The hyperbolic cosine function, denoted $\(\cosh\]\left\)(x\(\right\))$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}$.


b. Evaluate $\cosh \left(0\right)$. Use symmetry and part (a) to sketch a plausible graph for $y=\cosh \left(x\right)$.

Verified step by step guidance

1. To evaluate \( \cosh(0) \), substitute \( x = 0 \) into the definition of the hyperbolic cosine function: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
2. Calculate \( e^0 \) and \( e^{-0} \). Since any number to the power of 0 is 1, both \( e^0 \) and \( e^{-0} \) equal 1.
3. Substitute these values into the expression: \( \cosh(0) = \frac{1 + 1}{2} \).
4. To sketch the graph of \( y = \cosh(x) \), note that \( \cosh(x) \) is an even function, meaning it is symmetric about the y-axis. This is because \( \cosh(-x) = \cosh(x) \).
5. The graph of \( y = \cosh(x) \) resembles a U-shape, similar to a parabola, but it is not a parabola. It has a minimum value at \( x = 0 \) and increases exponentially as \( x \) moves away from zero in both directions.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as hyperbolic cosine (cosh), are analogs of trigonometric functions but for a hyperbola instead of a circle. They are defined using exponential functions, with cosh(x) = (e^x + e^(-x))/2. These functions are useful in various applications, including modeling shapes like hanging cables and in solving certain differential equations.

Symmetry in Functions

Symmetry in functions refers to the property where a function exhibits a certain balance around a point or axis. For example, the hyperbolic cosine function is even, meaning cosh(-x) = cosh(x). This symmetry can be used to simplify calculations and sketch graphs, as it indicates that the graph will mirror itself across the y-axis.

Graphing Techniques

Graphing techniques involve methods to visually represent mathematical functions. For the hyperbolic cosine function, understanding its key points, such as cosh(0) = 1, and its symmetry helps in sketching its graph accurately. Recognizing the shape of the graph, which resembles a parabola opening upwards, is essential for visualizing the behavior of the function across different values of x.