Problem 8 - 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. $$x=t+1, \quad y=\frac{t}{t+1}$$

Step-by-Step Solution

Answer

Curve: Plot points of \( (x, y) \) from the parametrics. Equation: \( y = \frac{x-1}{x} \).

Step 1: Understand Parametric Equations

The given parametric equations are \( x = t + 1 \) and \( y = \frac{t}{t+1} \). These equations express both \( x \) and \( y \) in terms of the parameter \( t \). Our goal is to sketch the curve and then eliminate the parameter to find a rectangular-coordinate (Cartesian) equation.

Step 2: Sketching the Parametric Curve

To sketch the curve, select values of \( t \) and calculate the corresponding \( x \) and \( y \) coordinates. For example:- \( t = -2 \): \( x = -1 \), \( y = \frac{-2}{-1} = 2 \)- \( t = 0 \): \( x = 1 \), \( y = 0 \)- \( t = 1 \): \( x = 2 \), \( y = \frac{1}{2} \)- \( t = 2 \): \( x = 3 \), \( y = \frac{2}{3} \)Plot these points and sketch the curve through them.

Step 3: Solve for the Parameter t

From the equation \( x = t + 1 \), solve for \( t \):\[ t = x - 1 \]

Step 4: Substitute to Eliminate the Parameter

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

Step 5: Final Rectangular-Coordinate Equation

The rectangular-coordinate equation for the curve is \( y = \frac{x-1}{x} \). This equation relates \( x \) and \( y \) without the parameter \( t \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, provide a systematic way to locate points in a plane using two quantities, typically labeled as \(x\) and \(y\). This system is widely preferred because it offers a clear and straightforward representation of geometric shapes and curves. When you're given parametric equations like \(x = t + 1\) and \(y = \frac{t}{t+1}\), both \(x\) and \(y\) depend on a third variable, \(t\), known as the parameter. By converting these parametric equations to a single relation between \(x\) and \(y\), you obtain what is called a rectangular-coordinate equation. This process gives you a direct relationship, making it easier to plot points on a graph without needing to consider \(t\).
Understanding rectangular coordinates is crucial because it allows you to interpret equations in the way most familiar from school mathematics: as functions or relations graphed in the \(xy\)-plane. This method simplifies sketching and understanding geometric concepts.

Eliminating the Parameter

To transform parametric equations into a rectangular-coordinate equation, you need to eliminate the parameter \(t\). This process involves expressing one of the parametric equations in terms of \(t\) and then substituting it into the other equation. For instance, from the given exercise, we start with the parametric equations \(x = t + 1\) and \(y = \frac{t}{t+1}\). By solving for \(t\) in terms of \(x\) using the first equation, we get:
\[ t = x - 1 \]
This substitution allows us to remove \(t\) from the system. Substituting \(t = x - 1\) into the second parametric equation, we achieve the rectangular-coordinate equation:
\[ y = \frac{x-1}{x} \]
By eliminating the parameter, we simplify the relationship to just \(x\) and \(y\), making the curve much easier to analyze and plot. Eliminating parameters is fundamental, especially when analyzing complex motion or shapes derived from parametric models.

Sketching Curves

Sketching curves from parametric equations is all about understanding how \(x\) and \(y\) change as \(t\) varies. Start by selecting several values for \(t\) and calculate corresponding \(x\) and \(y\) values. Using these points, you can begin sketching a rough outline of the curve. For example:

• When \(t = -2\), you get \(x = -1\) and \(y = 2\).
• For \(t = 0\), \(x = 1\) and \(y = 0\).
• When \(t = 1\), \(x = 2\) and \(y = \frac{1}{2}\).
• At \(t = 2\), \(x = 3\) and \(y = \frac{2}{3}\).

Plotting these points and smoothly connecting them will help you visualize the curve. The essence of sketching with parametric equations is to see how the shape behaves as each point is plotted. Keep an eye on the overall motion the points suggest, such as curves bending or changing direction. Visualization in sketching not only aids in understanding the relationship between \(x\) and \(y\), but also reinforces the concept of how curves are formed from parametric relationships.