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

Step-by-Step Solution

Answer

The curve is a hyperbola, given by the equation \( y = 1 - \frac{1}{x} \).

Step 1: Understand the Parametric Equations

We are given parametric equations with \( x = t + 1 \) and \( y = \frac{t}{t+1} \). The variable \( t \) is the parameter, which describes a curve in the Cartesian plane when \( t \) varies over all real numbers.

Step 2: Sketch the Curve

To sketch the curve, compute several points by substituting values of \( t \) into the parametric equations to get the corresponding \( (x, y) \) points. For example, for \( t = 0 \), \( x = 1 \) and \( y = 0 \); for \( t = 1 \), \( x = 2 \) and \( y = \frac{1}{2} \); for \( t = -1 \), \( x = 0 \) and \( y = 1 \). Plot these points on a graph to create a smooth curve.

Step 3: Eliminate Parameter \( t \)

To find a rectangular-coordinate equation, solve the equation for \( t \) in terms of \( x \): \( x = t + 1 \), so \( t = x - 1 \). Substitute this value for \( t \) in the equation for \( y \), yielding \( y = \frac{x-1}{x} \).

Step 4: Simplify Rectangular Equation

Simplify the expression \( y = \frac{x-1}{x} \) by dividing each term by \( x \). This gives \( y = 1 - \frac{1}{x} \). This is the rectangular form of the curve.

Step 5: Analyze the Rectangular Equation

The equation \( y = 1 - \frac{1}{x} \) describes a hyperbola with an asymptote along the x-axis as \( x \to \infty \) and another asymptote at the line \( y = 1 \). The graph has a discontinuity (undefined point) at \( x = 0 \), which corresponds to \( t = -1 \), as the denominator in the expression \( t+1 \) becomes zero.

Key Concepts

Rectangular-Coordinate Equation

The concept of converting parametric equations into rectangular-coordinate equations involves finding a relationship between the Cartesian coordinates

• \( x \)
• \( y \)

directly, without involving the parameter \( t \).
This is quite valuable as it often gives us a clearer picture of the curve's characteristics within a standard xy-plane.
In our case, the given parametric equations were \( x = t + 1 \) and \( y = \frac{t}{t+1} \).
To eliminate the parameter \( t \) and find the rectangular-coordinate equation, we manipulate the equation for \( x \):

• \( t = x - 1 \)

Substituting into the equation for \( y \) gives us:

• \( y = \frac{(x - 1)}{x} \)

Simplifying this expression by dividing the numerator by the denominator results in:

• \( y = 1 - \frac{1}{x} \)

This is the rectangular form of the curve, which succinctly describes the relationship between \( x \) and \( y \).
By eliminating the parameter, we've gained a more direct understanding of how \( y \) varies with \( x \).

Eliminating the Parameter

Eliminating the parameter is a process used to convert parametric equations into a single equation in rectangular form. This involves solving one of the parametric equations for the parameter \( t \).
Here, we started with the equation \( x = t + 1 \).
By rearranging this equation:

• \( t = x - 1 \)

Once \( t \) is expressed in terms of \( x \), substitute it back into the equation for \( y \) to yield:

• \( y = \frac{x-1}{x} \)

This process allows us to eliminate the extra variable \( t \), effectively reducing two equations into one.
By focusing on a single equation, analyzing and understanding the shape of the curve becomes simpler and more straightforward.
This equation, \( y = 1 - \frac{1}{x} \), is more accessible for sketching and interpreting the details of the curve.

Curve Sketching

Curve sketching with parametric equations involves plotting points on a graph and connecting them to form a smooth curve. When sketching a curve from parametric equations, you compute several \((x, y)\) coordinates by choosing values for the parameter \( t \).
For example, let's consider some specific values:

• When \( t = 0 \): \( x = 1 \), \( y = 0 \)
• When \( t = 1 \): \( x = 2 \), \( y = \frac{1}{2} \)
• When \( t = -1 \): \( x = 0 \), \( y = 1 \)

You plot these points and extend to other values of \( t \) to construct the full curve.
Curve sketching in this way provides a visual representation of the path traced by the equations as \( t \) changes.
Studying characteristics like where the curve might be undefined or approach certain lines (asymptotes) helps in understanding the complete behavior.
For the curve represented by \( y = 1 - \frac{1}{x} \), we notice it behaves like a hyperbola:

• There is a discontinuity at \( x = 0 \).
• The curve approaches the line \( y = 1 \) as \( x \to \infty \).

Sketching based on these insights makes the curve's nature and progression more intuitive.