Problem 21 - A Graphical Approach to Precalculus with Limits

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. $$x=t+2, y=\frac{1}{t+2} ; \text { for } t \neq-2$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{1}{x} \), for \( x \neq 0 \).

Step 1: Identify the Parametric Equations

The given parametric equations are \( x = t + 2 \) and \( y = \frac{1}{t+2} \). These equations describe how both \( x \) and \( y \) depend on the parameter \( t \).

Step 2: Express \( t \) in terms of \( x \)

Since \( x = t + 2 \), we can solve for \( t \) to get \( t = x - 2 \). This allows us to transform the parametric equations into a single rectangular equation.

Step 3: Substitute \( t \, \text{in} \, y \)

Substitute \( t = x - 2 \) into the equation for \( y \):\[y = \frac{1}{(x-2) + 2} = \frac{1}{x}\]This gives us the rectangular, or Cartesian, equation of the curve.

Step 4: Specify Domain Restrictions

Since the original equation requires \( t eq -2 \), the restriction becomes "\( x eq 0 \)" in the rectangular form, because \( t + 2 eq 0 \) implies \( x = t + 2 eq 0 \).

Step 5: Sketch the Graph of \( y = \frac{1}{x} \)

The graph of \( y = \frac{1}{x} \) is a hyperbola with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \), respectively. Remember, the graph will be symmetric about the origin and will approach these asymptotes but never touch them.

Key Concepts

Rectangular Equation

Parametric equations like \( x = t + 2 \) and \( y = \frac{1}{t+2} \) are often converted into rectangular or Cartesian form. This is done to express the relationship between \( x \) and \( y \) without the parameter \( t \). To derive the rectangular equation, substitute \( t \) after solving one of the parametric equations for \( t \). Here, from \( x = t + 2 \), we get \( t = x - 2 \). Substitute this into \( y = \frac{1}{t+2} \) to obtain a single equation in terms of \( x \) and \( y \): \[ y = \frac{1}{x} \]. This equation describes the same curve in the Cartesian plane, providing a direct and simplified relationship between the \( x \) and \( y \)-coordinates.

Domain Restrictions

When converting parametric equations to a rectangular form, consider the original restrictions of the parameters. The parametric equation \( y = \frac{1}{t+2} \) has a restriction \( t eq -2 \) because division by zero is undefined. transforming this into the rectangular form \( y = \frac{1}{x} \), the restriction becomes \( x eq 0 \) because substituting \(-2\) into \( t + 2 \) yields zero, resulting in division by zero. Always carry over these restrictions into the rectangular equation. They are vital to ensure the rectangular equation accurately represents the same curve.

Graphing Plane Curves

Graphing plane curves from rectangular equations allows for visual interpretation of their behavior. For \( y = \frac{1}{x} \), graphing reveals a hyperbolic shape. Key aspects to note:

• The curve has asymptotes at \( x = 0 \) and \( y = 0 \), indicating where the curve approaches but never touches.
• It appears in two branches, one in quadrant I and another in quadrant III, reflecting symmetry about the origin.
• The curve does not exist where the domain restriction excludes, i.e., at \( x = 0 \).

Understanding this transformation from equations into visual graphs helps strengthen comprehension of the equations' implications.

Cartesian Equation

The Cartesian equation represents relationships between two variables in the \( xy \)-plane without any parameter. After converting from parametric form, these equations simplify analysis and graphing. Consider \( y = \frac{1}{x} \): it is now free from the parameter \( t \) and directly defines \( y \) based on \( x \). Such equations provide a straightforward way to assess the position and nature of the curve exclusively based on \( x \) and \( y \). This conversion offers insight into algebraic properties and helps in sketching graphs. Cartesian equations are preferred for quick evaluations because they distinctly outline the behavior of curves.