Edexcel IGCSE Maths 复习笔记 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

• 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)

   

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

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)

• STEP 1 Draw a tangent to the curve at the required x-coordinate
• STEP 2 Turn the tangent into a right-angled 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”)