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 |
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
