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=2 t-5, \quad y=4 t-7, \quad-\infty

Short answer

The curve represented by the parametric equations is a line given by the Cartesian equation \(y = 4x + 7\), with no terminal points. The direction in which the curve is traced is from left to right as \(t\) increases.

Step-by-step solution

Graphing the Parametric Equations

Given a parametric representation of a curve: \(x=2t-5\) and \(y=4t-7\), where \(t\) ranges from \(-\infty\) to \(\infty\), we graph the curve by plotting values of \(x\) and \(y\) by substituting different values of \(t\). As \(t\) ranges from \(-\infty\) to \(\infty\), there are no initial or terminal points, and the curve extends indefinitely in both directions. The direction in which the curve is traced can be determined by substituting increasing values of \(t\) to verify that both \(x\) and \(y\) increase as \(t\) increases.

Finding a Cartesian Equation

A Cartesian equation can be obtained by eliminating the parameter \(t\). Since \(t = (x+5)/2\) and \(t = (y+7)/4\), setting these two expressions equal to each other gives the Cartesian equation of the curve as \(2y = 8x + 14\), which simplifies to \(y = 4x + 7\). This is a linear equation representing a straight line.

Identify Portion of the Cartesian Graph Traced by the Parametric Curve

Since the parameter \(t\) ranges from \(-\infty\) to \(\infty\), the entire line represented by the Cartesian equation is traced. Specifically, as \(t\) increases, the curve moves from left to right.