Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Match the ordered pair from Column II with the pair of parametric equations in Column I on whose graph the point lies. In each case, consider the given value of \(t\). I $$x=3 t+6, y=-2 t+4 ; \quad t=2$$ II A. \((5,25)\) B. \((7,2)\) C. \((12,0)\) D. \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

C. \((12, 0)\)

Step 1: Substitute Parameter t

We are given the parametric equations \( x = 3t + 6 \) and \( y = -2t + 4 \). First, substitute the given \( t = 2 \) into both equations.

Step 2: Calculate x-coordinate

Substituting \( t = 2 \) into the equation for \( x \): \[ x = 3(2) + 6 = 6 + 6 = 12 \].

Step 3: Calculate y-coordinate

Substituting \( t = 2 \) into the equation for \( y \): \[ y = -2(2) + 4 = -4 + 4 = 0 \].

Step 4: Identify Matching Ordered Pair

From the calculated coordinates \((12, 0)\), match it to the options given in Column II. The matching pair is option C, \((12, 0)\).

Key Concepts

Ordered Pair

In mathematics, an ordered pair is a set of two elements grouped together in a specific sequence, typically written as \((x, y)\). Ordered pairs are crucial when working with coordinate systems, particularly in the context of graphs and functions.
This concept allows us to uniquely specify a point on the Cartesian coordinate system, where the first element is the x-coordinate and the second is the y-coordinate.
Ordered pairs are essential in describing the location of a point in a two-dimensional plane, making them a fundamental part of graphing any equation.
Let's consider the parametric equations given in the original exercise:

• The equations: \(x = 3t + 6\) and \(y = -2t + 4\)
• For \(t = 2\), substituting gives the ordered pair \((x,y) = (12, 0)\)

Identifying ordered pairs requires careful substitution of the parameter \(t\) and calculation to confirm the positions in a list or graph.

Graph

A graph in mathematics provides a visual representation of equations, inequalities, or functions. With parametric equations, graphs are often used to illustrate how a pair of equations interact over a range of values.
This is particularly true in the plane, where parametric equations describe curves made of points at specific parameter values.
In our exercise, the parametric equations \(x = 3t + 6\) and \(y = -2t + 4\) sketch a line on the Cartesian plane using ordered pairs derived from substituting values for \(t\).
Here's what happens:

• The value of \(t\) varies, generating different points \((x,y)\)
• Each of these points lies on the curve described by the parametric equations
• When \(t = 2\), our point \((12, 0)\) is derived, which we would plot on this graph

Graphs allow us to better understand the relationship and behavior of equations by visualizing the progression of points.

t-parameter

The \(t\)-parameter in parametric equations is a variable that enables us to describe the position of a point along a path or curve in a graph.
In essence, it acts as a slider that helps determine the coordinates in the equations at particular instances.
By adjusting the \(t\)-value, we determine different points on a curve by computing the corresponding \(x\) and \(y\) values.
For instance, in our exercise:

• The equations \(x = 3t + 6\) and \(y = -2t + 4\) define the point generation dependent on \(t\)
• With \(t = 2\), find point \((x, y) = (12, 0)\)
• Varying \(t\) reveals a series of ordered pairs, illustrating the curve

The \(t\)-parameter thus allows a comprehensive exploration of points along the graph, anchoring the equations to specific instances and creating a dynamic view of how equations behave.