IGCSE Maths / Graphs and transformations
Finding the line to draw revision notes
Edexcel International GCSE Maths (4MA1), chapter 26: Graphs and transformations, specification point 3.3e. Free lesson.
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To solve an equation using a printed curve, rearrange it until the curve’s exact rule is on the left. Draw the remaining right-hand side as a straight line, then read the x-values where the line and curve cross.
Match the printed curve exactly
The curve is already drawn, so its rule must stay unchanged. Rearrange the equation until its left side matches the printed curve exactly. Whatever remains on the right gives the line to draw. Draw only the line, not a new version of the curve. The line can be horizontal or sloping.
Find a horizontal line
For the curve y = x² + 2x − 3, rearrange x² + 2x − 8 = 0 until the curve’s rule is on the left. The right-hand side is 5, so draw y = 5, a horizontal line five units up. The line meets the curve at (−4, 5) and (2, 5). The solutions are the x-values, −4 and 2. The height 5 is not an answer.
Find a sloping line
Sometimes the right-hand side still contains x. It then gives a sloping straight line. For the cubic y = x³ − 3x, split −4x into −3x − x so the printed curve’s rule can be collected on the left. Rearranging x³ − 4x + 1 = 0 gives y = x − 1. Draw the straight line y = x − 1, not a new cubic.
Read the crossings and avoid redrawing
Each crossing of the line and the printed curve gives a solution. Read the x-coordinate of each crossing. The printed curve never changes, even when the equation you are solving looks like a different curve. If your rearranged line still contains x² or x³, you have not matched the printed curve yet. For the curve y = x² − 5x and equation x² − 5x + 3 = 0, the line is y = −3.