Question

In Exercises \(5-22,\) a parametrization is given for a curve. (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? $$x=1-t, \quad y=1+t, \quad-\infty

Short answer

The parametric curve is a line which extends indefinitely in both directions (no initial and terminal points), and is traced from (2,0) to (0,2). The equivalent Cartesian equation is \(x + y = 2\) and the parametrized curve traces the entire line.

Step-by-step solution

Sketch the Parametric Curve

Plot the points for values of \(t\) to sketch the parametric curve. To determine the direction, let's assume three points: \(t=-1, t=0, t=1\). For \(t=-1\), we have \(x=2, y=0\). For \(t=0\), we get \(x=1, y=1\). And for \(t=1\), we have \(x=0, y=2\). The points plotted are in the direction from (2,0) to (0,2).

Determine Initial and Terminal Points

The range of \(t\) is given as \(-Infinity < t < Infinity\). This means that the line will extend in both directions indefinitely, there are no initial and terminal points.

Convert Parametric Equations into Cartesian Equation

To find a Cartesian equation from the parametric equations, eliminate the parameter. From \(x=1-t\) and \(y=1+t\), by adding these two equations, we have: \(x+y=2\). This is the Cartesian representation of the parametrized curve.

Determine the Portion of the Cartesian GRAPH traced by parametrized Curve

Since the values of \(t\) ranges from -Infinity to Infinity, the entire line given by \(x + y = 2\) will be traced.