Solutions to Odd-Numbered Exercises
1. A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, [latex]x=f\left(t\right)[/latex] and [latex]y=f\left(t\right)[/latex].
3. Choose one equation to solve for [latex]t[/latex], substitute into the other equation and simplify.
5. Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.
7. [latex]y=-2+2x[/latex]
9. [latex]y=3\sqrt{\frac{x - 1}{2}}[/latex]
11. [latex]x=2{e}^{\frac{1-y}{5}}[/latex] or [latex]y=1 - 5ln\left(\frac{x}{2}\right)[/latex]
13. [latex]x=4\mathrm{log}\left(\frac{y - 3}{2}\right)[/latex]
15. [latex]x={\left(\frac{y}{2}\right)}^{3}-\frac{y}{2}[/latex]
17. [latex]y={x}^{3}[/latex]
19. [latex]{\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{5}\right)}^{2}=1[/latex]
21. [latex]{y}^{2}=1-\frac{1}{2}x[/latex]
23. [latex]y={x}^{2}+2x+1[/latex]
25. [latex]y={\left(\frac{x+1}{2}\right)}^{3}-2[/latex]
27. [latex]y=-3x+14[/latex]
29. [latex]y=x+3[/latex]
31. [latex]\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)=2\sin t+1\hfill \end{array}[/latex]
33. [latex]\begin{array}{l}x\left(t\right)=\sqrt{t}+2t\hfill \\ y\left(t\right)=t\hfill \end{array}[/latex]
35. [latex]\begin{array}{l}x\left(t\right)=4\cos t\hfill \\ y\left(t\right)=6\sin t\hfill \end{array}[/latex]; Ellipse
37. [latex]\begin{array}{l}x\left(t\right)=\sqrt{10}\cos t\hfill \\ y\left(t\right)=\sqrt{10}\sin t\hfill \end{array}[/latex]; Circle
39. [latex]\begin{array}{l}x\left(t\right)=-1+4t\hfill \\ y\left(t\right)=-2t\hfill \end{array}[/latex]
41. [latex]\begin{array}{l}x\left(t\right)=4+2t\hfill \\ y\left(t\right)=1 - 3t\hfill \end{array}[/latex]
43. yes, at [latex]t=2[/latex]
45.
| [latex]t[/latex] | [latex]x[/latex] | [latex]y[/latex] |
|---|---|---|
| 1 | -3 | 1 |
| 2 | 0 | 7 |
| 3 | 5 | 17 |
47. answers may vary: [latex]\begin{array}{l}x\left(t\right)=t - 1\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}\text{ and }\begin{array}{l}x\left(t\right)=t+1\hfill \\ y\left(t\right)={\left(t+2\right)}^{2}\hfill \end{array}[/latex]
49. answers may vary: , [latex]\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)={t}^{2}-4t+4\hfill \end{array}\text{ and }\begin{array}{l}x\left(t\right)=t+2\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}[/latex]
