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, $x=f\left(t\right)$ and $y=f\left(t\right)$.

3. Choose one equation to solve for $t$, 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. $y=-2+2x$

9. $y=3\sqrt{\frac{x - 1}{2}}$

11. $x=2{e}^{\frac{1-y}{5}}$ or $y=1 - 5ln\left(\frac{x}{2}\right)$

13. $x=4\mathrm{log}\left(\frac{y - 3}{2}\right)$

15. $x={\left(\frac{y}{2}\right)}^{3}-\frac{y}{2}$

17. $y={x}^{3}$

19. ${\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{5}\right)}^{2}=1$

21. ${y}^{2}=1-\frac{1}{2}x$

23. $y={x}^{2}+2x+1$

25. $y={\left(\frac{x+1}{2}\right)}^{3}-2$

27. $y=-3x+14$

29. $y=x+3$

31. $\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)=2\sin t+1\hfill \end{array}$

33. $\begin{array}{l}x\left(t\right)=\sqrt{t}+2t\hfill \\ y\left(t\right)=t\hfill \end{array}$

35. $\begin{array}{l}x\left(t\right)=4\cos t\hfill \\ y\left(t\right)=6\sin t\hfill \end{array}$; Ellipse

37. $\begin{array}{l}x\left(t\right)=\sqrt{10}\cos t\hfill \\ y\left(t\right)=\sqrt{10}\sin t\hfill \end{array}$; Circle

39. $\begin{array}{l}x\left(t\right)=-1+4t\hfill \\ y\left(t\right)=-2t\hfill \end{array}$

41. $\begin{array}{l}x\left(t\right)=4+2t\hfill \\ y\left(t\right)=1 - 3t\hfill \end{array}$

43. yes, at $t=2$

45.

47. answers may vary: $\begin{array}{l}x\left(t\right)=t - 1\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}\text{ and }\begin{array}{l}x\left(t\right)=t+1\hfill \\ y\left(t\right)={\left(t+2\right)}^{2}\hfill \end{array}$

49. answers may vary: , $\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)={t}^{2}-4t+4\hfill \end{array}\text{ and }\begin{array}{l}x\left(t\right)=t+2\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}$