Solving Equations using Graphs (AQA GCSE Further Maths): Revision Note

Solving Equations using Graphs

How do we use graphs to solve equations?

• Solutions are always read off the x-axis

• Solutions of f(x) = 0 are where the graph of y = f(x) crosses the x-axis

• If asked to use the graph of y = f(x) to solve a different equation (the question will say something like “by drawing a suitable straight line”) then:

  • Rearrange the equation to be solved into f(x) = mx + c and draw the line y = mx + c

  • Solutions are the x-coordinates of where the line (y = mx + c) crosses the curve (y = f(x))

  • E.g. if given the curve for y = x³ + 2x² + 1 and asked to solve x³ + 2x²  − x − 1 = 0, then;

    1. rearrange x³ + 2x²  − x − 1 = 0 to x³ + 2x² + 1 = x + 2

    2. draw the line y = x + 2 on the curve y = x³ + 2x² + 1

    3. read the x-values of where the line and the curve cross (in this case there would be 3 solutions, approximately x = -2.2, x = -0.6 and x = 0.8);

• Note that solutions may also be called roots