11–14. Working with parametric equations Consider the following parametric equations.
a. Make a brief table of values of t, x, and y.
b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).


x=−t+6, y=3t−3; −5≤t≤5 

Verified step by step guidance

1. Identify the parametric equations given: \(x = -t + 6\) and \(y = 3t - 3\), with the parameter \(t\) ranging from \(-5\) to \(5\).
2. Choose several values of \(t\) within the interval \([-5, 5]\). For example, select \(t = -5, -3, 0, 2, 5\) to get a good spread of points.
3. Calculate the corresponding \(x\) and \(y\) values for each chosen \(t\) by substituting into the parametric equations: \(x = -t + 6\) and \(y = 3t - 3\).
4. Create a table listing each \(t\) value alongside its calculated \(x\) and \(y\) values. This table will help visualize how the points move as \(t\) changes.
5. Plot the points \((x, y)\) from the table on the coordinate plane. Then, sketch the curve formed by the parametric equations over the interval \(-5 \leq t \leq 5\), indicating the direction of increasing \(t\) to show the positive orientation of 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.

Creating a Table of Values

To analyze parametric equations, select values of the parameter t within the given interval, then compute corresponding x and y values. This table helps visualize points on the curve and understand how the curve evolves as t changes.

Plotting and Orientation of Parametric Curves

Plotting the (x, y) pairs from the table reveals the shape of the parametric curve. The positive orientation indicates the direction in which the curve is traced as t increases, which is important for understanding motion or direction along the curve.