Problem 7 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\frac{1}{t}, \quad y=t+1$$

Step-by-Step Solution

Answer

(a) Plot points for different \(t\) to sketch the curve. (b) The rectangular equation is \(y = \frac{1}{x} + 1\).

Step 1: Graph the Parametric Equations

To sketch the curve represented by the parametric equations, choose several values of the parameter \(t\), and then compute corresponding \(x\) and \(y\) values. For example, if \(t = 1\), then \(x = 1\) and \(y = 2\). Create a table of these values and use them to plot points on a graph. By plotting multiple points, observe any patterns that form to outline the overall shape of the curve.

Step 2: Understand the Parametric Relationships

Inspect the parametric equations: \(x = \frac{1}{t}\) and \(y = t + 1\). Notice that \(x\) is inversely proportional to \(t\) and \(y\) is a linear function of \(t\). This can help predict the behavior of the curve such that as \(t\) increases, \(x\) decreases and \(y\) increases. This information helps to understand the direction and progression of the graph over different intervals of \(t\).

Step 3: Eliminate the Parameter

To eliminate the parameter \(t\), solve one of the parametric equations for \(t\) and substitute it into the other equation. From \(x = \frac{1}{t}\), solve for \(t\): \(t = \frac{1}{x}\). Substitute this expression for \(t\) into the equation for \(y\): \[ y = \frac{1}{x} + 1. \] This gives the rectangular equation \(y = \frac{1}{x} + 1\).

Step 4: Verify the Rectangular Equation

Check a few values from your parametric table or graph to ensure that they satisfy the rectangular equation \(y = \frac{1}{x} + 1\). For example, for \(t = 1\), \(x = 1\) and \(y = 2\); substituting \(x = 1\) into \(y = \frac{1}{x} + 1\) gives \(y = 2\), which matches our parametric calculations, confirming the correctness of the rectangular equation.

Key Concepts

Rectangular-Coordinate Equation

In mathematics, transforming a pair of parametric equations into a rectangular-coordinate equation can simplify the representation of a curve. Here, we explore how a given set of parametric equations, like \(x = \frac{1}{t}\) and \(y = t + 1\), can be represented in a standard \(x, y\) coordinate form. To convert, we often need to eliminate the parameter—in this case, \(t\). By solving one of the parametric equations for \(t\) and substituting it into the other equation, we connect the two equations directly in terms of \(x\) and \(y\). For example, solving \(x = \frac{1}{t}\) for \(t\) gives \(t = \frac{1}{x}\). Substituting this into \(y = t + 1\) results in the rectangular-coordinate equation: \[ y = \frac{1}{x} + 1. \] This equation is now free of the parameter \(t\) and describes the relationship between \(x\) and \(y\) on a two-dimensional plane.

Parameter Elimination

Eliminating the parameter involves manipulating parametric equations to remove the variable that isn't \(x\) or \(y\). This transformation allows us to express curves traditionally described with parametric equations as standard functions or relations between \(x\) and \(y\). ### Why Eliminate Parameters?- **Simplicity:** It often simplifies understanding and comparing the equation by having it in a traditional form.- **Versatility:** Rectangular-coordinate equations are easily compatible with various graphing tools and software.- **Ease of Analysis:** Analytical operations such as differentiation and integration require equations in \(x\) and \(y\).### How It WorksTo eliminate the parameter, usually:

• Solve one equation for the parameter \(t\).
• Substitute this solved parameter back into the other equation.

In the given problem, converting \(x = \frac{1}{t}\) to \(t = \frac{1}{x}\) and substituting into \(y = t + 1\) transforms the set into \(y = \frac{1}{x} + 1\), thus eliminating \(t\). This straightforward approach highlights the hidden interplay between \(x\) and \(y\).

Graphing Parametric Equations

Graphing parametric equations involves plotting points on a coordinate plane based on sequential values of the parameter \(t\). This method effectively represents curves that may not be easily expressible using only \(x\) and \(y\).### Steps to Graph1. **Select Values for \(t\):** Choose a range of \(t\) values, such as integers or fractions, to comprehensively capture the shape of the curve.2. **Compute \(x\) and \(y\):** For each \(t\) value, calculate corresponding \(x\) and \(y\) using the parametric equations.3. **Plot the Points:** Place the calculated points \((x, y)\) onto a graph. Connect the points smoothly if the relationship suggests continuity.4. **Observe Patterns:** As you plot more points, a pattern or shape should emerge, revealing the curve's trajectory.In the problem example, begin by selecting values such as \(t = -1, 0, 1, 2\), compute \(x = \frac{1}{t}\) and \(y = t + 1\), and then plot these points on the graph. Observing the changes as \(t\) increases helps demonstrate how the curve behaves across its domain, offering insights into its - direction,- trends, and- restrictions.