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

Short 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-by-step solution

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.

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.

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.