Problem 7 - Essential Calculus

Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t-1, \quad y=\frac{t}{t-1} $$

Step-by-Step Solution

Answer

The rectangular equation corresponding to the given parametric equations is \( y = 1 + \frac{1}{x} \). The graph shows the function approaches the line \( y = 1 \) from above when \( x > 0 \) and from below when \( x < 0 \), but never reaches it.

Step 1: Write \( y \) as a Function of \( x \)

The two given parametric equations are \( x = t - 1 \) and \( y = \frac{t}{t - 1} \). Express \( t \) as a function of \( x \) in the first equation to get \( t = x + 1 \). Then, substitute this into the second equation in the place of \( t \) to obtain \( y = \frac{x + 1}{x} \). This is the rectangular equation corresponding to the given parametric equations.

Step 2: Simplifying the Equation

Simplify the equation further to get \( y = 1 + \frac{1}{x} \). In order to properly graph this function, it is necessary to note that \( x \neq 0 \) because the denominator of the fraction cannot be zero.

Step 3: Graphing the Equation

Now, plot the function \( y = 1 + \frac{1}{x} \). When \( x > 0 \), \( y \) tends to 1 from above, whereas when \( x < 0 \), \( y \) tends to 1 from below. The graph shows that the curve approaches the line \( y = 1 \) as \( x \) increases or decreases without bound, but does not touch the line.

Key Concepts

Rectangular Equations

Rectangular equations are expressions that relate the variables in the Cartesian coordinate system, typically involving the variables \( x \) and \( y \). These equations represent the relationship between \( x \) and \( y \) directly, without any intermediary parameter, such as the variable \( t \) in parametric equations. To convert a parametric equation to a rectangular equation, the parameter must be eliminated. This process often involves solving one of the parametric equations for the parameter and substituting this expression into the other equation. For example, given the parametric equations \( x = t - 1 \) and \( y = \frac{t}{t-1} \), we can express \( t \) as \( t = x + 1 \). Substituting \( x + 1 \) for \( t \) in the equation for \( y \) gives the rectangular equation \( y = \frac{x+1}{x} \), which simplifies to \( y = 1 + \frac{1}{x} \). This equation now expresses \( y \) in terms of \( x \), allowing us to analyze and graph the relationship directly in the \( xy \)-plane.

Curve Sketching

Curve sketching is a technique used to understand the shape and behavior of a graph by analyzing its mathematical equation. When sketching the curve for an equation like \( y = 1 + \frac{1}{x} \), several features need to be considered:

• Intercepts: Determine where the graph intersects the axes by setting \( y = 0 \) or \( x = 0 \), if possible. For this equation, \( x = 0 \) leads to a division by zero, so it presents a vertical asymptote instead.
• Asymptotes: Identify any lines that the graph approaches but never touches. Here, \( y \rightarrow 1 \) as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \), establishing a horizontal asymptote at \( y = 1 \). The vertical asymptote is already noted at \( x = 0 \).
• Behavior at bounds: Examine what happens as \( x \) approaches large positive or negative values. As analyzed, \( y \) comes close to 1 as \( \vert x \vert \) becomes large.

These properties guide us to sketch an accurate representation of the curve on the graph, indicating the direction the curve takes as the parameter \( x \) increases or decreases.

Graphing Functions

Graphing functions is the process of representing equations on a coordinate plane to visualize their shape and behavior. For the function \( y = 1 + \frac{1}{x} \), important steps to graphing include:

• Plotting key points: Choose specific values for \( x \), such as \( x = 1 \) and \( x = -1 \), to calculate \( y \) and place points on the graph.
• Identifying asymptotes: As previously mentioned, these include vertical asymptotes at \( x = 0 \) and a horizontal asymptote at \( y = 1 \).
• Drawing the curve: Use the plotted points and asymptotes to sketch the curve, ensuring that it approaches the asymptotes at large values of \( x \) and \( x = 0 \) without crossing them.

By following these steps, the graph of \( y = 1 + \frac{1}{x} \) is illustrated accurately, showing its two branches—one in the first quadrant and one in the third—reflecting its symmetry and approach towards the line \( y = 1 \). This visually demonstrates the mathematical relationship of the function.

Parameter Elimination

Parameter elimination is a vital process in converting parametric equations to rectangular form, allowing for straightforward graphing and analysis. Parametric equations involve a third variable, such as \( t \), that complicates direct visualization. To eliminate this parameter, follow these steps:

• Solve one equation for the parameter: Choose one of the parametric equations, solve for the parameter, \( t \), in terms of \( x \, or \, y \). For \( x = t - 1 \), express \( t \) as \( t = x + 1 \).
• Substitute into the other equation: Replace \( t \) in the second equation with the expression found, here substituting \( t = x + 1 \) into \( y = \frac{t}{t-1} \).
• Simplify the resulting equation: Often, simplifying the new equation is necessary, yielding the comprehensive rectangular equation \( y = 1 + \frac{1}{x} \).

Through parameter elimination, equations are translated into forms that yield greater insight into graphing and understanding the shape of the curve on a coordinate plane. It streamlines the process of determining key characteristics like intercepts and asymptotes, essential for accurate curve sketching.