Gradient of Straight Lines
Measuring the steepness of a line
What is Gradient?
The gradient (also called slope) measures how steep a line is. It tells us how much the line rises or falls for each unit we move horizontally.
Gradient is represented by the letter m.
The Gradient Formula
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}\]
Where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line
Types of Gradients
Positive Gradient
Line goes UP from left to right
m > 0
Negative Gradient
Line goes DOWN from left to right
m < 0
Zero Gradient
Horizontal line
m = 0
Undefined Gradient
Vertical line
m = undefined
Calculating Gradient
Example 1: Using Two Points
Find the gradient of the line passing through A(2, 3) and B(6, 11).
\((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (6, 11)\)
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2\]
Answer: The gradient is 2. This means for every 1 unit moved right, the line rises 2 units.
Example 2: Negative Gradient
Find the gradient of the line through P(-1, 5) and Q(3, -3).
\[m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2\]
Answer: The gradient is -2. The negative sign indicates the line slopes downward from left to right.
Gradient from an Equation
Slope-Intercept Form
\[y = mx + c\]
m = gradient (coefficient of x)
c = y-intercept (where line crosses y-axis)
Example 3: Finding Gradient from Equation
(a) \(y = 3x + 5\) → Gradient = 3
(b) \(y = -2x + 1\) → Gradient = -2
(c) \(y = \frac{1}{2}x - 4\) → Gradient = \(\frac{1}{2}\)
(d) \(2y = 6x + 4\) → First rearrange: \(y = 3x + 2\) → Gradient = 3
Interactive Gradient Explorer
Calculate Gradient from Two Points
Calculation
\(m = \frac{6 - 2}{5 - 1} = \frac{4}{4} = 1\)
Gradient = 1
The line rises 1 unit for every 1 unit moved right.
Parallel and Perpendicular Lines
Key Rules
Parallel lines have the same gradient: \(m_1 = m_2\)
Perpendicular lines have gradients that multiply to -1: \(m_1 \times m_2 = -1\)
Or equivalently: \(m_2 = -\frac{1}{m_1}\) (negative reciprocal)
Example 4: Parallel and Perpendicular
Line L has equation \(y = 2x + 3\).
(a) Find the gradient of a line parallel to L.
(b) Find the gradient of a line perpendicular to L.
Practice Problems
Question 1: Find the gradient of the line through (3, 7) and (5, 13).
Show Solution
\(m = \frac{13 - 7}{5 - 3} = \frac{6}{2} = 3\)
Question 2: What is the gradient of the line \(3y = 9x - 6\)?
Show Solution
Rearrange to \(y = mx + c\) form:
\(y = 3x - 2\)
Gradient = 3
Question 3: Line A has gradient 4. What is the gradient of a line perpendicular to A?
Show Solution
Perpendicular gradient = \(-\frac{1}{4}\)
Check: \(4 \times (-\frac{1}{4}) = -1\) ✓
Question 4: The line through (2, k) and (5, 12) has gradient 2. Find k.
Show Solution
\(\frac{12 - k}{5 - 2} = 2\)
\(\frac{12 - k}{3} = 2\)
\(12 - k = 6\)
\(k = 6\)
Quick Reference
Gradient formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
From equation \(y = mx + c\): gradient = m
Parallel: \(m_1 = m_2\)
Perpendicular: \(m_1 \times m_2 = -1\)