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Study Guides > Precalculus II

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]

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