Section A — Chapters 1–22

The Cartesian Plane

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to identify and label the x-axis, y-axis, and origin on the Cartesian plane.
✓By the end of this lesson students will be able to understand the concept of an ordered pair (x, y) as a set of coordinates.
✓By the end of this lesson students will be able to plot points accurately on the Cartesian plane given their coordinates.
✓By the end of this lesson students will be able to determine the coordinates of a given point on the Cartesian plane.
✓By the end of this lesson students will be able to identify the four quadrants of the Cartesian plane.

Key concepts

The Cartesian Plane

The Cartesian plane is a two-dimensional flat surface formed by two perpendicular number lines, used to locate points. It is also sometimes called the coordinate plane.

x-axis

The x-axis is the horizontal number line on the Cartesian plane. Positive numbers are to the right of the origin, and negative numbers are to the left.

y-axis

The y-axis is the vertical number line on the Cartesian plane. Positive numbers are above the origin, and negative numbers are below.

Origin

The origin is the point where the x-axis and y-axis intersect. Its coordinates are (0, 0).

Coordinates

Coordinates are a set of values that show an exact position of a point on the Cartesian plane. They are always written as an ordered pair.

Ordered Pair

An ordered pair is a pair of numbers (x, y) that represent the coordinates of a point. The first number, x, is the x-coordinate (horizontal position), and the second number, y, is the y-coordinate (vertical position).

Quadrants

The four regions into which the Cartesian plane is divided by the x-axis and y-axis. They are numbered anti-clockwise using Roman numerals, starting from the top-right: Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), Quadrant IV (+,-).

Key facts to remember

1The Cartesian plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis.
2The point where the x-axis and y-axis intersect is called the origin, with coordinates (0, 0).
3Coordinates are always written as an ordered pair (x, y), where x is the horizontal position and y is the vertical position.
4The x-coordinate tells you how far to move left (negative) or right (positive) from the origin.
5The y-coordinate tells you how far to move down (negative) or up (positive) from the origin.
6Points on the x-axis always have a y-coordinate of 0 (e.g., (5, 0)).
7Points on the y-axis always have an x-coordinate of 0 (e.g., (0, -2)).
8The four quadrants are numbered anti-clockwise from I to IV, starting from the top-right.

Worked examples

Example 1

Plot the points A(3, 2), B(-2, 4), C(1, -3), and D(-4, -1) on the Cartesian plane.

IDraw and label the x-axis and y-axis, marking the origin (0, 0) and a suitable scale on both axes.

IIFor point A(3, 2): Start at the origin. Move 3 units to the right along the x-axis. Then, move 2 units up parallel to the y-axis. Mark this point as A.

IIIFor point B(-2, 4): Start at the origin. Move 2 units to the left along the x-axis. Then, move 4 units up parallel to the y-axis. Mark this point as B.

IVFor point C(1, -3): Start at the origin. Move 1 unit to the right along the x-axis. Then, move 3 units down parallel to the y-axis. Mark this point as C.

VFor point D(-4, -1): Start at the origin. Move 4 units to the left along the x-axis. Then, move 1 unit down parallel to the y-axis. Mark this point as D.

Answer

The points A, B, C, and D are correctly plotted on the Cartesian plane.

Always use a ruler and pencil for drawing axes and plotting points. Ensure your axes are clearly labelled 'x', 'y', and '0' for the origin.

Example 2

Identify the coordinates of the points E, F, G, and H shown on a Cartesian plane where E is 4 units right of the origin on the x-axis, F is 3 units below the origin on the y-axis, G is 3 units left and 2 units up from the origin, and H is 2 units right and 2 units down from the origin.

IFor point E: Starting from the origin (0,0), move 4 units right along the x-axis. There is no vertical movement. So, the coordinates are (4, 0).

IIFor point F: Starting from the origin (0,0), there is no horizontal movement. Move 3 units down along the y-axis. So, the coordinates are (0, -3).

IIIFor point G: Starting from the origin (0,0), move 3 units left along the x-axis (x = -3). Then, move 2 units up parallel to the y-axis (y = 2). So, the coordinates are (-3, 2).

IVFor point H: Starting from the origin (0,0), move 2 units right along the x-axis (x = 2). Then, move 2 units down parallel to the y-axis (y = -2). So, the coordinates are (2, -2).

Answer

E(4, 0), F(0, -3), G(-3, 2), H(2, -2).

Example 3

For each of the following points, state the quadrant it lies in or if it lies on an axis: P(5, 7), Q(-3, 6), R(0, -4), S(-2, -5), T(8, -1), U(0, 0).

IFor P(5, 7): The x-coordinate (5) is positive and the y-coordinate (7) is positive. Points with positive x and positive y are in Quadrant I.

IIFor Q(-3, 6): The x-coordinate (-3) is negative and the y-coordinate (6) is positive. Points with negative x and positive y are in Quadrant II.

IIIFor R(0, -4): The x-coordinate is 0, meaning the point is on the y-axis. Since the y-coordinate (-4) is negative, it is on the negative y-axis.

IVFor S(-2, -5): The x-coordinate (-2) is negative and the y-coordinate (-5) is negative. Points with negative x and negative y are in Quadrant III.

VFor T(8, -1): The x-coordinate (8) is positive and the y-coordinate (-1) is negative. Points with positive x and negative y are in Quadrant IV.

VIFor U(0, 0): This point is the origin. It does not lie in any specific quadrant but is the intersection of both axes.

Answer

P is in Quadrant I, Q is in Quadrant II, R is on the negative y-axis, S is in Quadrant III, T is in Quadrant IV, U is the origin.

Points on the axes (where either x or y is zero) are not considered to be in any quadrant.

Common mistakes

✗Mixing up the x and y coordinates when plotting or identifying points (e.g., plotting (y, x) instead of (x, y)).
✗Forgetting to label the axes ('x' and 'y') and the origin ('0') on the Cartesian plane.
✗Using inconsistent scales on the x-axis and y-axis, which can distort shapes and distances.
✗Incorrectly identifying the quadrants or numbering them in the wrong order (they are anti-clockwise).
✗Plotting negative coordinates in the wrong direction (e.g., a negative x-coordinate to the right of the y-axis).

Exam tips

★Always use a ruler and a sharp pencil to draw your axes and plot points accurately.
★Clearly label your x-axis, y-axis, and the origin (0) on any graph you draw.
★Remember the order of coordinates: (x, y). A helpful memory aid is 'x comes before y in the alphabet'.
★Double-check your plotting by tracing the path from the origin to the point using the x-coordinate first, then the y-coordinate.

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