In Exercises 9–20, use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = 2t, y = |t − 1|; −∞ < t < ∞

Verified step by step guidance

1. Understand the parametric equations given: \(x = 2t\) and \(y = |t - 1|\), where \(t\) ranges over all real numbers from \(-\infty\) to \(\infty\). The goal is to plot points \((x, y)\) on the plane as \(t\) changes and show the direction of increasing \(t\).
2. Recognize that \(y = |t - 1|\) is a piecewise function. For \(t < 1\), \(y = 1 - t\), and for \(t \geq 1\), \(y = t - 1\). This will affect how the curve behaves on either side of \(t = 1\).
3. Create a table of values by choosing several values of \(t\) (for example, \(t = 0, 0.5, 1, 1.5, 2\) and also some negative values like \(t = -1, -0.5\)). For each \(t\), calculate \(x = 2t\) and \(y = |t - 1|\) to get points \((x, y)\) to plot.
4. Plot the points on the coordinate plane using the calculated \((x, y)\) pairs. Connect the points smoothly, keeping in mind the piecewise nature of \(y\) and that the curve will have a 'V' shape due to the absolute value.
5. Add arrows along the curve to indicate the orientation corresponding to increasing \(t\). Since \(x = 2t\) increases as \(t\) increases, the arrows should point from left to right along the curve.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted 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.

Graphing Parametric Curves

To graph parametric curves, plot points (x(t), y(t)) for various values of t and connect them smoothly. This method helps visualize the shape and behavior of the curve, especially when y is not a single-valued function of x.

Orientation and Direction of Parametric Curves

Orientation indicates the direction in which the curve is traced as the parameter t increases. Using arrows on the graph shows this direction, which is important for understanding the curve's progression and any time-dependent phenomena.