0580

Further Graphs and Tangents

Coordinate Geometry and Graphs

Reading the answers off

  • The solutions of an equation are the x values where two graphs meet
  • Where the equation is set equal to zero, the answers are where the curve crosses the x-axis
  • Where it is set equal to a number, draw that horizontal line and read across
  • Where two curves are given, the meeting points solve them simultaneously, and there the y values are wanted too
A cubic curve plotted on a grid from x = −3 to 3, crossing the x-axis three times, with a peak near x = −1.3 and a trough near x = 0.8
Source: Solving equations using graphs by Save My Exams
  • Read to the accuracy the scale supports and give the x values only, unless the question asks to solve simultaneously

When the equation is not the one plotted

  • A question often asks you to solve an equation that is not the one drawn
  • Rearrange the new equation until one side is exactly the expression that was plotted
  • Whatever is left on the other side is the line you must draw on the grid
  • Then read off the x values where that line cuts the curve
The same cubic with a straight line drawn across it: the two graphs meet at three points, and dashed arrows drop from each meeting point to the x-axis to give the three solutions
Source: Solving equations using graphs by Save My Exams
Exam tip

Read values to within half of the smallest square, which is the accuracy the syllabus states, and give the x values only unless coordinates are asked for. These parts follow through from the curve you drew, so an earlier plotting slip does not cost the reading marks — the schemes use "their" wording for exactly this.

Worked example

Choosing the line to draw

The graph of y = x² − 3x + 1 is drawn on a grid. Explain how to use it to solve x² − 3x − 2 = 0.

Solution:

  • The plotted expression is x² − 3x + 1, so make that appear on one side of the new equation
  • Start from x² − 3x − 2 = 0
  • Add 3 to both sides: x² − 3x + 1 = 3
  • The left side is now exactly the plotted curve
  • The line to add to the grid is therefore y = 3
  • The solutions are the x values where that line meets the curve