Introduction: Why Linear Graphs Matter!
Welcome to the world of Coordinate Geometry! This chapter is all about drawing and understanding straight lines, which mathematicians call linear graphs. Think of them as maps for constant change—like calculating the cost of a taxi ride (fixed fee + price per kilometre) or tracking how fast a bottle empties (constant rate of flow).
Mastering this topic is essential because linear relationships form the backbone of Algebra and show up everywhere in physics, economics, and real-life problem-solving. Don't worry if you find graphs tricky; we'll break down the process into simple, repeatable steps.
Section 1: The Coordinate Plane – Your Map
1.1 Understanding Coordinates (C4.1)
A straight line is drawn on a Cartesian plane (named after René Descartes). This plane uses two main lines called axes to locate every point.
- The Horizontal Axis: This is the x-axis. Movement along this axis is always stated first in a coordinate pair.
- The Vertical Axis: This is the y-axis. Movement up or down is stated second.
A point is always written as an ordered pair: \((x, y)\).
Memory Trick: To plot a point, you must Walk Before You Climb! (x then y)
1.2 Plotting Points
To draw a graph, you must first accurately plot points. Remember the rule:
- Start at the Origin \((0, 0)\).
- Move horizontally (left or right) according to the x-coordinate.
- Move vertically (up or down) according to the y-coordinate.
Quick Review:
\(x\) = Horizontal movement (left/right)
\(y\) = Vertical movement (up/down)
Key Takeaway: The coordinate plane allows us to turn algebraic equations into visual straight lines.
Section 2: The Linear Equation \(y = mx + c\) (C4.4)
Every non-vertical straight line can be perfectly described by a single, powerful equation: \(y = mx + c\).
2.1 Interpreting the Equation
This formula tells us everything we need to know about the line's position and its steepness.
-
\(m\) is the Gradient (The Steepness)
This determines how steep the line is and in which direction it slopes. It represents the rate of change.
-
\(c\) is the y-intercept (The Starting Point)
This is the point where the line crosses the y-axis. When \(x=0\), the value of \(y\) is \(c\). The coordinate of the y-intercept is always \((0, c)\).
Example: For the line \(y = 3x - 5\), the gradient (\(m\)) is 3 and the y-intercept (\(c\)) is -5. The line crosses the y-axis at \((0, -5)\).
2.2 Finding \(m\) and \(c\) from other forms
Sometimes the equation is given in a different form, like \(ax + by = c\). You must rearrange it into the form \(y = mx + c\) to find \(m\) and \(c\).
Example: Find the gradient and y-intercept of \(2y - 4x = 6\).
- Add \(4x\) to both sides: \(2y = 4x + 6\)
- Divide every term by 2: \(y = 2x + 3\)
Now it is clear: \(m = 2\) and \(c = 3\).
Key Takeaway: The form \(y = mx + c\) gives you the two vital pieces of information: the steepness (\(m\)) and the crossing point on the y-axis (\(c\)).
Section 3: Calculating the Gradient (m) (C4.2)
The gradient (\(m\)) is a measure of the line's steepness and direction. It is the ratio of the vertical change (the rise) to the horizontal change (the run).
3.1 Gradient from a Grid ("Rise over Run")
For Core students, finding the gradient often involves reading it directly from a grid:
\[m = \frac{\text{Rise}}{\text{Run}}\]
Step-by-Step Method:
- Identify two clear points on the line, \(A\) and \(B\).
- Draw a right-angled triangle connecting these two points.
- Count the vertical change (the Rise) – how far up or down did you move?
- Count the horizontal change (the Run) – how far across did you move?
- Calculate \(m = \frac{\text{Rise}}{\text{Run}}\).
Did you know? A gradient of \(m = 1\) means the line slopes up at a perfect 45-degree angle!
3.2 Types of Gradient
- Positive Gradient (\(m > 0\)): The line slopes up from left to right. (e.g., \(y = 2x + 1\))
- Negative Gradient (\(m < 0\)): The line slopes down from left to right. (e.g., \(y = -3x + 4\))
- Zero Gradient (\(m = 0\)): The line is perfectly horizontal. (e.g., \(y = 5\))
- Undefined Gradient: The line is perfectly vertical. (e.g., \(x = -2\))
Common Mistake to Avoid:
Don't confuse positive and negative slopes! If you are reading a graph from left to right, and the line is going downhill, the gradient must be negative.
Key Takeaway: Gradient measures vertical change divided by horizontal change. A steep hill has a large gradient, a flat road has a zero gradient.
Section 4: Drawing a Linear Graph
There are two main reliable ways to draw a straight line graph from its equation, such as \(y = 2x - 1\).
4.1 Method 1: Using a Table of Values (The Universal Method)
This method works for *any* function (linear, quadratic, or otherwise) and is the safest option if you are unsure.
Step-by-Step Process:
- Choose \(x\)-values: Select a range of \(x\)-values (e.g., -2, -1, 0, 1, 2).
- Calculate \(y\)-values: Substitute each \(x\)-value into the equation \(y = mx + c\) to find the corresponding \(y\)-value.
- Create Coordinates: Write down the resulting coordinate pairs \((x, y)\). You should aim for at least three points to ensure accuracy.
- Plot: Mark the coordinates clearly on the grid (use small crosses 'x').
- Draw the Line: Use a ruler to connect the points. Ensure the line extends across the full range of the graph paper provided.
Example: Draw \(y = 3x + 1\)
| \(x\) | Calculation \(y = 3x + 1\) | \(y\) | Point |
|---|---|---|---|
| -2 | \(3(-2) + 1 = -6 + 1\) | -5 | \((-2, -5)\) |
| 0 | \(3(0) + 1 = 0 + 1\) | 1 | \((0, 1)\) |
| 2 | \(3(2) + 1 = 6 + 1\) | 7 | \((2, 7)\) |
4.2 Method 2: Using the Gradient and Intercept (The Quick Method)
If you have the equation in the form \(y = mx + c\), you can quickly draw the graph using just \(m\) and \(c\).
Step-by-Step Process:
- Plot \(c\): Identify the y-intercept, \(c\), and plot the point \((0, c)\). This is your starting point.
- Use the Gradient (\(m\)): Write \(m\) as a fraction \(\frac{\text{Rise}}{\text{Run}}\).
- If \(m = 2\), use \(\frac{2}{1}\) (Rise 2, Run 1).
- If \(m = -\frac{1}{2}\), use \(\frac{-1}{2}\) (Rise -1, Run 2, meaning go down 1).
- Find the Next Points: Starting from \((0, c)\), use the Rise and Run values to find at least one or two more points.
- Draw the Line: Connect the points with a ruler, extending the line across the grid.
Example: Draw \(y = -\frac{2}{3}x + 4\)
- Start: \(c = 4\). Plot \((0, 4)\).
- Gradient: \(m = -\frac{2}{3}\). This means Rise = -2 (go down 2 units) and Run = 3 (go right 3 units).
- From \((0, 4)\), go down 2 and right 3 to reach \((3, 2)\). Plot this point.
- From \((3, 2)\), go down 2 and right 3 to reach \((6, 0)\). Plot this point.
- Rule the straight line through \((0, 4)\), \((3, 2)\), and \((6, 0)\).
Key Takeaway: Always use a ruler for linear graphs and plot enough points to check your accuracy (three is ideal).
Section 5: Special Lines and Parallelism
5.1 Horizontal and Vertical Lines (C4.4)
Some straight lines do not fit the traditional \(y = mx + c\) form, especially vertical lines.
1. Horizontal Lines (Gradient \(m=0\))
- Equation Form: \(y = k\) (where \(k\) is a constant number).
- Description: The line is parallel to the x-axis.
- Reason: Every point on the line has the same \(y\)-coordinate.
- Example: The line \(y = 5\) passes through \((1, 5)\), \((-2, 5)\), and \((0, 5)\).
2. Vertical Lines (Gradient is Undefined)
- Equation Form: \(x = k\) (where \(k\) is a constant number).
- Description: The line is parallel to the y-axis.
- Reason: Every point on the line has the same \(x\)-coordinate.
- Example: The line \(x = -3\) passes through \((-3, 1)\), \((-3, 7)\), and \((-3, 0)\).
5.2 Parallel Lines (C4.5)
Lines that never meet are parallel. They maintain the same steepness relative to each other.
The Golden Rule for Parallel Lines:
If two lines are parallel, their gradients (\(m\)) are equal.
\[m_1 = m_2\]
Example: A line parallel to \(y = 4x + 9\) must also have a gradient of 4. Its equation will be \(y = 4x + c\), where \(c\) could be any number except 9.
5.3 Finding the Equation of a Parallel Line (C4.5)
This is a common exam question that combines the concepts of gradient and coordinates.
Scenario: Find the equation of a line parallel to \(y = 4x - 1\) that passes through the point \((1, -3)\).
Step-by-Step Solution:
- Determine \(m\): Since the new line is parallel to \(y = 4x - 1\), its gradient is \(m = 4\).
- Start the Equation: We know the new equation looks like \(y = 4x + c\).
- Find \(c\): Substitute the given point \((1, -3)\) into the new equation.
\(y = 4x + c\)
\(-3 = 4(1) + c\)
\(-3 = 4 + c\)
\(c = -3 - 4\)
\(c = -7\)
- Write the Final Equation: Substitute \(m\) and \(c\) back into the general form.
The equation is \(y = 4x - 7\).
Key Takeaway: Parallel lines share the same gradient. If you know the gradient and one point, you can always find the full equation.