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=2 t,y=3t−4;−10≤d≤10 

Verified step by step guidance

1. Identify the parametric equations given: \(x = 2t\) and \(y = 3t - 4\), with the parameter \(t\) ranging from \(-10\) to \$10$.
2. Create a table of values by choosing several values of \(t\) within the interval \([-10, 10]\). For each chosen \(t\), calculate the corresponding \(x\) and \(y\) values using the parametric equations: \(x = 2t\) and \(y = 3t - 4\).
3. Plot the points \((x, y)\) from the table on the Cartesian plane. Each point corresponds to a specific value of \(t\).
4. Draw the curve by smoothly connecting the plotted points, representing the parametric curve defined by the equations. Make sure the curve reflects the continuous change of \(t\) from \(-10\) to \$10$.
5. Indicate the positive orientation on the curve by adding arrows showing the direction of increasing \(t\), which corresponds to moving from points with smaller \(t\) values to larger \(t\) values.

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 representation 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 Parametric Curves and Orientation

Plotting involves graphing the (x, y) pairs from the table to form the curve. The positive orientation shows the direction of increasing t, indicating how the curve is traced over time, which is important for understanding motion or directionality.