Problem 10 - Algebra and Trigonometry

Question

\(3-24=A\) 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. $$ x=t+1, \quad y=\frac{t}{t+1} $$

Step-by-Step Solution

Answer

The curve is \(y = \frac{x-1}{x}\).

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = t + 1\) and \(y = \frac{t}{t+1}\). \(t\) is the parameter that changes to produce the curve in the plane.

Step 2: Sketch the Parametric Curve

To sketch the curve, select various values for \(t\), calculate the corresponding \(x\) and \(y\) values, and plot them on a graph. This process reveals the curve's shape.

Step 3: Express 't' in Terms of 'x'

From the equation \(x = t + 1\), rearrange to find \(t\): \(t = x - 1\).

Step 4: Substitute 't' in the 'y' Equation

Substitute \(t = x - 1\) into \(y = \frac{t}{t+1}\): \[ y = \frac{x - 1}{x} \].

Step 5: Simplify the Rectangular Equation

The resulting equation \(y = \frac{x-1}{x}\) is already simplified as a rectangular coordinate equation, representing the original parametric curve.

Key Concepts

Rectangular Coordinate Equation

When working with parametric equations, transforming them into a rectangular coordinate equation can simplify the analysis of the curve. The rectangular coordinate equation relates the variables directly without using a parameter. Consider the given parametric equations: \(x = t + 1\) and \(y = \frac{t}{t+1}\). Here, the goal is to eliminate the parameter \(t\) to express \(y\) solely in terms of \(x\).
First, identify how \(t\) is expressed in terms of \(x\). From \(x = t + 1\), solve for \(t\), resulting in \(t = x - 1\). Then, substitute \(t = x - 1\) into the \(y\)-equation to obtain:

• \(y = \frac{x - 1}{x}\)

This equation, \(y = \frac{x-1}{x}\), is the rectangular equation. It provides a direct relationship between \(x\) and \(y\) without involving \(t\), offering a clearer picture of the curve's behavior in the Cartesian plane.

Parameter Elimination

Eliminating the parameter in parametric equations allows us to see how the two variables relate directly. To achieve this, we follow a systematic approach:

• Start by expressing the parameter \(t\) in terms of one of the variables from its respective equation. For the given equations, \(t\) is already expressed as \(x - 1\) when derived from \(x = t + 1\).
• Next, substitute this expression for \(t\) into the second parametric equation. This links \(y\) directly to \(x\), thereby eliminating the parameter.
• In our example, substituting \(t = x - 1\) into \(y = \frac{t}{t+1}\) simplifies to: \(y = \frac{x - 1}{x}\).

The result is a rectangular coordinate equation that represents the original parametric curve, making it easier to interpret and analyze.

Curve Sketching

Sketching curves from parametric equations involves plotting points that the parameter \(t\) generates. This visualizes the path traced by the equations as \(t\) varies. Here's how you can sketch from the given parametric equations \(x = t + 1\) and \(y = \frac{t}{t+1}\):

• Choose several values for \(t\), such as \(-2, -1, 0, 1, 2\).
• Calculate the respective \(x\) and \(y\) coordinates for each \(t\): for example, \(t = -1\) gives \(x = 0, y = \frac{-1}{0} = 0\); \(t = 0\) gives \(x = 1, y = 0\).
• Plot these \((x, y)\) points on a graph. Draw a smooth curve through the points, showing the shape of the path described by the parametric equations.

By connecting these points, you reveal the full path of the curve as it appears on the Cartesian plane. This method provides valuable insight into the curve's geometry and helps to visualize how the curve behaves as \(t\) changes.