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Door 7 · Grade 8

📐 Coordinate Geometry

The Cartesian plane has X-axis (horizontal) and Y-axis (vertical)

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Cartesian Plane and Ordered Pairs

What is the Coordinate System? The Cartesian coordinate system (named after mathematician René Descartes) is a method of locating points on a plane using two perpendicular number lines called axes.

The Axes:

• X-axis: The horizontal number line (left to right)
• Y-axis: The vertical number line (up and down)
• Origin: The point where the X-axis and Y-axis intersect, labeled as \(O(0, 0)\)

What are Ordered Pairs? An ordered pair \((x, y)\) represents the coordinates of a point, where:

• \(x\) = the distance from the Y-axis (called the x-coordinate or abscissa)
• \(y\) = the distance from the X-axis (called the y-coordinate or ordinate)

Important Rules:

• The order matters! \((2, 3)\) is different from \((3, 2)\)
• The first number always refers to the X-axis (horizontal position)
• The second number always refers to the Y-axis (vertical position)

How to Plot a…

Worked example

Problem: Write the coordinates of the points: A point that is 5 units right of origin and 3 units up.

  1. - Right of origin → positive x-coordinate: \(x = 5\)
  2. - Up from origin → positive y-coordinate: \(y = 3\)
  3. - **
  4. *Example 2:
  5. Plot the point \((-3, 4)\) on the coordinate plane.

Answer: ** \((5, 3)\)

Quadrants and Distance Interpretation

What are the Four Quadrants? The two perpendicular axes divide the plane into four quadrants, numbered counterclockwise starting from the top-right.

Quadrantx-signy-signExample pointQuadrant I (Q1)++(3, 4)Quadrant II (Q2)-+(-3, 4)Quadrant III (Q3)--(-3, -4)Quadrant IV (Q4)+-(3, -4)
Points on Axes:

• On X-axis: \(y = 0\), e.g., \((5, 0)\), \((-2, 0)\)
• On Y-axis: \(x = 0\), e.g., \((0, 3)\), \((0, -4)\)
• Origin: \((0, 0)\) (on both axes)

Distance Interpretation:

• The x-coordinate tells how far a point is from the Y-axis
• The y-coordinate tells how far a point is from the X-axis
• The distance from the origin = \(√x² + y²\) (Pythagorean theorem)

Distance between two points:

\[ Distance = √(x_2 - x_1)² + (y_2 - y_1)² \]
Horizontal and Vertical Distances:

• Same y-coordinate → horizontal distance = \(|x_2 - x_1|\)
• Same x-coordinate → vertical distance = \(|y_2 - y_1|\)

Worked example

Problem: In which quadrant does the point \((-5, 7)\) lie?

  1. - x = -5 (negative)
  2. - y = 7 (positive)
  3. - Negative x, positive y → Quadrant II
  4. - **
  5. *Example 2:

Answer: ** Quadrant II

Graphing Simple Relations

What is Graphing a Relation? A relation is a set of ordered pairs. Graphing a relation means plotting all points that satisfy a given condition or equation.

Common Relations to Graph:

RelationDescriptionExample\(x = k\)Vertical line at \(x = k\)\(x = 3\) (vertical line)\(y = k\)Horizontal line at \(y = k\)\(y = 2\) (horizontal line)\(y = x\)Diagonal line (slope 1)Points like (0,0), (1,1), (2,2)\(y = mx + c\)Straight line\(y = 2x + 1\)\(x = y\)Line at 45°Points where coordinates are equal
How to Graph a Simple Equation:

• Make a table of values (choose 3-5 x-values)
• Calculate corresponding y-values
• Plot each ordered pair
• Connect the points (if they form a line)

Identifying Patterns:

• Linear relations (\(y = mx + c\)) form straight lines
• The line \(y = x\) passes through origin at 45°
• Vertical lines: \(x = constant\) (all points share same x)
•…

Worked example

Problem: Make a table of values for \(y = x + 2\) for \(x = 0, 1, 2, 3\).

  1. - \(x = 0\) → \(y = 0 + 2 = 2\) → \((0, 2)\)
  2. - \(x = 1\) → \(y = 1 + 2 = 3\) → \((1, 3)\)
  3. - \(x = 2\) → \(y = 2 + 2 = 4\) → \((2, 4)\)
  4. - \(x = 3\) → \(y = 3 + 2 = 5\) → \((3, 5)\)
  5. - **

Answer: | x | y |