In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

Verified step by step guidance

1. Start with the given parametric equations: \(x = 2^{t}\) and \(y = 2^{-t}\), where \(t \geq 0\).
2. Express \(y\) in terms of \(x\) by eliminating the parameter \(t\). Since \(x = 2^{t}\), take the logarithm base 2 of both sides to get \(t = \log_{2} x\).
3. Substitute \(t = \log_{2} x\) into the equation for \(y\): \(y = 2^{-t} = 2^{-\log_{2} x}\).
4. Use the property of exponents and logarithms: \(2^{-\log_{2} x} = x^{-1} = \frac{1}{x}\). So the rectangular equation is \(y = \frac{1}{x}\).
5. Determine the domain for \(x\) based on \(t \geq 0\). Since \(x = 2^{t}\) and \(t \geq 0\), \(x \geq 1\). Sketch the curve \(y = \frac{1}{x}\) for \(x \geq 1\), and add arrows indicating the orientation as \(t\) increases (which corresponds to \(x\) increasing).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y being directly related to x, both x and y depend on t, allowing the description of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation relating x and y directly. This is done by solving one equation for t and substituting into the other, enabling analysis and graphing in the rectangular coordinate system.

Graphing and Orientation of Parametric Curves

Graphing parametric curves requires plotting points (x(t), y(t)) for values of t in the given interval. Orientation is shown by arrows indicating the direction of increasing t, which helps understand the curve's traversal and behavior over the parameter range.