Coordinate Geometry

The One Thing

Coordinate geometry represents each point on a Cartesian plane by an ordered pair $(x,y)$. The first coordinate determines horizontal position and the second determines vertical position; their signs identify the point’s quadrant or whether it lies on an axis.

Definitions and Results

• Cartesian Plane: A two-dimensional plane formed by two perpendicular number lines used to locate points.
• Coordinate Axes: The horizontal line is the $x$-axis and the vertical line is the $y$-axis.
• Origin: The point where the $x$-axis and $y$-axis intersect, represented by $(0,0)$.
• Ordered Pair: A pair of numbers $(x,y)$ giving the location of a point. The first number is the $x$-coordinate and the second is the $y$-coordinate. The order is essential: $(x,y)$ and $(y,x)$ generally represent different points.
• Abscissa: The $x$-coordinate of a point, showing its horizontal distance and direction from the $y$-axis.
• Ordinate: The $y$-coordinate of a point, showing its vertical distance and direction from the $x$-axis.
• Quadrants: The four regions into which the coordinate axes divide the Cartesian plane:

- Quadrant I: $x>0$ and $y>0$. - Quadrant II: $x<0$ and $y>0$. - Quadrant III: $x<0$ and $y<0$. - Quadrant IV: $x>0$ and $y<0$.

• Plotting a Point: Locating a point by moving first in the horizontal direction according to its $x$-coordinate and then vertically according to its $y$-coordinate.
• Sign of the $x$-coordinate: A positive $x$-coordinate places a point to the right of the $y$-axis; a negative $x$-coordinate places it to the left.
• Sign of the $y$-coordinate: A positive $y$-coordinate places a point above the $x$-axis; a negative $y$-coordinate places it below.
• Points on the axes: A point on the $x$-axis has the form $(x,0)$. A point on the $y$-axis has the form $(0,y)$.
• Origin condition: The origin is the only point common to both axes and is $(0,0)$.
• Coordinate alignment: Points with the same $x$-coordinate lie on a vertical line, while points with the same $y$-coordinate lie on a horizontal line.
• Interpretation of a point: Read the horizontal position first and the vertical position second.
• Scale requirement: Accurate plotting requires a suitable scale on both axes.

Worked Methods

Plotting a point

• Write the coordinates in the form $(x,y)$, preserving the order.
• Use the $x$-coordinate to determine the horizontal movement:

- move right if $x$ is positive; - move left if $x$ is negative.

• From that position, use the $y$-coordinate to determine the vertical movement:

- move upward if $y$ is positive; - move downward if $y$ is negative.

• Mark the resulting point and label it with its ordered pair.

Example: Plot $(3,-2)$.

• The $x$-coordinate is $3$, so move 3 units right from the origin.
• The $y$-coordinate is $-2$, so move 2 units downward.
• The point lies in Quadrant IV because $x>0$ and $y<0$.

Identifying the quadrant or axis

• Read the $x$-coordinate and determine whether it is positive, negative, or zero.
• Read the $y$-coordinate and determine whether it is positive, negative, or zero.
• If both coordinates are non-zero, use their signs to identify the quadrant:

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

• If $y=0$, the point lies on the $x$-axis and has the form $(x,0)$.
• If $x=0$, the point lies on the $y$-axis and has the form $(0,y)$.
• If $x=0$ and $y=0$, the point is the origin $(0,0)$.

Reading a plotted point

• Read the horizontal position of the point relative to the $y$-axis to obtain the abscissa, or $x$-coordinate.
• Read the vertical position relative to the $x$-axis to obtain the ordinate, or $y$-coordinate.
• Write the result in the order $(x,y)$.
• Use the signs to verify the point’s quadrant or axis.

Using coordinate relationships

• Compare the $x$-coordinates of points.
• Points with the same $x$-coordinate lie on a vertical line.
• Compare the $y$-coordinates of points.
• Points with the same $y$-coordinate lie on a horizontal line.
• Remember that the Cartesian plane connects numerical information with visual location, allowing relationships and positions to be identified geometrically.

Where It Goes Wrong

• Reversing the coordinates: $(x,y)$ must be read as horizontal position first and vertical position second; $(x,y)$ and $(y,x)$ generally represent different points.
• Moving vertically before horizontally: plotting requires following the $x$-coordinate first and then the $y$-coordinate.
• Ignoring signs: positive and negative coordinates determine direction and therefore the quadrant or axis.
• Forgetting axis conditions: $y=0$ places a point on the $x$-axis, while $x=0$ places it on the $y$-axis.
• Misidentifying the origin: the origin is specifically $(0,0)$ and is the only point common to both axes.
• Using an unsuitable scale: accurate plotting depends on choosing a suitable scale on the coordinate axes.

What Gets Asked

• Define the Cartesian plane, coordinate axes, origin, ordered pair, abscissa, ordinate, quadrants, or plotting a point.
• State the coordinates of the origin.
• Identify the $x$-coordinate and $y$-coordinate of a given ordered pair.
• Determine the quadrant in which a point lies from the signs of its coordinates.
• State whether a point lies on the $x$-axis, $y$-axis, or at the origin.
• Plot a specified point, including $(3,-2)$, on a Cartesian plane.
• Read the coordinates of a plotted point.
• Identify whether points lie on a common vertical or horizontal line by comparing their coordinates.
• Explain why coordinate order, coordinate signs, scale, and the sequence of horizontal then vertical movement are essential for accurate plotting.