Hyperbolic Functions & Graphs

$\sin h x = \frac{1}{2} (e^{x} - e^{- x})$

$\cosh x = \frac{1}{2} (e^{x} + e^{- x})$

$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{2 x} - 1}{e^{2 x} + 1}$

$y = \sinh x$
- Domain: $x \in ℝ$
- Range: $y \in ℝ$
- Non-stationary point of inflection at (0, 0)
- Its shape is similar to the graph of $y = x^{3}$

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

$y = \cosh x$
- Domain: $x \in ℝ$
- Range: $y \geq 1$
- Global minimum point at (0, 1)
- Its shape is similar to the graph of $y = x^{2}$

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

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

The graphs of y=sinhx and y=tanhx have rotational symmetry about the origin
- This means that
  - $\sinh (- x) = - \sinh x$
  - $\tanh (- x) = - \tanh x$
- $\sinh x$ and are therefore odd functions
The graph of y=coshx is symmetrical in the y-axis
- This means that
  - $\cosh (- x) = \cosh x$
- $\cosh x$ is therefore an even function

Sketch graphs and transformations
- e.g. $y = \sinh (2 x) - 4$
  - Write as a transformation of $y = \sinh x$ 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 $\sinh (\ln (5))$
  - Use the definitions to write in terms of e
  - Use $e^{\ln k} = k$ and $e^{- \ln k} = \frac{1}{k}$

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

Find the exact values of

(i)

2 \cosh (\ln (8)) - 3 \sinh (2 \ln (2))

(ii)

3 - \tanh (2 \ln (3)) + \tanh (- 2 \ln (3))

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

Sketch the graph of

y = 3 \cosh (\frac{1}{4} x) - 1

, labelling any points where the graph crosses the coordinate axes and any turning points.

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

Hyperbolic Functions & Graphs (Edexcel A Level Further Maths: Core Pure)