Gradient-Intercept Form

The most common way to express a linear function is in gradient-intercept form (also called slope-intercept form):

y = mx + c

where m is the gradient (slope) and c is the y-intercept

The gradient (m) measures the steepness of the line. It tells us the rate of change: for every 1 unit increase in x, y changes by m units. A positive gradient slopes upward (left to right), while a negative gradient slopes downward.

The y-intercept (c) is the point where the line crosses the y-axis, occurring at coordinates (0, c).

Gradient Comparison

Positive gradient (m > 0)

Line rises left to right

Zero gradient (m = 0)

Horizontal line

Negative gradient (m < 0)

Line falls left to right

Parallel and Perpendicular Lines

Two important relationships between lines are defined by their gradients:

Parallel Lines

Two lines are parallel if and only if they have the same gradient.

m₁ = m₂

Example: y = 3x + 1 and y = 3x - 5 are parallel (both have m = 3).

Perpendicular Lines

Two lines are perpendicular if the product of their gradients is -1.

m₁ × m₂ = -1

Example: y = 2x + 1 and y = -½x + 3 are perpendicular (2 × -½ = -1).

The gradient formula for a line passing through two points (x₁, y₁) and (x₂, y₂) is:

m = y₂ - y₁x₂ - x₁

Finding the Equation of a Line

You can determine the equation of a line if you know:

The gradient and y-intercept — substitute directly into y = mx + c
The gradient and a point — use the point-gradient form: y - y₁ = m(x - x₁)
Two points — first calculate the gradient, then use the point-gradient form

Point-Gradient Form

y - y₁ = m(x - x₁)

This form is especially useful when you know the gradient and any point on the line, not necessarily the y-intercept.

Key Vocabulary

Gradient

The rate of change of y with respect to x; the steepness and direction of a line. Also called slope.

Y-Intercept

The point where the line crosses the y-axis, found by setting x = 0 in the equation.

Parallel

Lines that have the same gradient and never intersect. They remain equidistant apart.

Perpendicular

Lines that intersect at right angles (90°). Their gradients multiply to give -1.

Worked Examples

1

Find the gradient and y-intercept of y = -3x + 7.

Step 1: Compare with y = mx + c.

Step 2: The coefficient of x is the gradient: m = -3.

Step 3: The constant term is the y-intercept: c = 7.

Answer: Gradient = -3, y-intercept = 7. The line crosses the y-axis at (0, 7) and falls 3 units for every 1 unit to the right.

2

Find the equation of the line through (2, 5) and (6, 13).

Step 1: Calculate the gradient: m = (13 - 5) / (6 - 2) = 8/4 = 2.

Step 2: Use point-gradient form with (2, 5): y - 5 = 2(x - 2).

Step 3: Expand: y - 5 = 2x - 4, so y = 2x + 1.

Answer: y = 2x + 1.

3

Find the equation of the line perpendicular to y = 4x - 1 that passes through (8, 3).

Step 1: The gradient of y = 4x - 1 is m₁ = 4.

Step 2: The perpendicular gradient is m₂ = -1/4 (since 4 × -1/4 = -1).

Step 3: Use point-gradient form: y - 3 = -¼(x - 8).

Step 4: Expand: y - 3 = -¼x + 2, so y = -¼x + 5.

Answer: y = -¼x + 5.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is the gradient of the line y = -5x + 2?

-5 5 2 -2

Question 2

What is the gradient of a line perpendicular to y = 3x + 4?

3 -1/3 -3 1/3

Question 3

What is the gradient of the line passing through (1, 4) and (3, 10)?

2 6 3 7

Question 4

Which equation represents a line parallel to y = 2x - 3?

y = -2x + 1 y = -½x + 4 y = ½x - 3 y = 2x + 7

Question 5

A line passes through (0, -2) with gradient 4. What is its equation?

y = 4x - 2 y = -2x + 4 y = 4x + 2 y = -4x - 2

Linear Functions