Gradients and intercepts

Reading off m and c from y = mx + c is one of OCR J560's bread-and-butter algebra skills. Beyond mechanics, examiners expect interpretation — what the gradient and intercept mean in real contexts.

y = mx + c

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

Examples:

• y = 2x + 3: gradient 2, y-intercept (0, 3).
• y = −x + 5: gradient −1, y-intercept (0, 5).
• y = 4: gradient 0 (horizontal), y-intercept (0, 4).

Reading the gradient from a graph

Pick two clear lattice points on the line. Compute (change in y) / (change in x). A line going up left-to-right has positive gradient; down has negative.

Equations not in y = mx + c form

Rearrange first.

• 2y = 6x + 4 → divide by 2 → y = 3x + 2 → gradient 3, intercept 2.
• 2x + 3y = 12 → 3y = −2x + 12 → y = −2/3 x + 4 → gradient −2/3, intercept 4.

x-intercept

Set y = 0 and solve for x. For y = 2x − 6, x-intercept is at x = 3, so the point is (3, 0).

Interpreting in context (the OCR Higher hook)

If the line plots distance d (km) against time t (h):

• Gradient = speed in km/h.
• y-intercept = starting distance.

If the line plots cost £C against number of items n:

• Gradient = cost per item.
• y-intercept = fixed/setup charge.

OCR mark schemes routinely award B1 for "with units and meaning" — saying "gradient is 5" loses a mark when "the cost is £5 per ticket" was needed.

OCR mark scheme conventions

• M1 for rearranging into y = mx + c.
• B1 for stating gradient.
• B1 for stating intercept (as a value or as a coordinate).
• B1 for an interpretation in context (units + meaning) where required.