Linear Graphs
Understand the equation of a line in the form y = mx + b, and learn how to find the gradient and y-intercept from a graph or equation.
The Equation y = mx + b
Every straight-line graph can be written in the form y = mx + b, where m is the gradient (steepness) and b is the y-intercept (where the line crosses the y-axis).
y = mx + b
Gradient
(slope)
y-intercept
(constant)
Graph of y = 2x + 1
Understanding Gradient
The gradient (m) measures the steepness of a line and is calculated as:
m = rise / run = (y2 − y1) / (x2 − x1)
Positive Gradient
Line goes up left to right
Negative Gradient
Line goes down left to right
Zero Gradient
Horizontal line
Finding the Equation from a Graph
- 1 Identify the y-intercept (b): read the value where the line crosses the y-axis.
- 2 Choose two clear points on the line and calculate the gradient (m) = rise ÷ run.
- 3 Substitute m and b into y = mx + b to write the equation.
Key Vocabulary
Gradient (m)
The steepness of a line. Calculated as rise ÷ run. A larger m means a steeper line.
y-intercept (b)
The point where the line crosses the y-axis. Found by setting x = 0 in the equation.
x-intercept
The point where the line crosses the x-axis. Found by setting y = 0 and solving for x.
Linear Equation
An equation whose graph is a straight line. The highest power of x is 1.
Worked Examples
State the gradient and y-intercept of y = 3x − 5
Comparing to y = mx + b: m = 3, b = −5
Answer: Gradient = 3; y-intercept = −5
Find the gradient of the line passing through (1, 4) and (3, 10).
Step 1: Gradient = (y2 − y1) ÷ (x2 − x1)
Step 2: m = (10 − 4) ÷ (3 − 1) = 6 ÷ 2 = 3
Answer: m = 3
Sketch the graph of y = −2x + 4.
Step 1: y-intercept: b = 4 → plot point (0, 4)
Step 2: Gradient: m = −2 = −2/1 → from (0,4), go right 1, down 2 to (1, 2)
Step 3: x-intercept: set y = 0: 0 = −2x + 4, so x = 2 → plot (2, 0)
Answer: Plot (0, 4) and (2, 0), draw a line through them with negative slope.
Knowledge Check
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Key Concepts Summary
- ●The equation of a line is y = mx + b, where m is the gradient and b is the y-intercept.
- ●Gradient = rise ÷ run = (y2 − y1) ÷ (x2 − x1).
- ●Positive m: line goes up; Negative m: line goes down; m = 0: horizontal line.
- ●The y-intercept is found by setting x = 0; the x-intercept by setting y = 0.
- ●To sketch a line, plot the y-intercept and use the gradient to find a second point.