Linear graphs and y = mx + c

Linear graphs underpin half of OCR J560 algebra. Every paper has at least one question on plotting, gradient, intercepts, or parallel/perpendicular lines.

The form y = mx + c

Every non-vertical straight line can be written as y = mx + c where:

• m = gradient (slope) = how much y changes per unit increase in x.
• c = y-intercept = y-value when x = 0.

Plotting a linear graph

Method 1 — table of values:

• Pick three or more x values (include negatives and zero).
• Compute y = mx + c for each.
• Plot the points and join with a straight line, extended.

Method 2 — gradient and intercept:

• Mark (0, c) on the y-axis.
• From there, go right 1 and up m (or down |m| if m < 0).
• Plot a second point and draw the line.

Calculating gradient from two points

Given (x₁, y₁) and (x₂, y₂):

m = (y₂ − y₁) / (x₂ − x₁)

"Rise over run."

Example: from (1, 3) to (4, 9). m = (9 − 3)/(4 − 1) = 6/3 = 2.

Special lines

Equation	Type
y = c	horizontal (gradient 0)
x = a	vertical (undefined gradient)
y = x	gradient 1, through origin
y = −x	gradient −1, through origin

Parallel lines

Two lines are parallel ⟺ they have the same gradient.

If y = 3x + 2 is parallel to y = mx + c, then m = 3 (any c).

Perpendicular lines (Higher)

Two lines are perpendicular ⟺ the product of their gradients is −1.

If a line has gradient m, the perpendicular has gradient −1/m (the negative reciprocal).

Example: perpendicular to y = 2x + 1 has gradient −1/2.

Finding the equation of a line

Given gradient m and a point (x₀, y₀):

• y − y₀ = m(x − x₀), then rearrange to y = mx + c.

Given two points: find m first, then use either point.

OCR mark scheme conventions

• B1 for the gradient.
• B1 for the y-intercept.
• A1 for the equation in the requested form.
• For graph plots: B1 for accurate plotting (within tolerance), B1 for ruled line through plotted points.