The gradient of the line equals m and the Y-intercept equals c in Y = mX + c. Your job is to read those two numbers, then use what m and c stand for to find the constants in the original relation.
Earlier lessons in this linear law chapter covered reducing a relation and choosing the axes. This lesson covers reading the result.
How do I get m and c from the line?
Pick two clear points on the drawn line, far apart, and read their coordinates. Calculate the gradient m = (Y₂ − Y₁) ÷ (X₂ − X₁). Then read the intercept c where the line meets the vertical axis, or solve for it by substituting one point.
Worked example: simple constants
A graph of y/x against x is a straight line through (2, 7) and (8, 25). The relation is y = hx² + kx. Find h and k.
The linear form is y/x = hx + k, so Y = y/x, X = x, m = h and c = k.
Gradient: m = (25 − 7) ÷ (8 − 2) = 18 ÷ 6 = 3.
Substitute (2, 7) into Y = 3X + c: 7 = 6 + c, so c = 1.
So h = 3 and k = 1, and the relation is y = 3x² + x. Check: when x = 2, y = 12 + 2 = 14 and y/x = 7, which matches the point (2, 7).
Worked example: constants inside logarithms
The relation is y = p·q^x, and the graph of lg y against x passes through (0, 0.48) and (5, 1.98).
From the lesson on reducing relations, lg y = (lg q)x + lg p. So m = lg q and c = lg p.
The line meets the vertical axis at 0.48, so lg p = 0.48 and p = 10^0.48 = 3.02.
Gradient: (1.98 − 0.48) ÷ (5 − 0) = 0.30, so lg q = 0.30 and q = 10^0.30 = 2.00 (3 s.f.). The relation is approximately y = 3.02 × 2.00^x.
The mistake that costs marks
A common slip is writing q = 0.30 because that is the gradient. The gradient of this line is lg q, so q needs one more step: 10 raised to the gradient.
A second slip is reading c from a graph whose horizontal axis starts at 4, not 0. The line has not yet reached the vertical axis, so any “intercept” you read is wrong.
Check yourself
The graph of xy against x² is a straight line through (1, 6) and (4, 15). The relation is y = ax + b/x. Find a and b.
Answer
Multiply by x: xy = ax² + b. So Y = xy, X = x², m = a, c = b.
Gradient: (15 − 6) ÷ (4 − 1) = 3, so a = 3.
Substitute (1, 6): 6 = 3 + c, so c = 3 and b = 3.
Check: when x = 2, y = 6 + 1.5 = 7.5, so xy = 15. On the line, X = 4 gives Y = 15. They match.
What to study next
Next, look at what the constants mean and how to use the line to predict. Continue with interpreting a linearised model in its original variables. The word-problem structure worksheet helps break long question stems into steps.
If you would like a teacher to rehearse these final steps with you, see online one-to-one Additional Mathematics tuition.