Coordinate Geometry

The One Thing

Coordinate geometry locates every point in the Cartesian plane using a unique ordered pair $(x,y)$. The first coordinate determines horizontal position, the second determines vertical position, and their signs identify the relevant quadrant or axis.

Definitions and Results

• Cartesian plane: A flat surface formed by two mutually perpendicular number lines used to represent the positions of points.
• Coordinate axes: The two perpendicular lines forming the Cartesian plane: the horizontal $x$-axis and the vertical $y$-axis.
• Origin: The point where the $x$-axis and $y$-axis intersect, represented by $O(0,0)$.
• x-coordinate: The first number in an ordered pair; it shows the horizontal distance and direction from the $y$-axis. It is also called the abscissa.
• y-coordinate: The second number in an ordered pair; it shows the vertical distance and direction from the $x$-axis. It is also called the ordinate.
• Ordered pair: A pair of numbers written as $(x,y)$ that uniquely identifies the position of a point. Coordinates must be written in the order $(x,y)$, never $(y,x)$.
• Quadrants: The four regions into which the coordinate axes divide the Cartesian plane. They are numbered anticlockwise, beginning with the upper-right quadrant:

- Quadrant I: $(+,+)$ - Quadrant II: $(-,+)$ - Quadrant III: $(-,-)$ - Quadrant IV: $(+,-)$

• Plotting a point: Locating a point by moving from the origin according to its $x$-coordinate and then its $y$-coordinate. Positive $x$-values indicate movement to the right, negative $x$-values movement to the left, positive $y$-values upward, and negative $y$-values downward.
• Collinear points: Points that lie on the same straight line.
• Graphical interpretation: Understanding information, positions, patterns, or relationships by examining points and graphs.

Further results:

• A point on the $x$-axis has the form $(x,0)$.
• A point on the $y$-axis has the form $(0,y)$.
• The origin is the only point lying on both axes.
• Points with the same $x$-coordinate lie on a vertical line.
• Points with the same $y$-coordinate lie on a horizontal line.
• The distance of a point from the $x$-axis is the absolute value of its $y$-coordinate.
• The distance of a point from the $y$-axis is the absolute value of its $x$-coordinate.
• A suitable scale must be chosen on both axes before plotting; axes must be labelled clearly, and points must be marked and named accurately.
• Every point in the Cartesian plane has one and only one ordered pair of coordinates, and every ordered pair represents one point.

Worked Methods

Plotting an ordered pair

• Draw two mutually perpendicular axes.
• Label the horizontal axis $x$ and the vertical axis $y$.
• Mark the origin as $O(0,0)$.
• Choose a suitable scale on both axes.
• Read the ordered pair as $(x,y)$, not $(y,x)$.
• Starting at the origin, move $x$ units horizontally:

- right for a positive $x$-coordinate; - left for a negative $x$-coordinate.

• From that position, move $y$ units vertically:

- upward for a positive $y$-coordinate; - downward for a negative $y$-coordinate.

• Mark and name the resulting point accurately.

Identifying the quadrant of a point

• Read the signs of the $x$- and $y$-coordinates.
• Match the signs to the quadrant:

- $(+,+)$ gives Quadrant I; - $(-,+)$ gives Quadrant II; - $(-,-)$ gives Quadrant III; - $(+,-)$ gives Quadrant IV.

• If one coordinate is zero, identify the axis instead:

- $(x,0)$ lies on the $x$-axis; - $(0,y)$ lies on the $y$-axis; - $(0,0)$ is the origin.

Interpreting positions and relationships

• Read each point as an ordered pair.
• Compare the $x$-coordinates to determine horizontal positions.
• Compare the $y$-coordinates to determine vertical positions.
• Points sharing an $x$-coordinate form a vertical line.
• Points sharing a $y$-coordinate form a horizontal line.
• Use the absolute value of the $y$-coordinate to determine distance from the $x$-axis.
• Use the absolute value of the $x$-coordinate to determine distance from the $y$-axis.
• Examine the plotted points for locations, patterns, relationships, or simple graphical information.

Coordinate geometry therefore connects numerical information with visual positions and is applicable to maps, graphs, geometry, science, and everyday location-based applications.

Where It Goes Wrong

• Reversing the coordinates by reading $(x,y)$ as $(y,x)$; the first coordinate must always be read horizontally along the $x$-axis.
• Confusing the axes by treating the $x$-axis as vertical or the $y$-axis as horizontal.
• Ignoring signs when plotting; negative $x$-values move left, negative $y$-values move downward, and the signs determine the quadrant.
• Forgetting that a zero coordinate places a point on an axis: $(x,0)$ is on the $x$-axis and $(0,y)$ is on the $y$-axis.
• Failing to distinguish the origin $O(0,0)$, which is the only point lying on both axes, from other axis points.
• Using an unsuitable or unmarked scale, unclear axis labels, or inaccurately named plotted points.

What Gets Asked

This material supports questions that require students to:

• Define the Cartesian plane, coordinate axes, origin, ordered pair, abscissa, ordinate, quadrants, collinear points, and graphical interpretation.
• State the coordinates of the origin and identify the $x$- and $y$-coordinates of a point.
• Plot points from given ordered pairs using a stated or selected scale.
• Identify the quadrant or axis containing a point from the signs or zero coordinates.
• Determine whether points lie on a horizontal or vertical line by comparing their coordinates.
• Calculate a point’s distance from the $x$-axis or $y$-axis using the absolute value of the relevant coordinate.
• Interpret locations, patterns, relationships, and simple graphs represented by plotted points.