Coordinate Geometry in the (x, y) Plane

To form the cartesian equation, eliminate t from the equation.

The expression of function asked to be shown in the question is in terms of x and y or g(x). This means we need to form the cartesian equation of the curve by eliminating t from the equations.

1. Solve for \( t \) in the x-equation:
   
   \( x = \frac{1}{t} + 2 \)
   \( x – 2 = \frac{1}{t} \)
   \( t = \frac{1}{x-2} \)
   
2. Now, substitute the value of \( t \) in the parametric equation of curve of y variable.
   \( y = \frac{1-2t}{3+t} \)
   As \( y = g(x) \):
   \( g(x) = \frac{1 – 2\left(\frac{1}{x-2}\right)}{3 + \frac{1}{x-2}} \)
3. Simplify the expression
   
   \( g(x) = \frac{1 – \frac{2}{x-2}}{3 + \frac{1}{x-2}} \)
   \( g(x) = \frac{(x-2) – 2}{3(x-2) + 1} \)
   \( g(x) = \frac{x-4}{3x-5} \)
   

5. Identify the Parametric Equation for y and its Restriction
   The curve C is defined by the parametric equation \( y = \frac{1-2t}{3+t} \), with the restriction that t is greater than zero (\( t > 0 \)).
6. Determine Asymptotes of the y vs. t Graph
   To understand the behavior of this reciprocal function, we identify its asymptotes:
   • Horizontal Asymptote: This is found by dividing the coefficients of t in the numerator and denominator. In \( y = \frac{-2t+1}{t+3} \), the coefficients of t are -2 and 1, respectively. Therefore, the horizontal asymptote is \( y = -2 \).
   • Vertical Asymptote: This occurs where the denominator is zero. Setting \( 3 + t = 0 \) gives \( t = -3 \).
7. Sketch the Graph of y against t
   • To determine the general shape of the graph (which quadrants relative to the asymptotes it occupies), we can find where it crosses an axis. Let’s find the Y-intercept by setting \( t = 0 \): \( y = \frac{1-2(0)}{3+0} = \frac{1}{3} \).
   • Since the graph crosses the y-axis at \( y = \frac{1}{3} \) (which is above the horizontal asymptote \( y = -2 \)), we know the graph must lie in the regions shown below, relative to its asymptotes:
     • It will pass through \( (0, \frac{1}{3}) \).
     • It will approach \( y = -2 \) as \( y \) approaches infinity.
     • It will approach \( t = -3 \) as \( y \) approaches infinity.
     • The two main branches of the reciprocal graph will be in the top-right and bottom-left sections formed by the asymptotes.

4. Apply the Condition \( t > 0 \): The crucial restriction is that t must be greater than zero. This means we only consider the part of the graph where t is positive.
   • We discard all parts of the graph where t is zero or negative (to the left of the y-axis).
   • The relevant part of the graph starts at the point where \( t = 0 \), which we found to be \( y = 1/3 \).
   • As t increases from 0 (moving to the right on the t-axis), the graph for y will move downwards, approaching the horizontal asymptote \( y = -2 \) but never actually reaching it.