Chapter 1.6, Problem 68E

Solution

a.

To graph the parametric equations $x=\cos \left(t\right)$ and $y=\sin \left(t\right)$ .

Given the settings of the grapher:

  $\begin{array}{ccc}{\text{T}}_{\min }=0,& {\text{T}}_{\max }=7,& {\text{T}}_{\text{step}}=0.1\\ {\text{X}}_{\min }=-4.7,& {\text{X}}_{\max }=4.7,& {\text{X}}_{\text{scl}}=1\\ {\text{Y}}_{\min }=-3.1,& {\text{Y}}_{\max }=3.1,& {\text{Y}}_{\text{scl}}=1\end{array}$.

Graph the parametric equations $x=\cos \left(t\right)$ and $y=\sin \left(t\right)$

on the grapher:

  

b.

To determine a transformation of the parametric function $x$ that produces a $4$ units wide ellipse that is $2$ units tall.

The transformation $x=2\cos \left(t\right)$ produces a $4$ units wide ellipse that is $2$ units tall.

Determination of the transformation using the grapher:

The function $x=\cos \left(t\right)$ is transformed to $x=2\cos \left(t\right)$ and the equations $x=2\cos \left(t\right)$ and $y=\sin \left(t\right)$ are graphed:

  

The transformation $x=2\cos \left(t\right)$ seems to produce a $4$ units wide ellipse that is $2$ units tall.

Thus, the transformation $x=2\cos \left(t\right)$ produces a $4$ units wide ellipse that is $2$ units tall.

Conclusion:

The transformation $x=2\cos \left(t\right)$ produces a $4$ units wide ellipse that is $2$ units tall.

c.

To determine the transformations of both the functions, $x$ and $y$ , that produce a circle of radius $3$ units.

The transformations $x=3\cos \left(t\right)$ and $y=3\sin \left(t\right)$ produce a circle of radius $3$ .

Determination of the transformations using the grapher:

The transformations $x=3\cos \left(t\right)$ and $y=3\sin \left(t\right)$ are applied and are graphed:

  

The transformations $x=3\cos \left(t\right)$ and $y=3\sin \left(t\right)$ seem to produce a circle of radius $3$ about the origin.

Thus, the transformations $x=3\cos \left(t\right)$ and $y=3\sin \left(t\right)$ produce a circle of radius $3$ .

Conclusion:

The transformations $x=3\cos \left(t\right)$ and $y=3\sin \left(t\right)$ produce a circle of radius $3$ .

d.

To determine the transformations of both the functions that produce an $8$ units wide ellipse that is $4$ units tall.

The transformations $x=4\cos \left(t\right)$ and $y=2\sin \left(t\right)$ produce an ellipse that is $8$ units wide and $4$ units tall.

Determination of the transformations using the grapher:

The transformations $x=4\cos \left(t\right)$ and $y=2\sin \left(t\right)$ are applied and are graphed:

  

The transformations $x=4\cos \left(t\right)$ and $y=2\sin \left(t\right)$ seem to produce an ellipse that is $8$ units wide and $4$ units tall.

Thus, the transformations $x=4\cos \left(t\right)$ and $y=2\sin \left(t\right)$ produce an ellipse that is $8$ units wide and $4$ units tall.

Conclusion:

The transformations $x=4\cos \left(t\right)$ and $y=2\sin \left(t\right)$ produce an ellipse that is $8$ units wide and $4$ units tall.