Chapter 1.4, Problem 5E

(a)

To determine

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

Answer to Problem 5E

The graph of the parametric equations with direction is shown in figure (1) and there are no initial or terminal points.

Explanation of Solution

Given information: The parametric equations are $x=3t$ , $y=9t^2$ such that $-\infty <t<\infty$.

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 then quit the window.

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

Step 3: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-3,3\right]$ by $\left[-1,3\right]$ as shown below.

  

Figure (1)

Since, $-\infty <t<\infty$ , so there are no initial or terminal points.

Therefore, the graph of the parametric equations with direction is shown in figure (1) and there are no initial or terminal points.

(b)

To determine

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

Answer to Problem 5E

The Cartesian equation is $y=x^2$ and the whole portion of the graph of the Cartesian equation is traced by the parameterized curve.

Explanation of Solution

Given information: The equations are $x=3t$ and $y=9t^2$ , $-\infty <t<\infty$.

Calculation:

Square both sides of the equation $x=3t$ and solve.

  $\begin{array}{c}{x}^2={\left(3t\right)}^2\\ =9t^2\end{array}$

Substitute $x^2$ for $9t^2$ in the equation $y=9t^2$.

  $y=x^2$

Since the domain of the parametric equation is $-\infty <t<\infty$ , the whole portion of the graph of the Cartesian equation is traced by the parameterized curve.

Therefore, the Cartesian equation is $y=x^2$ and the whole portion of the graph of the Cartesian equation is traced by the parameterized curve.