GRADE 8 Mathematics GEOMETRY – Coordinates and Graphs Notes

1. The Coordinate Plane

We use two number lines that meet at right angles. The horizontal line is the x-axis and the vertical line is the y-axis. The meeting point is the origin, written (0, 0).

• Right on the x-axis → positive x values.
• Left on the x-axis → negative x values.
• Up on the y-axis → positive y values.
• Down on the y-axis → negative y values.

2. Ordered Pairs (x, y)

An ordered pair (x, y) tells you where to put a point:

1. Start at the origin (0,0).
2. Move x units: right if x is positive, left if x is negative.
3. Then move y units: up if y is positive, down if y is negative.

3. Quadrants

The plane is divided into four quadrants. The usual numbering:

• Quadrant I: (x positive, y positive)
• Quadrant II: (x negative, y positive)
• Quadrant III: (x negative, y negative)
• Quadrant IV: (x positive, y negative)

4. Example: A Simple Graph

Below is a simple grid from x = −5 to 5 and y = −5 to 5 with four example points:

Reading example: Point A(3,2) means 3 to the right, 2 up.

5. Plotting Steps (quick)

To plot (x, y):

1. Move horizontally x units from origin.
2. Then move vertically y units.
3. Mark the point and write its label.

6. Graphing a Straight Line (simple)

Equation: y = mx + c, where m is the gradient (slope) and c is the y-intercept (where the line crosses the y-axis).

Example: y = 1x + 1 (that is y = x + 1)

• Find two easy points: when x = 0 → y = 1 gives (0,1). When x = 1 → y = 2 gives (1,2).
• Plot these two points and draw the straight line through them.

Note: For school work, often you use graph paper so each square is 1 unit. Draw the line neatly through plotted points.

7. Distance and Midpoint (useful)

For two points A(x1, y1) and B(x2, y2):

Midpoint: M = ((x1 + x2)/2, (y1 + y2)/2)

Distance (straight line): d = sqrt[(x2 − x1)^2 + (y2 − y1)^2]

Simple quick case: If two points are on the same horizontal line, distance = difference in x. If on same vertical line, distance = difference in y.

Example: A(3,2), D(0,4).

• Midpoint M = ((3 + 0)/2, (2 + 4)/2) = (1.5, 3).
• Distance d = sqrt[(0 − 3)^2 + (4 − 2)^2] = sqrt[9 + 4] = sqrt(13) ≈ 3.6 units.

8. Tips for Exams

• Label axes and the scale clearly (each square = 1 unit unless told otherwise).
• Write ordered pairs as (x, y) — x first, y second.
• Check signs: negative values move left or down.
• When asked to draw a line, find at least two points then draw through them neatly.

9. Practice Exercises

1. Plot these points: P(2,3), Q(-3,2), R(-1,-4). Which quadrant is each point in?
2. Find the midpoint of P(2,3) and Q(-3,2).
3. Calculate the distance between P and Q (leave as √ form).
4. Draw the graph of y = x − 2 using two points and state the y-intercept.