Problem 7 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\frac{1}{t}, \quad y=t+1 $$

Step-by-Step Solution

Answer

The curve is a hyperbola with the equation \(y = \frac{1}{x} + 1\).

Step 1: Sketch the Parametric Curve

To sketch the curve, let's generate some points by choosing values for the parameter \(t\). For example, take \(t = -2, -1, 0.5, 1, 2\). Calculate \(x\) and \(y\) as follows: - For \(t = -2\), \((x, y) = (-0.5, -1)\).- For \(t = -1\), \((x, y) = (-1, 0)\).- For \(t = 0.5\), \((x, y) = (2, 1.5)\).- For \(t = 1\), \((x, y) = (1, 2)\).- For \(t = 2\), \((x, y) = (0.5, 3)\). Plot these points on the Cartesian coordinate system and connect them to get the curve. The graph will resemble a hyperbolic curve.

Step 2: Eliminate the Parameter

To convert the parametric equations to a rectangular form, solve for \(t\) in terms of one parameter. From \(x = \frac{1}{t}\), we have \(t = \frac{1}{x}\). Substitute this into the equation for \(y\): \[ y = t + 1 \Rightarrow y = \frac{1}{x} + 1 \].The rectangular-coordinate equation is \(y = \frac{1}{x} + 1\).

Key Concepts

Rectangular-Coordinate Equation

The idea of a rectangular-coordinate equation comes from the need to represent curves and shapes using the standard Cartesian coordinate system. In this system, we have the familiar x-axis and y-axis. Instead of using a third variable often called the parameter, we express both the x and y coordinates directly. Think of it like translating a unique language of parametric equations into something universally understood by the x, y coordinate system.

In this exercise, we started with the parametric equations x = \( \frac{1}{t} \) and y = t + 1. By eliminating the parameter t, we transformed these equations into the rectangular form y = \( \frac{1}{x} + 1 \). This allows you to plot the curve using only the x and y coordinates, enabling easier analysis and graphing.

Eliminating the Parameter

To eliminate the parameter means to rewrite the parametric equations in a way that does not involve the parameter at all. This task involves expressing one variable in terms of another, getting rid of the parameter. It's like solving a puzzle where you remove the middle piece to directly connect the two remaining pieces.

Here's how it works in our example:

• We have x = \( \frac{1}{t} \). From this, solve for t as t = \( \frac{1}{x} \).
• Substitute this t-value into the other equation, y = t + 1.
• You get y = \( \frac{1}{x} + 1 \) as the rectangular-coordinate equation.

This new equation no longer has the parameter. Instead, it shows a direct relationship between x and y, making it easier to graph and analyze. This skill is useful for converting parametric curves into forms suitable for various mathematical and practical applications.

Hyperbolic Curve

Hyperbolic curves can be a bit mysterious if you're new to them. They are formed by hyperbolas, which come from the x and y relationships that look a bit different from circles or parabolas. A hyperbola opens along the x-axis or y-axis, often resembling two gentle slopes heading in opposite directions.

When we transformed our parametric equations into the rectangular form y = \( \frac{1}{x} + 1 \), the resulting graph was a hyperbolic curve. To understand this, remember:

• A hyperbola shapes itself like two separate arches that open around the coordinate axes.
• In our exercise, the curve exhibits this characteristic hyperbolic shape, reflecting the reciprocal relationship of the y and x coordinates.

Hyperbolic curves are notable for their distinctive shape and are often used to model real-world phenomena like radio waves or certain economic processes. Understanding this curve helps in recognizing how different equation forms can give rise to unique geometric figures.