?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
In which quadrant is the point (−4, 2)?
- Negative x means move left.
- Positive y means move up.
- The upper-left region is Quadrant II.
End of lesson
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