1. What is a system of parametric equations?

2. Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

3. Explain how to eliminate a parameter given a set of parametric equations.

4. What is a benefit of writing a system of parametric equations as a Cartesian equation?

5. What is a benefit of using parametric equations?

6. Why are there many sets of parametric equations to represent on Cartesian function?

For the following exercises, eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

7. $\begin{cases}x\left(t\right)=5-t\hfill \\ y\left(t\right)=8 - 2t\hfill \end{cases}$

8. $\begin{cases}x\left(t\right)=6 - 3t\hfill \\ y\left(t\right)=10-t\hfill \end{cases}$

9. $\begin{cases}x\left(t\right)=2t+1\hfill \\ y\left(t\right)=3\sqrt{t}\hfill \end{cases}$

10. $\begin{cases}x\left(t\right)=3t - 1\hfill \\ y\left(t\right)=2{t}^{2}\hfill \end{cases}$

11. $\begin{cases}x\left(t\right)=2{e}^{t}\hfill \\ y\left(t\right)=1 - 5t\hfill \end{cases}$

12. $\begin{cases}x\left(t\right)={e}^{-2t}\hfill \\ y\left(t\right)=2{e}^{-t}\hfill \end{cases}$

13. $\begin{cases}x\left(t\right)=4\text{log}\left(t\right)\hfill \\ y\left(t\right)=3+2t\hfill \end{cases}$

14. $\begin{cases}x\left(t\right)=\text{log}\left(2t\right)\hfill \\ y\left(t\right)=\sqrt{t - 1}\hfill \end{cases}$

15. $\begin{cases}x\left(t\right)={t}^{3}-t\hfill \\ y\left(t\right)=2t\hfill \end{cases}$

16. $\begin{cases}x\left(t\right)=t-{t}^{4}\hfill \\ y\left(t\right)=t+2\hfill \end{cases}$

17. $\begin{cases}x\left(t\right)={e}^{2t}\hfill \\ y\left(t\right)={e}^{6t}\hfill \end{cases}$

18. $\begin{cases}x\left(t\right)={t}^{5}\hfill \\ y\left(t\right)={t}^{10}\hfill \end{cases}$

19. $\begin{cases}x\left(t\right)=4\text{cos}t\hfill \\ y\left(t\right)=5\sin t \hfill \end{cases}$

20. $\begin{cases}x\left(t\right)=3\sin t\hfill \\ y\left(t\right)=6\cos t\hfill \end{cases}$

21. $\begin{cases}x\left(t\right)=2{\text{cos}}^{2}t\hfill \\ y\left(t\right)=-\sin t \hfill \end{cases}$

22. $\begin{cases}x\left(t\right)=\cos t+4\\ y\left(t\right)=2{\sin }^{2}t\end{cases}$

23. $\begin{cases}x\left(t\right)=t - 1\\ y\left(t\right)={t}^{2}\end{cases}$

24. $\begin{cases}x\left(t\right)=-t\\ y\left(t\right)={t}^{3}+1\end{cases}$

25. $\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)={t}^{3}-2\end{cases}$

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x\text{-}y$ table.

26. $\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)=t+4\end{cases}$

27. $\begin{cases}x\left(t\right)=4-t\\ y\left(t\right)=3t+2\end{cases}$

28. $\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)=5t\end{cases}$

29. $\begin{cases}x\left(t\right)=4t - 1\\ y\left(t\right)=4t+2\end{cases}$

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting $x\left(t\right)=t$ or by setting $y\left(t\right)=t$.

30. $y\left(x\right)=3{x}^{2}+3$

31. $y\left(x\right)=2\sin x+1$

32. $x\left(y\right)=3\mathrm{log}\left(y\right)+y$

33. $x\left(y\right)=\sqrt{y}+2y$

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x\left(t\right)=a\cos t$ and $y\left(t\right)=b\sin t$. Identify the curve.

34. $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$

35. $\frac{{x}^{2}}{16}+\frac{{y}^{2}}{36}=1$

36. ${x}^{2}+{y}^{2}=16$

37. ${x}^{2}+{y}^{2}=10$

38. Parameterize the line from $\left(3,0\right)$ to $\left(-2,-5\right)$ so that the line is at $\left(3,0\right)$ at $t=0$, and at $\left(-2,-5\right)$ at $t=1$.

39. Parameterize the line from $\left(-1,0\right)$ to $\left(3,-2\right)$ so that the line is at $\left(-1,0\right)$ at $t=0$, and at $\left(3,-2\right)$ at $t=1$.

40. Parameterize the line from $\left(-1,5\right)$ to $\left(2,3\right)$ so that the line is at $\left(-1,5\right)$ at $t=0$, and at $\left(2,3\right)$ at $t=1$.

41. Parameterize the line from $\left(4,1\right)$ to $\left(6,-2\right)$ so that the line is at $\left(4,1\right)$ at $t=0$, and at $\left(6,-2\right)$ at $t=1$.

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.

42. $\begin{cases}{x}_{1}\left(t\right)=3t\hfill \\ {y}_{1}\left(t\right)=2t - 1\hfill \end{cases}\text{ and }\begin{cases}{x}_{2}\left(t\right)=t+3\hfill \\ {y}_{2}\left(t\right)=4t - 4\hfill \end{cases}$

43. $\begin{cases}{x}_{1}\left(t\right)={t}^{2}\hfill \\ {y}_{1}\left(t\right)=2t - 1\hfill \end{cases}\text{ and }\begin{cases}{x}_{2}\left(t\right)=-t+6\hfill \\ {y}_{2}\left(t\right)=t+1\hfill \end{cases}$

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

44. $\begin{cases}{x}_{1}\left(t\right)=3{t}^{2}-3t+7\hfill \\ {y}_{1}\left(t\right)=2t+3\hfill \end{cases}$

45. $\begin{cases}{x}_{1}\left(t\right)={t}^{2}-4\hfill \\ {y}_{1}\left(t\right)=2{t}^{2}-1\hfill \end{cases}$

46. $\begin{cases}{x}_{1}\left(t\right)={t}^{4}\hfill \\ {y}_{1}\left(t\right)={t}^{3}+4\hfill \end{cases}$

47. Find two different sets of parametric equations for $y={\left(x+1\right)}^{2}$.

48. Find two different sets of parametric equations for $y=3x - 2$.

49. Find two different sets of parametric equations for $y={x}^{2}-4x+4$.