91Ó°ÊÓ

Parametric equations often use a third variable, usually called a parameter, to uniquely define the position of a point on a curve with respect to time or another dimension. In this particular problem, the given parametric equations are \( x = t + 1 \) and \( y = \frac{t}{t+1} \). Here, \( t \) is the parameter describing both \( x \) and \( y \).

To convert these parametric equations to a rectangular-coordinate equation, we must eliminate the parameter, \( t \). By expressing \( t \) in terms of \( x \) from the first equation, we get \( t = x - 1 \). Substitute this expression for \( t \) into the equation for \( y \) to express \( y \) in terms of \( x \) alone, resulting in the equation \( y = \frac{x - 1}{x} \). This form shows the relationship between \( x \) and \( y \) directly, without the parameter \( t \), which is characteristic of a rectangular-coordinate equation.

Parameter elimination is the process used to transform parametric equations into a rectangular form by removing the parameter. The step-by-step transition involves isolating the parameter in one of the given equations and substituting it back into the other.

In our example, starting from \( x = t + 1 \), we solved for the parameter to get \( t = x - 1 \). Once isolated, we substituted \( t \) in the second equation, \( y = \frac{t}{t+1} \), resulting in \( y = \frac{x - 1}{x} \). This approach effectively 'collapses' the parametric equations into a single equation involving only \( x \) and \( y \).

Here is a quick rundown of the importance of parameter elimination:

• It simplifies the expressions by reducing them to one variable.
• Facilitates easy graphing of curves.
• Provides a better understanding of how \( x \) and \( y \) are related.

Graph sketching from a rectangular-coordinate equation assists in visualizing the relationship between variables \( x \) and \( y \). Starting from our simplified equation \( y = 1 - \frac{1}{x} \), sketching relies on understanding the behavior of this rational function.

To sketch the graph, it's important to identify:

• **Asymptotes:** The horizontal asymptote at \( y = 1 \) tells us how the function behaves as \( x \) goes to infinity or negative infinity. The vertical asymptote at \( x = 0 \) indicates a division by zero, where the function is undefined.
• **Points:** By calculating specific values, like when \( x = 2 \) leading to \( y = 0.5 \), we can pinpoint locations on the graph.
• **Behavior:** Observing the equation further, note how it approaches the asymptotes but never touches them, particularly significant points like where \( x \) becomes large.

Using these features allows the curve to be accurately plotted and scrutinized in detail.

Rational functions are expressions that involve the ratio of two polynomials. In the form \( y = 1 - \frac{1}{x} \), the function is represented as a combination of constant terms and a rational expression. Understanding rational functions is crucial as they commonly appear in various mathematical analyses and applications.

Key characteristics of rational functions include:

• **Asymptotes:** Horizontal and vertical asymptotes help in determining the boundaries and behavior of the function at extreme values.
• **Domains and Ranges:** The study of where the function is undefined or limited, such as avoiding values that lead to division by zero.
• **Intercepts:** Points where the graph intersects the axes, important for sketching the graph.

In this instance, \( y = 1 - \frac{1}{x} \) has a vertical asymptote at \( x = 0 \) since division by zero makes the value undefined, and a horizontal asymptote at \( y = 1 \), denoting the leveling behavior of the graph as \( x \) becomes extremely positive or negative. Analyzing these aspects provides richer insight into how rational functions behave and interact graphically.