Coordinates and Graphs
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Master plotting coordinates, reading points, and understanding transformations across all four quadrants of the Cartesian plane.
The Gikomba Market Grid: Picture an open-air market laid out on a grid. The central East–West avenue is the \(x\)-axis, and the central North–South avenue is the \(y\)-axis. Their intersection is the Origin \((0,0)\). Every stall is identified by an ordered pair \((x, y)\): first move horizontally along \(x\) (East = positive, West = negative), then vertically along \(y\) (North = positive, South = negative).
The two perpendicular axes divide the plane into four quadrants:
\((+ , +)\)
Top-Right (East & North)
\((- , +)\)
Top-Left (West & North)
\((- , -)\)
Bottom-Left (West & South)
\((+ , -)\)
Bottom-Right (East & South)
Interactive Coordinate Explorer
Click anywhere on the grid or adjust the sliders to place a point. Observe its coordinates, quadrant, and distance from the origin.
Ordered Pair: (2, 3)
Location: Quadrant I
Key Formulas
\[(x, y)\]
\(x\) represents horizontal distance from the origin (East/West). \(y\) represents vertical distance (North/South).
\[(x, y) \xrightarrow{\text{shift } a \text{ right, } b \text{ up}} (x + a, y + b)\]
\[(x, y) \xrightarrow{\text{shift } a \text{ left, } b \text{ down}} (x - a, y - b)\]
Across \(x\)-axis: \((x, y) \to (x, -y)\)
Across \(y\)-axis: \((x, y) \to (-x, y)\)
Across the line \(y = x\): \((x, y) \to (y, x)\)
\(90^\circ\) Clockwise (or \(270^\circ\) Anticlockwise): \((x, y) \to (y, -x)\)
\(180^\circ\) Rotation: \((x, y) \to (-x, -y)\)
\(90^\circ\) Anticlockwise (or \(270^\circ\) Clockwise): \((x, y) \to (-y, x)\)
\[y = mx + c\]
Where \(m\) is the gradient (slope) and \(c\) is the \(y\)-intercept where the line crosses the \(y\)-axis at \((0, c)\).
Worked Examples
Problem: Plot the point \(P(-3, 4)\) and determine in which quadrant it lies.
- Start at the origin \((0, 0)\).
- The \(x\)-coordinate is \(-3\). Move 3 units to the left along the horizontal axis.
- The \(y\)-coordinate is \(+4\). Move 4 units upwards parallel to the vertical axis.
- Since \(x < 0\) (negative) and \(y > 0\) (positive), the point \(P(-3, 4)\) lies in Quadrant II.
Problem: A matatu station is located at \(A(4, -3)\).
(a) The station is translated \(3\) units to the left and \(5\) units up to a new position \(A'\). Find the coordinates of \(A'\).
(b) Find the image of \(A\) when reflected across the \(y\)-axis.
- Part (a): Left means subtract from \(x\); Up means add to \(y\).
\[x' = 4 - 3 = 1\]
\[y' = -3 + 5 = 2\]
So, \(A' = (1, 2)\) in Quadrant I. - Part (b): Reflection across the \(y\)-axis changes the sign of \(x\) while \(y\) remains the same: \((x, y) \to (-x, y)\).
\[(4, -3) \to (-4, -3)\]
Problem: The vertices of a triangular plot in a farm layout are \(A(1, 2)\), \(B(5, 2)\), and \(C(1, 6)\).
(a) Plot the points and identify the length of the base and height.
(b) Calculate the area of the triangular plot.
- Base \(AB\): Both \(A\) and \(B\) lie on the horizontal line \(y = 2\).
Length of base = \(x_B - x_A = 5 - 1 = 4\text{ units}\). - Height \(AC\): Both \(A\) and \(C\) lie on the vertical line \(x = 1\).
Length of height = \(y_C - y_A = 6 - 2 = 4\text{ units}\). - Triangle Area: Since \(AB\) is horizontal and \(AC\) is vertical, the angle between them is \(90^\circ\).
\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8\text{ square units}\]
Common Mistakes
Plotting \((2, 5)\) by going up \(2\) and right \(5\) leads to \((5, 2)\).
Correction: Always remember: "Walk along the corridor (horizontal \(x\)), then take the lift (vertical \(y\))." The first number is strictly horizontal, the second is vertical.
Assuming that any point with a negative sign must be in Quadrant III.
Correction: Only points where both coordinates are negative \((- , -)\) lie in Quadrant III. \((- , +)\) is in Quadrant II, and \((+ , -)\) is in Quadrant IV.
Thinking that \((0, 4)\) lies on the \(x\)-axis because \(0\) is the first coordinate.
Correction: When \(x = 0\), you move 0 steps horizontally, meaning you remain strictly on the \(y\)-axis. Conversely, \((4, 0)\) lies on the \(x\)-axis.
Real World
📱 Ride-Hailing & GPS in Nairobi
Apps like Uber and Little Cab use latitude and longitude coordinates, which act as a spherical Cartesian plane to track passenger locations and compute shortest routes along city avenues.
🌾 Smart Agriculture & Drones
Agricultural drones over tea and coffee estates in Kericho and Kiambu fly along pre-programmed \((x, y)\) coordinate waypoints to monitor crop health and spray fertilizers evenly.
🏗️ Architecture and Construction
Civil engineers laying out the foundation of the Nairobi Expressway or school buildings use Cartesian coordinate benchmarks to align concrete pillars with millimetre precision.
Practice