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.

Verified step by step guidance

1. Recall the definitions of the hyperbolic functions: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\), \(\sinh x = \frac{e^{x} - e^{-x}}{2}\), and \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).
2. 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 \(\tanh x\), check if \(\tanh(-x) = -\tanh x\) to determine if it is odd.
3. 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 \(\tanh x\), it passes through the origin and has horizontal asymptotes.
4. Identify asymptotes: \(\cosh x\) and \(\sinh x\) do not have asymptotes as they grow exponentially; \(\tanh x\) has horizontal asymptotes at \(y = 1\) and \(y = -1\) because as \(x \to \infty\), \(\tanh x \to 1\) and as \(x \to -\infty\), \(\tanh x \to -1\).
5. 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 \(\tanh x\) (an 'S'-shaped curve bounded by horizontal asymptotes at \(y=\pm 1\)).

Key Concepts

Hyperbolic Functions

Hyperbolic functions include sinh x, cosh x, and tanh x, defined using exponential functions: sinh x = (e^x - e^{-x})/2, cosh x = (e^x + e^{-x})/2, and tanh x = sinh x / cosh x. They resemble trigonometric functions but relate to hyperbolas rather than circles.

Even and Odd Functions

A function f(x) is even if f(-x) = f(x) for all x, meaning its graph is symmetric about the y-axis. It is odd if f(-x) = -f(x), showing symmetry about the origin. Determining this helps understand the symmetry properties of the given hyperbolic functions.

Asymptotes and Graph Behavior

Asymptotes are lines that a graph approaches but never touches. For tanh x, horizontal asymptotes occur at y = ±1 as x approaches ±∞. Understanding asymptotes helps in accurately sketching the behavior of hyperbolic functions at extreme values.