In Exercises 71–76, eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = 2t − 1, y = 1 − t; −∞ < t < ∞

Verified step by step guidance

1. Identify the given parametric equations: \(x = 2t - 1\) and \(y = 1 - t\) with parameter \(t\) ranging over all real numbers.
2. Solve one of the parametric equations for \(t\) in terms of \(x\) or \(y\). For example, from \(x = 2t - 1\), solve for \(t\): \(t = \frac{x + 1}{2}\).
3. Substitute the expression for \(t\) into the other parametric equation to eliminate the parameter. Substitute \(t = \frac{x + 1}{2}\) into \(y = 1 - t\) to get \(y\) in terms of \(x\).
4. Simplify the resulting equation to express \(y\) explicitly as a function of \(x\), which will give the Cartesian equation of the curve.
5. To determine the orientation of the curve, consider how \(x\) and \(y\) change as \(t\) increases. For example, calculate a few points for increasing values of \(t\) and plot arrows on the graph to indicate the direction of increasing \(t\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of 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 express y directly in terms of x, removing t. This is done by solving one equation for t and substituting into the other, resulting in a Cartesian equation that describes the same curve.

Orientation of Parametric Curves

Orientation indicates the direction in which the curve is traced as the parameter increases. Using arrows on the graph shows this direction, which is important for understanding the behavior and properties of the curve over the parameter's domain.