Question

. x=|t|, y=|1-| t|| A pair of parametric equations is given. Find a rectangular-coordinate equation for the curve by eliminating the parameter.

Answer 1

The rectangular-coordinate equation for the given parametric equations x=|t| and y=|1-|t|| consists of two lines: y = x - 1 for x ≥ 1, and y = 1 + x for x < 1.

Step-by-step explanation:

To find a rectangular-coordinate equation for the curve represented by the parametric equations x=|t| and y=|1-|t||, we need to eliminate the parameter t. First, since x=|t|, it means that t could be either x or -x, because taking the absolute value of either will give us x. Next, we tackle the equation for y. We rewrite it without the absolute value as a piecewise function, which means we look at two cases:

• If t ≥1, then y=|1-t| = t-1.
• If t<1, then y=|1-t| = 1-t.

Substituting the value of t in terms of x from our earlier conclusion (where t=x or t=-x), we apply both cases:

• If t ≥1 then t=x, which gives us y = x - 1.
• If t<1 then t=-x, which gives us y = 1 - (-x) = 1 + x.

This means the curve is represented by two lines in the rectangular coordinate system. The equation of the curve is:

• 
  For x ≥1, y = x - 1
• 
  For x<1, y = 1 + x