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 [latex]t[/latex] to rewrite the parametric equation as a Cartesian equation.
7. [latex]\begin{cases}x\left(t\right)=5-t\hfill \\ y\left(t\right)=8 - 2t\hfill \end{cases}[/latex]
8. [latex]\begin{cases}x\left(t\right)=6 - 3t\hfill \\ y\left(t\right)=10-t\hfill \end{cases}[/latex]
9. [latex]\begin{cases}x\left(t\right)=2t+1\hfill \\ y\left(t\right)=3\sqrt{t}\hfill \end{cases}[/latex]
10. [latex]\begin{cases}x\left(t\right)=3t - 1\hfill \\ y\left(t\right)=2{t}^{2}\hfill \end{cases}[/latex]
11. [latex]\begin{cases}x\left(t\right)=2{e}^{t}\hfill \\ y\left(t\right)=1 - 5t\hfill \end{cases}[/latex]
12. [latex]\begin{cases}x\left(t\right)={e}^{-2t}\hfill \\ y\left(t\right)=2{e}^{-t}\hfill \end{cases}[/latex]
13. [latex]\begin{cases}x\left(t\right)=4\text{log}\left(t\right)\hfill \\ y\left(t\right)=3+2t\hfill \end{cases}[/latex]
14. [latex]\begin{cases}x\left(t\right)=\text{log}\left(2t\right)\hfill \\ y\left(t\right)=\sqrt{t - 1}\hfill \end{cases}[/latex]
15. [latex]\begin{cases}x\left(t\right)={t}^{3}-t\hfill \\ y\left(t\right)=2t\hfill \end{cases}[/latex]
16. [latex]\begin{cases}x\left(t\right)=t-{t}^{4}\hfill \\ y\left(t\right)=t+2\hfill \end{cases}[/latex]
17. [latex]\begin{cases}x\left(t\right)={e}^{2t}\hfill \\ y\left(t\right)={e}^{6t}\hfill \end{cases}[/latex]
18. [latex]\begin{cases}x\left(t\right)={t}^{5}\hfill \\ y\left(t\right)={t}^{10}\hfill \end{cases}[/latex]
19. [latex]\begin{cases}x\left(t\right)=4\text{cos}t\hfill \\ y\left(t\right)=5\sin t \hfill \end{cases}[/latex]
20. [latex]\begin{cases}x\left(t\right)=3\sin t\hfill \\ y\left(t\right)=6\cos t\hfill \end{cases}[/latex]
21. [latex]\begin{cases}x\left(t\right)=2{\text{cos}}^{2}t\hfill \\ y\left(t\right)=-\sin t \hfill \end{cases}[/latex]
22. [latex]\begin{cases}x\left(t\right)=\cos t+4\\ y\left(t\right)=2{\sin }^{2}t\end{cases}[/latex]
23. [latex]\begin{cases}x\left(t\right)=t - 1\\ y\left(t\right)={t}^{2}\end{cases}[/latex]
24. [latex]\begin{cases}x\left(t\right)=-t\\ y\left(t\right)={t}^{3}+1\end{cases}[/latex]
25. [latex]\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)={t}^{3}-2\end{cases}[/latex]
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\text{-}y[/latex] table.
26. [latex]\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)=t+4\end{cases}[/latex]
27. [latex]\begin{cases}x\left(t\right)=4-t\\ y\left(t\right)=3t+2\end{cases}[/latex]
28. [latex]\begin{cases}x\left(t\right)=2t - 1\\ y\left(t\right)=5t\end{cases}[/latex]
29. [latex]\begin{cases}x\left(t\right)=4t - 1\\ y\left(t\right)=4t+2\end{cases}[/latex]
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\left(t\right)=t[/latex] or by setting [latex]y\left(t\right)=t[/latex].
30. [latex]y\left(x\right)=3{x}^{2}+3[/latex]
31. [latex]y\left(x\right)=2\sin x+1[/latex]
32. [latex]x\left(y\right)=3\mathrm{log}\left(y\right)+y[/latex]
33. [latex]x\left(y\right)=\sqrt{y}+2y[/latex]
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\left(t\right)=a\cos t[/latex] and [latex]y\left(t\right)=b\sin t[/latex]. Identify the curve.
34. [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1[/latex]
35. [latex]\frac{{x}^{2}}{16}+\frac{{y}^{2}}{36}=1[/latex]
36. [latex]{x}^{2}+{y}^{2}=16[/latex]
37. [latex]{x}^{2}+{y}^{2}=10[/latex]
38. Parameterize the line from [latex]\left(3,0\right)[/latex] to [latex]\left(-2,-5\right)[/latex] so that the line is at [latex]\left(3,0\right)[/latex] at [latex]t=0[/latex], and at [latex]\left(-2,-5\right)[/latex] at [latex]t=1[/latex].
39. Parameterize the line from [latex]\left(-1,0\right)[/latex] to [latex]\left(3,-2\right)[/latex] so that the line is at [latex]\left(-1,0\right)[/latex] at [latex]t=0[/latex], and at [latex]\left(3,-2\right)[/latex] at [latex]t=1[/latex].
40. Parameterize the line from [latex]\left(-1,5\right)[/latex] to [latex]\left(2,3\right)[/latex] so that the line is at [latex]\left(-1,5\right)[/latex] at [latex]t=0[/latex], and at [latex]\left(2,3\right)[/latex] at [latex]t=1[/latex].
41. Parameterize the line from [latex]\left(4,1\right)[/latex] to [latex]\left(6,-2\right)[/latex] so that the line is at [latex]\left(4,1\right)[/latex] at [latex]t=0[/latex], and at [latex]\left(6,-2\right)[/latex] at [latex]t=1[/latex].
For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
42. [latex]\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}[/latex]
43. [latex]\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}[/latex]
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.
44. [latex]\begin{cases}{x}_{1}\left(t\right)=3{t}^{2}-3t+7\hfill \\ {y}_{1}\left(t\right)=2t+3\hfill \end{cases}[/latex]
| [latex]t[/latex] | [latex]x[/latex] | [latex]y[/latex] |
|---|---|---|
| –1 | ||
| 0 | ||
| 1 |
45. [latex]\begin{cases}{x}_{1}\left(t\right)={t}^{2}-4\hfill \\ {y}_{1}\left(t\right)=2{t}^{2}-1\hfill \end{cases}[/latex]
| [latex]t[/latex] | [latex]x[/latex] | [latex]y[/latex] |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 |
46. [latex]\begin{cases}{x}_{1}\left(t\right)={t}^{4}\hfill \\ {y}_{1}\left(t\right)={t}^{3}+4\hfill \end{cases}[/latex]
| [latex]t[/latex] | [latex]x[/latex] | [latex]y[/latex] |
|---|---|---|
| -1 | ||
| 0 | ||
| 1 | ||
| 2 |
47. Find two different sets of parametric equations for [latex]y={\left(x+1\right)}^{2}[/latex].
48. Find two different sets of parametric equations for [latex]y=3x - 2[/latex].
49. Find two different sets of parametric equations for [latex]y={x}^{2}-4x+4[/latex].