Solutions for Parametric Equations

Solutions to Try Its

1.

[latex]t[/latex] [latex]x\left(t\right)[/latex] [latex]y\left(t\right)[/latex]
[latex]-1[/latex] [latex]-4[/latex] [latex]2[/latex]
[latex]0[/latex] [latex]-3[/latex] [latex]4[/latex]
[latex]1[/latex] [latex]-2[/latex] [latex]6[/latex]
[latex]2[/latex] [latex]-1[/latex] [latex]8[/latex]

2. [latex]\begin{array}{l}x\left(t\right)={t}^{3}-2t\\ y\left(t\right)=t\end{array}[/latex]

3. [latex]y=5-\sqrt{\frac{1}{2}x - 3}[/latex]

4. [latex]y=\mathrm{ln}\sqrt{x}[/latex]

5. [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1[/latex]

6. [latex]y={x}^{2}[/latex]

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]