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? $$x=4-\sqrt{t}, \quad y=\sqrt{t}, \quad 0 \leq t$$

Short answer

The initial point when \(t=0\) is \((4, 0)\) and the direction of the curve is from right to left, as \(t\) accounts for time and increases from 0 to infinity. The Cartesian equation of the curve is \(x + y^2 = 4\). The portion of the graph traced by the parametric curve is the graph of the Cartesian equation for \(x \leq 4\) and \(y \geq 0\), considering the parameter \(t \geq 0\).

Step-by-step solution

Plot the Parametric Curve

To graph this curve, the simplest method is to create a table of values for \(t\), compute the corresponding \(x\) and \(y\) values using the given equations, and then plot the points. Remember that \(t\) cannot be negative as given by the condition \(t \geq 0\). With each value of \(t\), calculate the corresponding values of \(x\) and \(y\). Then plot these points on a graph.

Identify Initial and Terminal Points

The initial point of the curve is the point where the curve starts which is when \(t=0\). Calculate the initial point by substituting \(t=0\) in the equations to find the \(x\) and \(y\) coordinates. The terminal point will be at the value of \(t\) at the limit, which in this case can be considered infinity since no end value for \(t\) was given.

Determine the Direction

The direction of the curve is determined by the parameter \(t\). As \(t\) increases from 0, we move along the curve in the direction of increasing \(t\). Trace the direction in the generated plot.

Finding a Cartesian Equation

To find out a Cartesian equation that describes the same curve, eliminate the parameter \(t\) from the given parametric equations. Solve one of the equations for \(t\) and substitute it into the other equation.

Determine the Portion of Graph

To determine what portion of the graph of the Cartesian equation is traced by the parametric curve, see if there is a limitation on the values of \(x\) and \(y\) due to the restriction on \(t\).