Problem 2 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=6 t-4, \quad y=3 t, \quad t \geq 0$$

Step-by-Step Solution

Answer

The curve is a line with equation \( x = 2y - 4 \).

Step 1: Understanding Parametric Equations

Parametric equations express a set of related quantities as explicit functions of a common variable, known as the parameter. Here, we have the parametric equations \( x = 6t - 4 \) and \( y = 3t \), with the parameter \( t \geq 0 \). These equations represent a curve in a 2D plane.

Step 2: Sketching the Curve

To sketch the curve, we can create a table of values for \( t \), calculate the corresponding \( x \) and \( y \) values, and plot these on a coordinate plane. For example, if \( t = 0 \), then \( x = -4 \) and \( y = 0 \), giving the point (-4, 0). If \( t = 1 \), then \( x = 2 \) and \( y = 3 \), giving the point (2, 3). Plotting several of these points will help visualize the curve.

Step 3: Eliminating the Parameter

To eliminate the parameter, we need to express \( y \) in terms of \( x \) using the given equations. From \( y = 3t \), solve for \( t \) to get \( t = \frac{y}{3} \). Substitute this into the equation for \( x \): \( x = 6\left(\frac{y}{3}\right) - 4 \), which simplifies to \( x = 2y - 4 \). This is the rectangular-coordinate equation of the curve.

Key Concepts

Rectangular-Coordinate Equation

A rectangular-coordinate equation is one that expresses the relationship between two variables only, typically in terms of the Cartesian coordinates x and y. In contrast to parametric equations, where each variable is often expressed in terms of a third variable (the parameter), rectangular-coordinate equations provide a more direct representation of the relationship between x and y.

When the task involves converting parametric equations into a rectangular form, you are essentially eliminating the parameter. This process results in an equation that can more easily be graphed and analyzed on the Cartesian plane. For example, in our exercise, we started with the parametric equations \( x = 6t - 4 \) and \( y = 3t \). By expressing t in terms of y using \( t = \frac{y}{3} \), and substituting into the equation for x, we successfully found the rectangular-coordinate equation \( x = 2y - 4 \). This equation now describes the same curve without needing to involve the parameter t.

Parameter Elimination

Parameter elimination is the process of removing the parameter from parametric equations to produce a single equation in rectangular form. It allows us to describe the same curve in the Cartesian coordinate system. This process is fundamental in converting parametric forms to Cartesian forms and enhances understanding by focusing only on the key variables, typically x and y in two dimensions.

To eliminate the parameter from our given equations \( x = 6t - 4 \) and \( y = 3t \), we begin by solving one of the equations for the parameter t. Here, we chose the equation for y, giving us \( t = \frac{y}{3} \). By substituting this expression for t back into the equation for x, we derive a direct relationship between x and y. Substitution leads us to \( x = 2y - 4 \), a straightforward linear equation. This sort of methodical substitution of parameters aids in understanding not just how values relate in the equation but their geometric representation as well.

2D Coordinate Plane

The 2D coordinate plane, also known as the Cartesian plane, is an essential concept for graphing functions and understanding equations like those in this exercise. It comprises two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is identified by an ordered pair (x, y), representing its distance from the intersection (origin) of the two axes.

Graphing on a 2D plane enables visualization of relationships between variables. When transitioning from parametric equations to a rectangular-coordinate form, this visual representation becomes clearer. For example, with our rectangular equation \( x = 2y - 4 \), we can plot a straight line by selecting various y-values to find corresponding x-values, forming points like (-4, 0) and (2, 3). As such, graphing translates numerical relationships into visible patterns, making it easier to grasp the behavior and characteristics of functions such as slopes and intercepts. In practice, the 2D plane serves as the foundational backdrop for exploring and understanding mathematical relationships.