Question

Sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph. \(x=\cosh t, \quad y=\sinh t, \quad-2 \leq t \leq 2\)

Short answer

The graph is one branch of a hyperbola, oriented from \( (1,0) \) outwards as \( t \) increases.

Step-by-step solution

Understand the Parametric Equations

The given parametric equations are expressed in terms of the hyperbolic functions: \( x = \cosh t \) and \( y = \sinh t \). These are similar to the expressions for a hyperbola in Cartesian form. The hyperbolic cosine \( \cosh t \) and hyperbolic sine \( \sinh t \) are defined for real numbers and create a hyperbolic shape.

Identify Key Points

Since the parameter \( t \) ranges from -2 to 2, calculate the values of \( x \) and \( y \) at the endpoints and midpoints of this interval. Evaluate at \( t = -2, -1, 0, 1, 2 \). This helps determine the shape and position of the curve. These points will be:- \( t = -2: (x, y) = (\cosh(-2), \sinh(-2)) \)- \( t = -1: (x, y) = (\cosh(-1), \sinh(-1)) \)- \( t = 0: (x, y) = (\cosh(0), \sinh(0)) = (1, 0) \)- \( t = 1: (x, y) = (\cosh(1), \sinh(1)) \)- \( t = 2: (x, y) = (\cosh(2), \sinh(2)) \)

Plot Points and Sketch the Graph

Using a graphing utility, plot the points determined in Step 2. The points describe a smooth, continuous, open curve that resembles one arm of a hyperbola (right branch). Each \( t \) value corresponds to a point on this hyperbola starting from the origin and moving outwards as \( t \) increases.

Determine Orientation

The orientation of the curve, or the direction in which the curve is traced as \( t \) increases, starts at the origin (\( t = 0 \)) and moves outward along the right branch of the hyperbola as \( t \) goes from 0 to 2. The trace of the curve will start at \( (1, 0) \) and move towards increasingly larger values of both \( x \) and \( y \), given that both \( \cosh(t) \) and \( \sinh(t) \) are increasing functions for positive \( t \).

Key concepts

Hyperbolic Functions

Hyperbolic functions are defined similarly to the more familiar trigonometric functions. However, instead of dealing with angles in a circle, hyperbolic functions map real numbers to a hyperbola. Two commonly used hyperbolic functions are the hyperbolic sine, denoted as \( \sinh(t) \), and the hyperbolic cosine, written as \( \cosh(t) \).

These functions resemble exponential functions and are expressed as follows:

• \( \cosh(t) = \frac{e^t + e^{-t}}{2} \)
• \( \sinh(t) = \frac{e^t - e^{-t}}{2} \)

These equations describe a relationship that is similar to, but distinct from, the Pythagorean identity known in circular trigonometry: \( \cos^2(\theta) + \sin^2(\theta) = 1 \). Instead, for hyperbolic functions, we have: \( \cosh^2(t) - \sinh^2(t) = 1 \).

When graphing parametric equations such as \( x = \cosh(t) \) and \( y = \sinh(t) \), the result is typically a hyperbolic curve. This appears different from the circular curves seen with regular sine and cosine, as these functions give an ever-increasing response to increasing \( t \).

Graph Orientation

Graph orientation refers to the direction in which a parametric curve is traced out as the parameter \( t \) increases. In the context of the given parametric equations \( x = \cosh(t) \) and \( y = \sinh(t) \), orientation is crucial to understanding how the curve develops over the interval.

As \( t \) varies from \(-2\) to \(2\), the curve starts at a point corresponding to \( t = -2 \) and traces towards \( t = 2 \). This often involves visualizing or plotting specific points for successive values of \( t \) to see the progression of the curve. Because both \( \cosh(t) \) and \( \sinh(t) \) increase as \( t \) becomes more positive, the graph will progress outward as \( t \) increases over the specified range.

Orientation is essential when calculating or predicting motion along the path, and any shift in orientation can fundamentally change the characteristics of motion. Knowing that the curve traces from left to right or right to left can affect everything from practical applications to theoretical examinations of such a parametric system.

Curve Sketching

Curve sketching involves creating a graph from given equations without an exhaustive list of individual calculations for each point. Using a set of known parametric equations, such as \( x = \cosh(t) \), \( y = \sinh(t) \), the task is to visualize and understand the overall shape of the graph.

Firstly, identify key points by evaluating the functions at significant values of \( t \), like the endpoints \(-2\), \(0\), and \(2\). This helps in establishing the general structure of the curve.

The plots of \( \cosh(t) \) and \( \sinh(t) \) point to a smooth curve which indicates one branch of a hyperbola.

• At \( t = 0 \), the coordinates are \((1, 0)\), where the curve intersects the x-axis.
• As \( t \) grows positive or negative, the curve stretches away from the origin in one direction, resembling branches of a hyperbola.

Once these principal points are plotted, employing a graphing utility can help connect the dots to better visualize the comprehensive trajectory. This aids in recognizing trends, overall direction, and theoretical connection with hyperbolic geometry.