This module covers two linked ideas. The first is coordinate geometry of straight lines: writing an equation from intercepts, using gradients, and recognising perpendicular lines. The second is linearisation: rewriting a curved relationship such as y = ax² + b so that a graph of two chosen quantities becomes a straight line, then reading the constants from its gradient and intercept.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content, given formulae and calculator rules in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to find the gradient between two points, rearrange an equation to make y the subject, and substitute values into a formula. If solving for a constant is slow, revisit simultaneous linear and nonlinear models first, because finding two constants from two pieces of information uses the same habit.
An orienting example
Variables x and y are connected by y = px² + q. A graph of y against x² is a straight line through (1, 8) and (4, 17). Find p and q, then find x when y = 53.
Step 1, name the straight line: compare y = px² + q with Y = mX + c. Here Y = y, X = x², m = p and c = q.
Step 2, gradient: m = (17 − 8) ÷ (4 − 1) = 9 ÷ 3 = 3. So p = 3.
Step 3, intercept: substitute (1, 8) into Y = 3X + c: 8 = 3 + c, so c = 5. So q = 5.
Step 4, use the model: y = 3x² + 5. When y = 53, 3x² = 48, so x² = 16 and x = 4 (taking the positive value, as the graph used positive x).
Check: 3(16) + 5 = 53. The point (4, 17) on the graph has X = 4, so x = 2, and the model gives 3(4) + 5 = 17.
The whole question depended on one decision: what to put on each axis.
In which order should you study it?
- Find a line from intercept information: builds an equation from where a line meets the axes.
- Determine a perpendicular relationship: uses the gradient product −1, which finishes the pure line skills.
- Transform data into a straight-line form: chooses the quantities to plot so a curve becomes a line.
- Recover model constants from gradient and intercept: turns a graph back into the original formula.
- Assess the limits of an extrapolated linear model: asks when a fitted line should not be trusted.
Then work through the mixed practice set. The lessons do not have to be taken strictly in this order, but the line skills make the linearisation lessons much lighter.
Which traps catch most students here?
- Mixing up the two intercepts, so a point on the x-axis is written with its coordinates the wrong way round.
- Subtracting coordinates in different orders when finding a gradient, which flips the sign.
- Plotting the original y and x for a curved relationship and expecting a line.
- Reading the gradient of the new graph as the constant of the original formula without checking what was plotted.
- Taking only the reciprocal, or only the negative, when finding a perpendicular gradient.
- Trusting a fitted line far outside the data without asking whether the situation still makes sense.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper first and write the line Y = mX + c explicitly before you read off m and c. Then substitute a second data point to check your constants. The non-calculator working trainer helps you keep fraction gradients exact, and the mistake log and retest queue gives you a place to record which slip you made so you can retest it later.