Introduction to Cartesian Co-ordinates and Straight-Line Graphs

Welcome! In this chapter, we are exploring the "map" of the mathematical world. Just as a GPS uses coordinates to find a location, mathematicians use Cartesian co-ordinates to pinpoint positions on a flat surface. We will also learn how a simple algebraic rule can create a perfectly straight line on a graph. This is a foundational skill for the Functions, graphs and calculus section of your Pearson Edexcel International GCSE (Specification B) course.

1. The Cartesian Coordinate System

A Cartesian grid is made up of two number lines that cross at right angles. These lines are called axes.

  • The \(x\)-axis is the horizontal line (running left to right).
  • The \(y\)-axis is the vertical line (running up and down).
  • The point where they meet is called the origin, and its coordinates are \((0, 0)\).

How to Plot Points

Every point on the graph has an "address" written as a pair of numbers: \((x, y)\). These are called coordinates.

  1. The First Number (\(x\)): Tells you how far to move horizontally from the origin. Positive moves right, negative moves left.
  2. The Second Number (\(y\)): Tells you how far to move vertically. Positive moves up, negative moves down.

Memory Trick: Remember "along the corridor, then up the stairs." You must go across the \(x\)-axis before you go up or down the \(y\)-axis.

2. The Equation of a Straight Line

In your exam, you will frequently see the equation of a straight line written in the form:

\(y = mx + c\)

Each letter in this "recipe" tells us something specific about how the line looks:

The Gradient (\(m\))

The letter \(m\) represents the gradient, which is the steepness of the line.

  • If \(m\) is positive, the line goes up from left to right.
  • If \(m\) is negative, the line goes down from left to right.
  • The larger the value of \(m\), the steeper the line.

To calculate the gradient between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

\(m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}\)

The Y-Intercept (\(c\))

The letter \(c\) represents the \(y\)-intercept. This is the point where the line crosses the \(y\)-axis.

At this point, the value of \(x\) is always \(0\). So, the coordinates of the intercept are always \((0, c)\).

Quick Example: In the equation \(y = 2x + 3\), the gradient is \(2\) and the line crosses the \(y\)-axis at \(3\).

3. Drawing a Straight-Line Graph

Don't worry if drawing lines seems difficult; there is a simple step-by-step process you can follow using a Table of Values.

Step-by-Step Guide:

Let's draw the graph of \(y = 3x - 1\).

  1. Pick some \(x\) values: Usually, small numbers like \(-1, 0, 1, 2\) are easiest.
  2. Calculate \(y\): Plug each \(x\) into the equation.
    If \(x = 0\), then \(y = 3(0) - 1 = -1\).
    If \(x = 1\), then \(y = 3(1) - 1 = 2\).
    If \(x = 2\), then \(y = 3(2) - 1 = 5\).
  3. Write the coordinates: Your points are \((0, -1), (1, 2),\) and \((2, 5)\).
  4. Plot and Join: Mark these points on your grid and use a ruler to draw a straight line through them.

Key Takeaway: If your points don't form a perfectly straight line, go back and check your calculations! Straight-line equations (linear equations) must never curve.

4. Horizontal and Vertical Lines

Sometimes equations look a bit different. These are special cases you should recognize immediately:

  • Vertical Lines: Written as \(x = a\). For example, \(x = 4\) is a vertical line where every point has an \(x\)-coordinate of \(4\). It passes through \((4, 0), (4, 1), (4, -5)\), etc.
  • Horizontal Lines: Written as \(y = b\). For example, \(y = -2\) is a horizontal line where every point has a \(y\)-coordinate of \(-2\). It passes through \((0, -2), (5, -2), (-3, -2)\), etc.

5. Common Mistakes to Avoid

Even the best students can make these simple slips. Keep an eye out for them!

  • Swapping \(x\) and \(y\): Always check that you have moved across first, then up/down.
  • Negative Gradients: If the gradient is \(-2\), the line must go downhill. If your drawn line goes uphill, check your signs.
  • Subtraction errors: When using the gradient formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), be very careful with double negatives (e.g., \(5 - (-2) = 7\)).

Summary Checklist

Quick Review Box:

  • Can you identify the gradient \(m\) and intercept \(c\) from an equation?
  • Do you know that the \(y\)-intercept occurs when \(x = 0\)?
  • Can you plot a line by creating a table of values?
  • Do you remember that \(y = mx + c\) is the standard form for any straight line?

Note: To learn about finding gradients of curves using tangents or differentiation, please refer to the later chapters in the "Functions, graphs and calculus" section.