Edexcel A Level Further Maths: Core Pure:复习笔记4.1.1 Hyperbolic Functions & Graphs

Hyperbolic Functions & Graphs

What are the definitions of the hyperbolic functions?

  • Hyperbolic sine

    • This can be pronounced "shine" or "sinch"
  • Hyperbolic cosine

    • This can be pronounced "cosh"
  • Hyperbolic tangent

    • This can be pronounced "than" or "tanch"

What are the graphs of the hyperbolic functions and their key features?

    • Domain:
    • Range:
    • Non-stationary point of inflection at (0, 0)
    • Its shape is similar to the graph of

edexcel-al-fm-cp-4-1-1-sinhx-graph

    • Domain:
    • Range:
    • Global minimum point at (0, 1)
    • Its shape is similar to the graph of

edexcel-al-fm-cp-4-1-1-coshx-graph

    • Domain:
    • Range:
    • Non-stationary point of inflection at (0, 0)
    • Asymptotes at y=1 and y=-1
    • Its shape is similar to the graph of

edexcel-al-fm-cp-4-1-1-tanhx-graph

What other features of the hyperbolic functions and graphs do I need to know?

  • The graphs of y=sinhx and y=tanhx have rotational symmetry about the origin
    • This means that
    • and  are therefore odd functions
  • The graph of y=coshx is symmetrical in the y-axis
    • This means that
    • is therefore an even function

What may I be asked to do with hyperbolic functions and their graphs?

  • Sketch graphs and transformations
    • e.g.
      • Write as a transformation of and apply the transformations in the correct order
    • Where possible label the key features of the transformed graph
      • Intersections with the coordinate axes
      • Equations of any asymptotes
      • Coordinates of any turning points
  • Find exact values
    • e.g. Find the exact value of
      • Use the definitions to write in terms of e
      • Use

Exam Tip

  • When using a calculator make sure you use sinh, cosh and tanh and NOT sin, cos and tan
  • Questions asking for values in exact form are often easier “to see” without a calculator, using the definitions of sinh and cosh, rather than trying to type in a complicated expression with e and ln

Worked Example

a) Find the exact values of
(i)
(ii)

al-fm-4-1-1-we-solution-a

 

b) Sketch the graph of , labelling any points where the graph crosses the coordinate axes and any turning points.

al-fm-4-1-1-we-solution-b

 

 

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