Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Fill in the blank to correctly complete each sentence. For the plane curve defined by \(x=-3 t+6, y=t^{2}-3, \quad\) for \(t\) in \([-5,5]\),the ordered pair that corresponds to \(t=4\) is __________.

Step-by-Step Solution

Answer

The ordered pair is (-6, 13).

Step 1: Identify the Given Functions

The plane curve is defined by the parametric equations: \( x = -3t + 6 \) and \( y = t^2 - 3 \). Our goal is to find the ordered pair \((x, y)\) corresponding to \( t = 4 \).

Step 2: Calculate the x-coordinate

Substitute \( t = 4 \) into the equation for \( x \): \[x = -3(4) + 6 = -12 + 6 = -6\]

Step 3: Calculate the y-coordinate

Substitute \( t = 4 \) into the equation for \( y \):\[y = (4)^2 - 3 = 16 - 3 = 13\]

Step 4: Form the Ordered Pair

Combine the x and y coordinates to form the ordered pair.The ordered pair for \( t = 4 \) is \((-6, 13)\).

Key Concepts

Plane Curve

A plane curve is a curve that lies on a plane, defined by equations or functions. In the context of parametric equations, a plane curve can be visualized by setting relationships between a parameter, often denoted as \( t \), and the variables \( x \) and \( y \). These are the positions on the curve, in terms of the parameter \( t \).

Using parametric equations, such as \( x = -3t + 6 \) and \( y = t^2 - 3 \), provides a way to define complex curves by expressing both \( x \) and \( y \) as functions of \( t \). This approach can illustrate movement or changes along the curve as \( t \) varies. Importantly, parametric equations offer a way to capture dynamic systems' behavior, rather than static snapshots, allowing us to understand how one variable affects another along the curve.

Ordered Pair

An ordered pair is a fundamental concept in mathematics, representing a pairing of two elements such as \( (x, y) \). Unlike simple lists of numbers, ordered pairs express a specific sequence, meaning the order in which each element appears is crucial.

Ordered pairs are used widely across different areas of math to locate points on a plane. For parametric curves, ordered pairs represent specific points on the curve corresponding to a particular value of the parameter \( t \).

• In our example, with the parametric equations given, for \( t = 4 \), the ordered pair found was \( (-6, 13) \).
• Each ordered pair provides the coordinates that identify a unique point.

X-Coordinate

The x-coordinate of a point in a plane curve indicates its horizontal position. By solving parametric equations, the x-coordinate is determined based on the given parameter \( t \).

For instance, in the equation \( x = -3t + 6 \), when substituting \( t = 4 \), we calculate the x-coordinate as:

• Start with the equation: \( x = -3t + 6 \)
• Substitute \( t = 4 \): \( x = -3(4) + 6 \)
• Simplify: \( x = -12 + 6 = -6 \).

This results in \( x = -6 \), showing the exact horizontal position of the point when \( t = 4 \).

Y-Coordinate

The y-coordinate represents the vertical position of a point on the plane. Like the x-coordinate, it is found using the parametric equations, but it represents the vertical movement or position of our point.

Consider the parametric equation \( y = t^2 - 3 \). To find the y-coordinate when \( t = 4 \):

• Begin with the equation: \( y = t^2 - 3 \)
• Substitute \( t = 4 \): \( y = (4)^2 - 3 \)
• Calculate: \( y = 16 - 3 = 13 \).

Thus, the y-coordinate for the parameter \( t = 4 \) is 13, situating the point vertically on the plane.