Welcome to Linear Graphs and Coordinates!

Linear graphs are the foundation of so much in our world. From architects designing buildings to scientists tracking the speed of a car, straight lines help us understand how one thing changes in relation to another. In this chapter, we will learn how to read these graphs, draw them, and use algebra to describe them perfectly.

Prerequisite Check: Before we start, remember that a linear graph is simply a straight line. If your graph has a curve, it isn't linear!

1. Coordinates in Four Quadrants

To draw or read a graph, we need a "map." In mathematics, this map is called the coordinate plane. It is made of two axes: the horizontal \(x\)-axis and the vertical \(y\)-axis.

Coordinates are written as a pair: \((x, y)\).

  • The first number (\(x\)) tells you how far to move left or right.
  • The second number (\(y\)) tells you how far to move up or down.

Memory Aid: A famous way to remember this is: "Along the corridor (horizontal) and up the stairs (vertical)."

The Four Quadrants

Because axes include negative numbers, we have four areas (quadrants):

  • 1st Quadrant: Both numbers are positive, e.g., \((3, 5)\).
  • 2nd Quadrant: \(x\) is negative, \(y\) is positive, e.g., \((-2, 4)\).
  • 3rd Quadrant: Both numbers are negative, e.g., \((-1, -6)\).
  • 4th Quadrant: \(x\) is positive, \(y\) is negative, e.g., \((5, -3)\).

Quick Review: The point where the axes cross is called the origin. Its coordinates are always \((0, 0)\).

2. The Equation of a Straight Line: \(y = mx + c\)

Every straight line can be written as an equation. The most common form is \(y = mx + c\). Each letter (other than \(x\) and \(y\)) represents a specific feature of the line:

The Gradient (\(m\))

The letter \(m\) stands for the gradient, which is a measure of how steep the line is.

  • 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.

The \(y\)-intercept (\(c\))

The letter \(c\) is the \(y\)-intercept. This is the point where the line crosses the vertical \(y\)-axis. At this point, the value of \(x\) is always \(0\).

Example: In the equation \(y = 2x + 3\):
The gradient is \(2\) (for every \(1\) unit you move right, you move \(2\) units up).
The \(y\)-intercept is \(3\) (the line crosses the vertical axis at the point \((0, 3)\)).

3. Calculating the Gradient

If you are given two points on a line, say \((x_1, y_1)\) and \((x_2, y_2)\), you can calculate the gradient using this formula:

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

Step-by-Step Example:

Find the gradient of the line passing through \((1, 2)\) and \((3, 10)\).

1. Label your points: \(x_1 = 1, y_1 = 2\) and \(x_2 = 3, y_2 = 10\).
2. Subtract the \(y\) values: \(10 - 2 = 8\).
3. Subtract the \(x\) values: \(3 - 1 = 2\).
4. Divide: \(m = \frac{8}{2} = 4\).
The gradient is \(4\).

Did you know? In real-life contexts, the gradient represents a rate of change. For example, on a distance-time graph, the gradient is the speed!

4. Parallel and Perpendicular Lines

Parallel Lines

Lines that never meet are parallel. For two lines to be parallel, they must have the exact same gradient.

Example: \(y = 3x + 5\) and \(y = 3x - 2\) are parallel because they both have a gradient of \(3\).

Perpendicular Lines (Higher Tier Only)

(H) Lines that meet at a right angle (\(90^{\circ}\)) are perpendicular. If two lines are perpendicular, the product of their gradients is \(-1\).

(H) If the gradient of one line is \(m\), the gradient of the perpendicular line is \(-\frac{1}{m}\) (the negative reciprocal).

Example: If line A has a gradient of \(4\), a line perpendicular to it will have a gradient of \(-\frac{1}{4}\).

5. Finding the Equation of a Line

Sometimes you are given the gradient and a point, or just two points, and asked to find the full equation.

Method 1: Given a Gradient (\(m\)) and a Point \((x, y)\)

1. Start with \(y = mx + c\).
2. Substitute the gradient (\(m\)) and the coordinates of the point into the equation.
3. Solve for \(c\).
4. Rewrite the equation with your values for \(m\) and \(c\).

Method 2: Given Two Points

1. Calculate the gradient (\(m\)) using the formula \(\frac{y_2 - y_1}{x_2 - x_1}\).
2. Follow the steps in Method 1 to find \(c\).

Key Takeaway: Always aim to get your final answer into the format \(y = mx + c\).

6. Common Pitfalls to Avoid

  • Swapping \(x\) and \(y\): Always double-check that you put the \(x\) value first in a coordinate and the \(y\) value on top in a gradient fraction.
  • Signs in Gradients: Be very careful with negative numbers. Subtracting a negative is the same as adding! \(10 - (-2) = 12\).
  • Intercept Confusion: The \(y\)-intercept is where the line crosses the vertical axis, not the horizontal one.

7. Summary Table

Variable: \(x\)
Role: The independent variable (horizontal position).

Variable: \(y\)
Role: The dependent variable (vertical position).

Variable: \(m\)
Role: Gradient (Steepness). Calculated as \(\frac{\text{Rise}}{\text{Run}}\).

Variable: \(c\)
Role: \(y\)-intercept. Where the line hits the \(y\)-axis.

Don't worry if this seems tricky at first! Like any map-reading, it takes a bit of practice to get used to the directions. Once you master the \(y = mx + c\) formula, you have the key to solving almost any linear graph problem.