MathematicsExam code: 0580

Further Graphs and Tangents

Coordinate Geometry and Graphs

PractiseFlashcards

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 within half of the smallest square, the accuracy the syllabus sets, 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

Write down the line before you draw it

When you are told to solve an equation by drawing a line on the graph, the line itself earns marks. Write its equation in your working, because that alone often earns one, then rule it right across the grid so it meets the curve at every solution. The answers depend on that line: correct values read off with no correct line drawn usually earn just one special-case mark.

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
The curve y = x² − 3x + 1 on a grid from x = −1 to 4 with the ruled line y = 3 across it; the two meeting points are marked with crosses and dashed lines drop from them to the x-axis at about −0.6 and 3.6

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