Question

Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.

Accepted Answer

Recall the definitions of the hyperbolic functions: $$\cosh x = \frac{e$$^{x}$$ + e$$^{-x}$$}{2}$$, $$\sinh x = \frac{e$$^{x}$$ - e$$^{-x}$$}{2}$$, and $$	anh x = \frac{\sinh x}{\cosh x} = \frac{e$$^{x}$$ - e$$^{-x}$$}{e$$^{x}$$ + e$$^{-x}$$}. $$Determine the symmetry of each function by checking $$f(-x)$$: For $$\cosh x$$, compute $$\cosh(-x)$$ and compare it to $$\cosh x to $$see if it is even; for $$\sinh x$$, compute $$\sinh(-x)$$ and compare it to $$-\sinh x to $$check if it is odd; for $$	anh x$$, check if $$	anh(-x) = -	anh x to $$determine if it is odd. Analyze the behavior and key points of each function: For $$\cosh x$$, note it has a minimum at $$x=0$$ with $$\cosh 0 = 1$$; for $$\sinh x$$, it passes through the origin with $$\sinh 0 = 0$$; for $$	anh x$$, it passes through the origin and has horizontal asymptotes. Identify asymptotes: $$\cosh x$$ and $$\sinh x do $$not have asymptotes as they grow exponentially; $$	anh x$$ has horizontal asymptotes at $$y = 1$$ and $$y = -1$$ because as $$x 	o \infty$$, $$	anh x 	o 1$$ and as $$x 	o -\infty$$, $$	anh x 	o -1. $$Sketch each graph using the above information: plot key points and symmetry, draw the shape of $$\cosh x (a$$ 'U'-shaped curve), $$\sinh x (an$$ 'S'-shaped curve through the origin), and $$	anh x (an$$ 'S'-shaped curve bounded by horizontal asymptotes at $$y=\pm 1).$$

Sketch the graphs of y = cosh x, y = sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.

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Key Concepts

Hyperbolic Functions

Even and Odd Functions

Asymptotes and Graph Behavior