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Study Guides > Precalculus II

Solutions for Parametric Equations

Solutions to Try Its

1.
tt x(t)x\left(t\right) y(t)y\left(t\right)
1-1 4-4 22
00 3-3 44
11 2-2 66
22 1-1 88
2. x(t)=t32ty(t)=t\begin{array}{l}x\left(t\right)={t}^{3}-2t\\ y\left(t\right)=t\end{array} 3. y=512x3y=5-\sqrt{\frac{1}{2}x - 3} 4. y=lnxy=\mathrm{ln}\sqrt{x} 5. x24+y29=1\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1 6. y=x2y={x}^{2}

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(t)x=f\left(t\right) and y=f(t)y=f\left(t\right). 3. Choose one equation to solve for tt, 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+2xy=-2+2x 9. y=3x12y=3\sqrt{\frac{x - 1}{2}} 11. x=2e1y5x=2{e}^{\frac{1-y}{5}} or y=15ln(x2)y=1 - 5ln\left(\frac{x}{2}\right) 13. x=4log(y32)x=4\mathrm{log}\left(\frac{y - 3}{2}\right) 15. x=(y2)3y2x={\left(\frac{y}{2}\right)}^{3}-\frac{y}{2} 17. y=x3y={x}^{3} 19. (x4)2+(y5)2=1{\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{5}\right)}^{2}=1 21. y2=112x{y}^{2}=1-\frac{1}{2}x 23. y=x2+2x+1y={x}^{2}+2x+1 25. y=(x+12)32y={\left(\frac{x+1}{2}\right)}^{3}-2 27. y=3x+14y=-3x+14 29. y=x+3y=x+3 31. x(t)=ty(t)=2sint+1\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)=2\sin t+1\hfill \end{array} 33. x(t)=t+2ty(t)=t\begin{array}{l}x\left(t\right)=\sqrt{t}+2t\hfill \\ y\left(t\right)=t\hfill \end{array} 35. x(t)=4costy(t)=6sint\begin{array}{l}x\left(t\right)=4\cos t\hfill \\ y\left(t\right)=6\sin t\hfill \end{array}; Ellipse 37. x(t)=10costy(t)=10sint\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. x(t)=1+4ty(t)=2t\begin{array}{l}x\left(t\right)=-1+4t\hfill \\ y\left(t\right)=-2t\hfill \end{array} 41. x(t)=4+2ty(t)=13t\begin{array}{l}x\left(t\right)=4+2t\hfill \\ y\left(t\right)=1 - 3t\hfill \end{array} 43. yes, at t=2t=2 45.
tt xx yy
1 -3 1
2 0 7
3 5 17
47. answers may vary: x(t)=t1y(t)=t2 and x(t)=t+1y(t)=(t+2)2\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: , x(t)=ty(t)=t24t+4 and x(t)=t+2y(t)=t2\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}

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.