IGCSE Maths / Graphs and transformations
Solving equations graphically
Edexcel International GCSE Maths (4MA1), chapter 26: Graphs and transformations, specification point 3.3e. Free lesson.
Read the revision notes · IGCSE Maths past papers
Solve an equation from a graph by finding where the curve crosses the x-axis or meets a horizontal line. The solutions are the x-values at the crossings, not their heights.
When the equation equals zero
Solving an equation that equals zero means finding where y = 0. The x-axis is the line y = 0, so the solutions are the x-values where the curve crosses it. For y = x² − 2x − 3, the curve crosses the x-axis at x = −1 and x = 3. These solutions are also called the roots of the equation. Read the x-values at the crossings. The y-values are both zero, so they are not the answers.
When the equation equals a number
If the equation equals a number other than zero, draw a second graph: a horizontal line at that height. At each crossing, the curve and line have the same y-value, so the equation is satisfied. The solutions are the x-values of the crossings. For x² − 4x + 7 = 4, draw the curve y = x² − 4x + 7 and the line y = 4. They cross at x = 1 and x = 3. These are the solutions. The height of both crossings is 4, but 4 is not a solution.
Why the crossings give the solutions
At a crossing, the curve and the line have equal heights. Subtracting the line's y-value from the curve's y-value moves the crossings down to the x-axis, but their x-positions stay the same. For y = x² − 4x + 7 and y = 4, set the rules equal at a crossing: x² − 4x + 7 = 4. Subtract 4 from both sides to get x² − 4x + 3 = 0. Factorise: (x − 1)(x − 3) = 0, so x = 1 or x = 3. These are the same x-values as the graph's crossings.
What to read from the graph
The solutions are x-values, which show positions along the bottom axis. The y-value shows the height, not the position. If there are two crossings, include both answers. If the equation equals zero, use the x-axis as the line to find crossings. If it equals another number, draw that horizontal line first. Check that you have written down every crossing's x-value.