Problem 66 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$x=3 y-2$$

Step-by-Step Solution

Answer

The parametric equations where \(t = x\) are \(x = t\) and \(y = \frac{t + 2}{3}\). The parametric equations where \(t = 2 - x\) are \(x = 2 - t\) and \(y = \frac{t + 4}{3}\).

Step 1: Solve for \(x\) and \(y\) given \(t = x\)

Given the equation \(x = 3y - 2\) , and parameter \(t = x\). This implies that our parametric equation would be \(x = t\) . To find the value of \(y\) in terms of \(t\), substitute \(t\) for \(x\) in the original equation. So, \(t = 3y - 2\). Solve this equation for \(y\) to get \(y = \frac{t + 2}{3}\) . This gives us the parametric equations when \(t = x\).

Step 2: Solve for \(x\) and \(y\) given \(t = 2 - x\)

Given the equation \(x = 3y - 2\) , and parameter \(t = 2 - x\). This implies that our parametric equation would be \(x = 2 - t\). To find the value of \(y\) in terms of \(t\), substitute \(2 - t\) for \(x\) in the original equation. So, this gives us \(2 - t = 3y - 2\). Solve this equation for \(y\) to get \(y = \frac{t + 4}{3}\). This gives us the parametric equations when \(t = 2 - x\).

Key Concepts

Rectangular Equations

A rectangular equation is typically written in terms of the two variables, usually known as \(x\) and \(y\). These equations describe curves or graphs on a plane using a direct relationship between these variables. For instance, the rectangular equation \(x = 3y - 2\) represents a line on the Cartesian plane.

To convert this into parametric form, one or both of the variables are expressed as functions of an independent variable, known as the parameter. This parameter often simplifies analysis and manipulation of the equation.

• This direct relationship allows us to quickly plot or derive properties like slopes and intercepts directly from the equation without extra transformation.
• Rectangular equations form the backbone for defining simple geometric shapes and linear transformations.

The simplicity of rectangular equations makes them crucial in introductory algebra and geometry, providing a straightforward method to visualize and interact with mathematical relationships.

Parameterization

Parameterization is the process of expressing a geometric figure or a function in terms of a parameter. This method is particularly useful in converting equations into a form where each coordinate is a separate function of a parameter \(t\). For example, in the exercise, we took the rectangular equation \(x = 3y - 2\) and represented it in parametric form as \(x = t\) and \(y = \frac{t + 2}{3}\) when \(t = x\).

Alternatively, when setting \(t = 2 - x\), we derive another set of parametric equations: \(x = 2 - t\) and \(y = \frac{t + 4}{3}\). Each parameterization provides a different perspective on the same curve or line.

• Parameterization can simplify the calculation of complex curves and is often used in physics and engineering.
• It provides a way to break a curve into its \(x\) and \(y\) components, expressed through a common parameter.
• This approach can highlight relationships and symmetries not initially obvious in the rectangular form.

In essence, parameterization enhances flexibility in mathematical modeling, helping to visualize and analyze equations in a variety of scenarios.

Solving Equations

Solving equations is the procedure of finding unknown variables that satisfy the equation under given conditions or constraints. In the exercise, given the rectangular equation \(x = 3y - 2\), setting \(t = x\) or \(t = 2-x\) allows us to re-express this relationship in parametric form to find \(y\) in terms of \(t\).

For the case \(t = x\):

• Substitute \(t\) into the equation: \(t = 3y - 2\)
• Solve for \(y\) to obtain \(y = \frac{t + 2}{3}\)

For the case \(t = 2-x\):

• Substitute \(2-t\) for \(x\): \(2 - t = 3y - 2\)
• Solve for \(y\) to obtain \(y = \frac{t + 4}{3}\)

The ability to solve for \(y\) in terms of \(t\) demonstrates flexibility in handling the same equation under different parameter conditions. Parametric equations offer alternative ways to solve and represent solutions, highlighting the adaptability required when working with various mathematical problems.