Problem 73 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=\frac{1}{x}$$

Step-by-Step Solution

Answer

The parametric equations for \(t=x\) are \(x=t, y=1/t\). And for \(t=2 - x\) the equations are \(x=2 - t, y=1/(2-t)\).

Step 1: Substitute Parameter for \(t=x\)

Starting with the given equation \(y = 1/x\), for a parametric equation, you can express \(x\) and \(y\) in terms of a third variable usually \(t\). Given \(t = x\), the new parametric equations are: \(x=t\) and \(y=1/t\).

Step 2: Substitute Parameter for \(t = 2 - x\)

Again, starting with \(y = 1/x\), but with a different parameter \(t = 2-x\), first, one must solve for \(x\) in terms of \(t\), which is \(x= 2-t\). Substituting into the original equation, one gets \(y=1/(2-t)\). The new parametric equations are: \(x=2-t\) and \(y=1/(2-t)\).

Key Concepts

Rectangular Equation

A rectangular equation represents a relationship between two variables, typically x and y, on a two-dimensional Cartesian plane. This means that for every value of x, there is a corresponding value of y, expressed directly through an equation such as \( y = \frac{1}{x} \). In this form, the equation paints a complete picture of a curve or a line within the plane. However, sometimes this depiction can be limiting when trying to express more complex curves or motions. That's where parametric equations come into play, allowing us to describe the graph using a third variable, often called a parameter, to obtain more flexibility in representation.

Substitution Method

The substitution method is a technique used to convert a rectangular equation into parametric form. This involves an extra step where we introduce a parameter \( t \) to express both x and y.

• Step 1: Assign a parameter, say \( t \), to one of the variables, x or y, or relate it to both via a simple equation.
• Step 2: Use this relationship to express the second variable, leading to a pair of equations, each dependent on \( t \).

For instance, in our problem, if \( t = x \), then by substitution, we have \( x = t \) and subsequently \( y = \frac{1}{t} \). If \( t = 2 - x \), solve for x as \( x = 2 - t \) and plug into the original equation to find \( y \). This way, the curve can also be expressed flexibly depending on how we choose \( t \). The substitution method greatly promotes exploring alternative structures of a known curve.

Variable Parameterization

Variable parameterization is a powerful concept in mathematics where a parameter, \( t \), is used to express variables such as x and y in terms of more manageable or meaningful equations. This tends to simplify the processes involving integration, differentiation, and even graphing.

• It allows for the separation of variables, making it easier to handle curves that may not be easy to manipulate in a single, static form.
• This can sometimes make complex curves more intuitive by breaking them down into simpler, linear paths when plotted over a range.

In our example, \( t \) became a bridge between expressions for x and y, making it easy to handle equations like \( y = \frac{1}{x} \). Ultimately, variable parameterization gives greater control and interpretation, enriching our understanding of how equations can describe movement and shape.