Mathematics (0580) | Gradient of linear graphs

Welcome to the chapter on the Gradient of Linear Graphs! Don't worry if the term "gradient" sounds complicated—it’s just a fancy word for "steepness."

Whenever you look at a straight line on a graph, it’s either flat, going up, or going down. The gradient is the number that tells us exactly how steep that line is and in which direction it is sloping.

This skill is vital for understanding linear equations (\(y = mx + c\)) and for solving problems related to speed, distance, and rates of change. Let's make sure you master it!

1. Understanding the Concept of Gradient

The gradient is a measure of the vertical change compared to the horizontal change between any two points on a straight line.

Think of it like walking up a hill:

• If the hill is steep, the gradient is a large number.
• If the hill is nearly flat, the gradient is close to zero.
• If you are walking downhill, the gradient is negative.

Key Definition: The Steepness Ratio

The gradient, usually represented by the letter \(m\), is defined as:

\[\text{Gradient } (m) = \frac{\text{Vertical Change}}{\text{Horizontal Change}}\]

Memory Aid (Mnemonic):

The easiest way to remember this is:
\(m\) = Rise over Run

\[m = \frac{\text{Rise}}{\text{Run}}\]

2. Finding the Gradient from a Grid (The "Rise over Run" Method)

If you are given a straight line drawn on a coordinate grid, you can find the gradient by counting the squares. (This method is particularly important for Core syllabus students, as C4.2 specifies finding the gradient "From a grid only").

Step-by-Step Guide:

1. Pick Two Clear Points: Choose any two points on the line where the coordinates are easy to read (they should be on grid intersections). Let’s call them Point A and Point B.
2. Calculate the Run (Horizontal Change): Move horizontally (left or right) from Point A until you are directly beneath (or above) Point B. Count how many units you moved.
3. Calculate the Rise (Vertical Change): Move vertically (up or down) from your new position until you reach Point B. Count how many units you moved.
4. Calculate the Gradient: Divide the Rise by the Run.

Important Rule for Signs:

• If the line goes up (from left to right), the Rise is Positive (\(+\)).
• If the line goes down (from left to right), the Rise is Negative (-\()).
• The Run (horizontal movement) is usually counted as Positive (\(+\)) by moving from left to right.

Key Takeaway from Graph Method

The gradient is always a ratio. For every unit you "run" horizontally, you "rise" vertically by the amount of the gradient, \(m\).

3. Calculating the Gradient using the Formula

For more advanced questions (or when a grid isn't provided, as required in the Extended syllabus E4.2), we must use the official gradient formula based on coordinates.

Suppose you have two points on the line:
Point 1: \((x_1, y_1)\)
Point 2: \((x_2, y_2)\)

The Gradient Formula

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

This formula just translates "Rise over Run" into coordinate language: the difference in the y-coordinates is the Rise, and the difference in the x-coordinates is the Run.

Step-by-Step Guide: Using Coordinates

Problem: Find the gradient of the line segment joining \(A(3, 8)\) and \(B(7, 2)\).

1. Label Your Points:

   Let \(A = (x_1, y_1) = (3, 8)\)
   Let \(B = (x_2, y_2) = (7, 2)\)

2. Apply the Formula:

   \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 8}{7 - 3}\]

3. Calculate the Result:

   \[m = \frac{-6}{4}\]

4. Simplify the Fraction:

   \[m = -\frac{3}{2} \text{ or } -1.5\]

Note: Since the gradient is negative (-1.5), the line slopes downwards from left to right.

4. Interpreting Different Types of Gradient

The sign and size of the gradient tell us instantly about the line's direction and steepness.

1. Positive Gradient (\(m > 0\))

   The line goes uphill (increases) from left to right. The larger the number, the steeper the line. Example: \(m = 5\) is much steeper than \(m = 1/2\).

2. Negative Gradient (\(m < 0\))

   The line goes downhill (decreases) from left to right. Example: \(m = -3\) is steeper going down than \(m = -1\).

3. Zero Gradient (\(m = 0\))

   This is a perfectly horizontal line. The Rise is zero (e.g., \(\frac{0}{5}\)). The equation of such a line is always in the form \(y = c\).

4. Undefined Gradient

   This is a perfectly vertical line. The Run is zero (e.g., \(\frac{5}{0}\)). Since division by zero is mathematically undefined, the gradient is undefined. The equation of such a line is always in the form \(x = k\).

5. Gradient and Parallel Lines (C4.5 and E4.5)

Gradients are especially useful when determining the relationship between two lines.

The Rule for Parallel Lines

If two straight lines are parallel, they have the same gradient.

This makes sense! If two ski slopes are parallel, they must have exactly the same steepness.

If Line A has gradient \(m_A\) and Line B has gradient \(m_B\), then if Line A is parallel to Line B:

\[m_A = m_B\]

Example of Parallel Lines

Line 1 has the equation \(y = 4x - 1\). The gradient is \(m_1 = 4\).
If Line 2 is parallel to Line 1, then the gradient of Line 2 must also be \(m_2 = 4\).

(Syllabus Tip: Questions often ask you to find the equation of a line parallel to a given line that passes through a specific point. You use the parallel gradient and the given point to solve for the y-intercept, \(c\).)

Summary and Quick Review

You have successfully learned how to quantify the steepness of any linear graph!

Keep practising these calculations—the more comfortable you are with the formula and the "Rise over Run" concept, the easier the rest of Coordinate Geometry will be!