Chapter 1, Problem 48RE

(a)

To determine

To find: The graph of the parametric equations, the initial and terminal points. Also, indicate the direction in which the curve is traced.

Answer to Problem 48RE

The graph of the parametric equations with the direction is shown in figure (1), the initial point is $\left(3,0\right)$ and there is no terminal point.

Explanation of Solution

Given information:The given equations are $x=1+t$ and $y=\sqrt{4-2t}$ such that $t\le 2$ .

Calculation:

Use the following step to graph the parametric equations by graphing calculator.

Step 1: First press the “ON” button graphical calculator.

Step 2: Press the $\boxed{\text{MODE}}$ button, under FUNC and choose PAR option.

Step 3: Press $\boxed{\text{Y=}}$ , and set the parametric equations under ${\text{X}}_{\text{1T}}$ and ${\text{Y}}_{\text{1T}}$ .

Step 4: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-5,5\right]$ by $\left[-2,2\right]$ as shown below.

  

Figure (1)

Substitute 2 for $t$ in the equation $x=1+t$ .

  $\begin{array}{c}x=1+2\\ =3\end{array}$

Substitute 2 for $t$ in the equation $x=1+t$ .

  $\begin{array}{c}y=\sqrt{4-2\times 2}\\ =0\end{array}$

The initial point is $\left(3,0\right)$ and there is no terminal point.

Therefore, the graph of the parametric equations with the direction is shown in figure (1), the initial point is $\left(3,0\right)$ and there is no terminal point.

(b)

To determine

To find:The Cartesian equation and the portion of the graph of the Cartesian equation that is traced by the parameterized curve.

Answer to Problem 48RE

The Cartesian equation is $y=\sqrt{6-2x}$ and the parameterization tracesthe entire curve.

Explanation of Solution

Given information:The given equations are $x=1+t$ and $y=\sqrt{4-2t}$ such that $t\le 2$ .

Calculation:

Solve both the equations.

  $\begin{array}{c}y=\sqrt{4-2\left(x-1\right)}\\ =\sqrt{4-2x+2}\\ =\sqrt{6-2x}\end{array}$

For all $x\in \left(-\infty ,\infty \right)$ . The Cartesian equation traced the entire graph.

Therefore, the Cartesian equation is $y=\sqrt{6-2x}$ and the parameterization traces the entire curve.