Edexcel IGCSE Maths

Revision Notes

3.9.1 Finding Gradients of Non-Linear Graphs

What is a non-linear graph?

  • A linear graph is a straight-line graph
  • These are easily identified as their equations can always be written in the form
    y = mx + c, where m is the gradient and c is the y-axis intercept

 

GoNL Notes fig1, downloadable IGCSE & GCSE Maths revision notes

  • All other graphs are non-linear – ie. curves
  • The equations of non-linear graphs take various forms
  • Here are a few you could plot quickly using graphing software
    • y = x2 – 4x + 3 (a quadratic graph – called a parabola)
    • y = sin x (a trigonometric graph)
    • y = x3 + 2x2 – 4 (a cubic graph)
    • y = 1 / x (a reciprocal graph)

 

GoNL Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What is a gradient?

  • Gradient means steepness
  • Another way of thinking about gradient is how y changes as x changes
  • On a graph this means how steep the graph is at a certain point on it
    • ie. how is y changing at a particular value of x
  • For a linear graph the gradient is constant – the value of x is irrelevant
  • For a non-linear graph, the gradient is dependent on the x-coordinate

 

GoNL Notes fig3, downloadable IGCSE & GCSE Maths revision notes

How do I find the gradient of a non-linear graph?

  • Using a copy of the graph it will only be possible to find an estimate of a gradient
  • Differentiation allows gradients to be found exactly for certain graphs
  • First, a tangent to the curve must be drawn
    • A tangent to a curve is a straight line that touches it at one point only
  • The gradient of a curve, at point (x , y) is equal to the gradient of the tangent at point (x , y)

 

GoNL Notes fig4, downloadable IGCSE & GCSE Maths revision notes

  • STEP 1 Draw a tangent to the curve at the required x-coordinate
  • STEP 2 Turn the tangent into a rightangled triangle
  • STEP 3 Measure/Read off (some estimating usually involved here) the rise and the run
  • STEP 4 The gradient is given by rise ÷ run
    (Alternatively this is “Change in y” ÷ “Change in x”)

GoNL Notes fig5eg, downloadable IGCSE & GCSE Maths revision notes

GoNL Notes fig6s1, downloadable IGCSE & GCSE Maths revision notes

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GoNL Notes fig9s4, downloadable IGCSE & GCSE Maths revision notes

Exam Tip

A sharp pencil helps – but not too sharp – pencil markings made with very sharp pencils are difficult for examiners to see once papers have been scanned into a computer.

Remember your answer is an estimate so can vary a fair amount from someone else’s attempt.

Make your working clear – your tangent, right-angled triangle and your rise/run values should all be clear in your working.

Worked Example

GoNL Example fig1 qu, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig2 sola1, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig3 sola2, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig4 sola3, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig5 sola4, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig6 solb1-3, downloadable IGCSE & GCSE Maths revision notes

GoNL Example fig7 solb4, downloadable IGCSE & GCSE Maths revision notes

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