Linear graphs and the form y = mx + c

Every straight-line graph can be written y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). Master this form and you can sketch, analyse, and find equations of straight lines fluently.

Reading a line from its equation

In y = 3x - 4:

• gradient m = 3 (line rises 3 for every 1 right)
• y-intercept c = -4 (line crosses the y-axis at (0, -4))

To plot, mark (0, -4), then go right 1 and up 3 to (1, -1), repeat to reach more points. Join with a ruler.

Finding the equation from a graph

1. Read c directly: where does the line meet the y-axis?
2. Pick two clear lattice points and compute m = (y₂ − y₁)/(x₂ − x₁).
3. Write y = mx + c.

If a line passes through (0, 5) and (4, 13): c = 5, m = (13 − 5)/(4 − 0) = 2, so y = 2x + 5.

Equation given a point and gradient

Use point-gradient form, then rearrange to y = mx + c: y - y₁ = m(x - x₁).

Example: line through (3, 1) with gradient 2. y - 1 = 2(x - 3) ⇒ y = 2x - 5.

Equation through two points

Compute m, then use one point. From (1, 2) and (4, 11): m = 9/3 = 3. Use (1, 2): y - 2 = 3(x - 1) ⇒ y = 3x - 1.

Parallel and perpendicular lines

• Parallel lines have the same gradient: y = 3x + 7 and y = 3x − 5 are parallel.
• Perpendicular gradients multiply to −1: if one line has gradient 2, a perpendicular line has gradient −1/2. Symbolically, m₁ × m₂ = −1 so m₂ = −1/m₁ (the negative reciprocal).

Worked example: find the equation of the line through (4, 1) perpendicular to y = 2x + 7. The given gradient is 2. Perpendicular gradient = −1/2. Use point-gradient: y − 1 = −½(x − 4) ⇒ y = −½x + 3.

Special cases

• Horizontal line: y = c (gradient 0).
• Vertical line: x = a (gradient undefined; not in the form y = mx + c).