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”)