Introduction to Graphs — Class 8 IMO

The Cartesian Plane

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 Point:

Start at the origin \((0, 0)\)
Move horizontally along the X-axis according to \(x\):
Positive \(x\) → move RIGHT
Negative \(x\) → move LEFT
Move vertically according to \(y\):
Positive \(y\) → move UP
Negative \(y\) → move DOWN

Example 1: What are the coordinates of the origin?

(0, 0).

Example 2: What does the point (3, 4) mean?

3 units right and 4 units up from the origin.

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

- Right of origin → positive x-coordinate: \(x = 5\) - Up from origin → positive y-coordinate: \(y = 3\) - **Answer:** \((5, 3)\) *Example 2: Plot the point \((-3, 4)\) on the coordinate plane. Solution: - Start at \((0, 0)\) - Move 3 units LEFT (because \(x = -3\) is negative) - Move 4 units UP (because \(y = 4\) is positive) - Mark the point and label it \(P(-3, 4)\) - **Answer:** Point plotted in Quadrant II *Example 3: A rectangle has vertices at \((1, 1)\), \((5, 1)\), \((5, 4)\), and \((1, 4)\). Find the length and width. Solution: - Length = distance along X-axis = \(5 - 1 = 4\) units - Width = distance along Y-axis = \(4 - 1 = 3\) units - **Answer:** Length = 4 units, Width = 3 units

Quick recap

x-axis is horizontal, y-axis is vertical, meeting at the origin (0, 0).
A point is (x, y), with x read first.
The Cartesian plane has X-axis (horizontal) and Y-axis (vertical)
Origin is \((0, 0)\) where axes intersect
Ordered pair \((x, y)\): first is x-coordinate, second is y-coordinate
Positive x → right; negative x → left
Positive y → up; negative y → down
The order of numbers in a pair is extremely important
---

✓ Quick check

What is the x-coordinate of the point (5, 2)?

The first number is the x-coordinate, 5.

The coordinates of the origin are ___ ?

The origin is (0, 0).

Plotting Points

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

Quadrant	x-sign	y-sign	Example point
Quadrant 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 = \(\sqrt{x^2 + y^2}\) (Pythagorean theorem)

Distance between two points:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Horizontal and Vertical Distances:

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

Example 1: Where is the point (2, 3)?

2 right and 3 up, in the first quadrant.

Example 2: What is the y-coordinate of a point on the x-axis?

0.

Example 3: In which quadrant does the point \((-5, 7)\) lie?

- x = -5 (negative) - y = 7 (positive) - Negative x, positive y → Quadrant II - **Answer:** Quadrant II *Example 2: Find the distance between the points \(A(2, 3)\) and \(B(5, 7)\). Solution: - Difference in x: \(x_2 - x_1 = 5 - 2 = 3\) - Difference in y: \(y_2 - y_1 = 7 - 3 = 4\) - Distance = \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) units - **Answer:** 5 units *Example 3: A point \(P\) lies on the X-axis at a distance of 6 units from the origin. What are its coordinates? Solution: - On X-axis → \(y = 0\) - Distance from origin = 6 units - Two possibilities: \((6, 0)\) or \((-6, 0)\) - **Answer:** \((6, 0)\) or \((-6, 0)\)

Quick recap

Plot (x, y): x along, then y up/down.
On the x-axis y = 0; on the y-axis x = 0.
Quadrant I: (+, +) | Quadrant II: (-, +) | Quadrant III: (-, -) | Quadrant IV: (+, -)
Points on axes are NOT in any quadrant
Distance from origin = \(\sqrt{x^2 + y^2}\)
Distance between points = \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
Points with same y-coordinate are horizontally aligned
Points with same x-coordinate are vertically aligned
---

✓ Quick check

A point on the y-axis has x-coordinate ___ ?

Points on the y-axis have x = 0.

The point (0, 5) lies on the ___ ?

x = 0, so it is on the y-axis.

Graphs of Linear Equations

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:

Relation	Description	Example
\(x = k\)	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: \(x = \text{constant}\) (all points share same x)
Horizontal lines: \(y = \text{constant}\) (all points share same y)

Example 1: On y = 2x, find y when x = 3.

y = 2 × 3 = 6.

Example 2: What does the steepness of a distance-time graph show?

The speed.

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

- \(x = 0\) → \(y = 0 + 2 = 2\) → \((0, 2)\) - \(x = 1\) → \(y = 1 + 2 = 3\) → \((1, 3)\) - \(x = 2\) → \(y = 2 + 2 = 4\) → \((2, 4)\) - \(x = 3\) → \(y = 3 + 2 = 5\) → \((3, 5)\) - **Answer: | x | y | |---|---| | 0 | 2 | | 1 | 3 | | 2 | 4 | | 3 | 5 | *Example 2: Plot the graph of \(y = 2x\) for \(x = -2, -1, 0, 1, 2\). Solution: - \(x = -2\) → \(y = -4\) → \((-2, -4)\) - \(x = -1\) → \(y = -2\) → \((-1, -2)\) - \(x = 0\) → \(y = 0\) → \((0, 0)\) - \(x = 1\) → \(y = 2\) → \((1, 2)\) - \(x = 2\) → \(y = 4\) → \((2, 4)\) - Plot all points and draw a straight line through them - **Answer:** A straight line passing through origin with slope 2 *Example 3: Write the equation for a horizontal line that passes through the point \((3, 5)\). Solution: - Horizontal line has constant y-value - The point \((3, 5)\) has \(y = 5\) - Every point on this line has \(y = 5\) - Equation: \(y = 5\) - **Answer:** \(y = 5\)

Quick recap

y = mx passes through the origin; y = mx + c cuts the y-axis at c.
Substitute x to find a point; steeper distance-time graph = faster.
To graph a relation, plot points that satisfy the condition
\(y = mx + c\) represents a straight line (m = slope, c = y-intercept)
\(x = \text{constant}\) → vertical line
\(y = \text{constant}\) → horizontal line
\(y = x\) is a diagonal line through origin (45° angle)
Use a table of values to find points before plotting
---

✓ Quick check

On y = x + 1, what is y when x = 2?

y = 2 + 1 = 3.

On y = 3x, what is y when x = 4?

y = 3 × 4 = 12.