Chapter 1.4, Problem 22E

(a)

To determine

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

Answer to Problem 22E

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

Explanation of Solution

Given information: The equations are $x=t^2-3$ and $y=t$ , $t\le 0$.

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 equal to ${\text{X}}_{\text{1T}}$ 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 $\left[-4,5\right]$ by $\left[-4,2\right]$ as shown below.

  

Figure (1)

To have an initial point, $t$ must have a lower bound. Since $t\le 0$ , it does not have lower bound so there is no initial point.

If $t=0$ , then $x=-3$ and $y=0$ . So, the terminal point is $\left(-3,0\right)$.

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

(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 22E

The Cartesian equation for a curve that contains the parameterized curve is $x=y^2-3$ and parameterized curve traces the lower half of the parabola defined by $x=y^2-3$.

Explanation of Solution

Given information: The equations are $x=t^2-3$ and $y=t$ , $t\le 0$.

Calculation:

Substitute $y$ for $t$ in the equation $x=t^2-3$.

  $x=y^2-3$

As shown in the graph, the parameterized curve traces the lower half of the parabola defined by $x=y^2-3$.

Therefore, the Cartesian equation for a curve that contains the parameterized curve is $x=y^2-3$ and parameterized curve traces the lower half of the parabola defined by $x=y^2-3$.