Edexcel A Level Further Maths: Core Pure:复习笔记4.1.1 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

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

• 
  • 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

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