Problem 63 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=2-x$$

Step-by-Step Solution

Answer

The parametric equations for (a) and (b) are respectively (a) \(x=t\), \(y=2-t\) and (b) \(x=2-t\), \(y=t\).

Step 1: Case (a): \(t=x\)

Since it is given that \(t=x\), replace \(x\) with \(t\) in the original equation. We would then represent the equation as \(y=2-t\). Therefore, the set of parametric equations is \(x=t\) and \(y=2-t\).

Step 2: Case (b): \(t=2-x\)

In this case, \(t\) is given as \(2-x\). Substituting \(x\) would give us \(x=2-t\). Also replace \(x\) as \(2-t\) in the original equation, \(y=2-(2-t)\). Simplifying this, we get \(y=t\). So, the set of parametric equations in this case is \(x=2-t\) and \(y=t\).

Key Concepts

Rectangular Equation

A rectangular equation is a standard way of expressing a curve on a coordinate plane. It typically involves two variables, usually denoting horizontal and vertical positions, such as \(x\) and \(y\). In such an equation, the relationship between \(x\) and \(y\) is directly established. A classic example is the equation of a line, such as \(y = 2 - x\), which represents a straight line on a graph. In this linear equation, for each value of \(x\), you can determine the corresponding \(y\) value, and thus draw the line by connecting these points.
The focus of rectangular equations is on simplifying visualization since they explicitly define the graph of a function or relation without needing to consider additional parameters. Rectangular equations are quite useful for understanding basic graph shapes like circles, ellipses, and lines.

Parameterization

Parameterization involves expressing a geometric or algebraic object using one or more independent variables, called parameters. The idea is to switch from a singular relationship between two variables to a pair of equations, each representing a coordinate in terms of another variable \(t\).
By introducing a parameter, you can often simplify the manipulation or understanding of a spatial curve or shape. For instance, in the given equation \(y = 2 - x\), a parameter \(t\) can be used to form parametric equations:

• Case (a): \(x=t\), \(y=2-t\)
• Case (b): \(x=2-t\), \(y=t\)

Each approach provides a new way to represent the same line, making it easier to handle in various mathematical operations, such as integration or differentiation, especially when dealing with more complex figures.

Variable Substitution

Variable substitution is a technique used to make complex equations easier to work with by introducing new variables. In the context of parameterization, it involves substituting one of the original variables, such as \(x\) or \(y\), with another variable, such as \(t\).
In the original exercise, the substitutions demonstrate how introducing a parameter \(t\) simplifies the handling of the equation \(y = 2 - x\). For example:

• In Case (a): substituting \(x = t\) yields the parametric form \(x = t, y = 2 - t\).
• In Case (b): using \(t = 2 - x\) results in \(x = 2 - t, y = t\) by substituting back into the equation.

This approach not only simplifies mathematical processes by reducing complexities and allowing the parameter \(t\) to mediate between \(x\) and \(y\), but also enhances computational efficiency and understanding of the relationships represented in the graph.