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What is a system of parametric equations?
A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, $\text{\hspace{0.17em}}x=f\left(t\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=f\left(t\right).$
Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.
Explain how to eliminate a parameter given a set of parametric equations.
Choose one equation to solve for $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ substitute into the other equation and simplify.
What is a benefit of writing a system of parametric equations as a Cartesian equation?
What is a benefit of using parametric equations?
Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.
Why are there many sets of parametric equations to represent on Cartesian function?
For the following exercises, eliminate the parameter $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ to rewrite the parametric equation as a Cartesian equation.
$\{\begin{array}{l}x\left(t\right)=5-t\hfill \\ y\left(t\right)=8-2t\hfill \end{array}$
$y=-2+2x$
$\{\begin{array}{l}x\left(t\right)=6-3t\hfill \\ y\left(t\right)=10-t\hfill \end{array}$
$\{\begin{array}{l}x\left(t\right)=2t+1\hfill \\ y\left(t\right)=3\sqrt{t}\hfill \end{array}$
$y=3\sqrt{\frac{x-1}{2}}$
$\{\begin{array}{l}x\left(t\right)=3t-1\hfill \\ y\left(t\right)=2{t}^{2}\hfill \end{array}$
$\{\begin{array}{l}x\left(t\right)=2{e}^{t}\hfill \\ y\left(t\right)=1-5t\hfill \end{array}$
$x=2{e}^{\frac{1-y}{5}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y=1-5ln\left(\frac{x}{2}\right)$
$\{\begin{array}{l}x\left(t\right)={e}^{-2t}\hfill \\ y\left(t\right)=2{e}^{-t}\hfill \end{array}$
$\{\begin{array}{l}x(t)=4\text{log}(t)\hfill \\ y(t)=3+2t\hfill \end{array}$
$x=4\mathrm{log}\left(\frac{y-3}{2}\right)$
$\{\begin{array}{l}x(t)=\text{log}(2t)\hfill \\ y(t)=\sqrt{t-1}\hfill \end{array}$
$\{\begin{array}{l}x\left(t\right)={t}^{3}-t\hfill \\ y\left(t\right)=2t\hfill \end{array}$
$x={\left(\frac{y}{2}\right)}^{3}-\frac{y}{2}$
$\{\begin{array}{l}x\left(t\right)=t-{t}^{4}\hfill \\ y\left(t\right)=t+2\hfill \end{array}$
$\{\begin{array}{l}x\left(t\right)={e}^{2t}\hfill \\ y\left(t\right)={e}^{6t}\hfill \end{array}$
$y={x}^{3}$
$\{\begin{array}{l}x\left(t\right)={t}^{5}\hfill \\ y\left(t\right)={t}^{10}\hfill \end{array}$
$\{\begin{array}{l}x(t)=4\text{cos}\text{\hspace{0.17em}}t\hfill \\ y(t)=5\mathrm{sin}\text{\hspace{0.17em}}t\hfill \end{array}$
${\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{5}\right)}^{2}=1$
$\{\begin{array}{l}x\left(t\right)=3\mathrm{sin}\text{\hspace{0.17em}}t\hfill \\ y\left(t\right)=6\mathrm{cos}\text{\hspace{0.17em}}t\hfill \end{array}$
$\{\begin{array}{l}x(t)=2{\text{cos}}^{2}t\hfill \\ y(t)=-\mathrm{sin}\text{\hspace{0.17em}}t\hfill \end{array}$
${y}^{2}=1-\frac{1}{2}x$
$\{\begin{array}{l}x(t)=\mathrm{cos}\text{\hspace{0.17em}}t+4\\ y(t)=2{\mathrm{sin}}^{2}t\end{array}$
$\{\begin{array}{l}x(t)=t-1\\ y(t)={t}^{2}\end{array}$
$y={x}^{2}+2x+1$
$\{\begin{array}{l}x(t)=-t\\ y(t)={t}^{3}+1\end{array}$
$\{\begin{array}{l}x(t)=2t-1\\ y(t)={t}^{3}-2\end{array}$
$y={\left(\frac{x+1}{2}\right)}^{3}-2$
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x\text{-}y$ table.
$\{\begin{array}{l}x(t)=2t-1\\ y(t)=t+4\end{array}$
$\{\begin{array}{l}x(t)=4-t\\ y(t)=3t+2\end{array}$
$y=-3x+14$
$\{\begin{array}{l}x(t)=2t-1\\ y(t)=5t\end{array}$
$\{\begin{array}{l}x(t)=4t-1\\ y(t)=4t+2\end{array}$
$y=x+3$
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting $x\left(t\right)=t$ or by setting $\text{\hspace{0.17em}}y(t)=t.$
$y\left(x\right)=3{x}^{2}+3$
$y\left(x\right)=2\mathrm{sin}\text{\hspace{0.17em}}x+1$
$\{\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)=2\mathrm{sin}t+1\hfill \end{array}$
$x\left(y\right)=3\mathrm{log}\left(y\right)+y$
$x\left(y\right)=\sqrt{y}+2y$
$\{\begin{array}{l}x\left(t\right)=\sqrt{t}+2t\hfill \\ y\left(t\right)=t\hfill \end{array}$
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using $x\left(t\right)=a\mathrm{cos}\text{\hspace{0.17em}}t$ and $\text{\hspace{0.17em}}y(t)=b\mathrm{sin}\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ Identify the curve.
$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$
$\frac{{x}^{2}}{16}+\frac{{y}^{2}}{36}=1$
$\{\begin{array}{l}x\left(t\right)=4\mathrm{cos}\text{\hspace{0.17em}}t\hfill \\ y\left(t\right)=6\mathrm{sin}\text{\hspace{0.17em}}t\hfill \end{array};\text{\hspace{0.17em}}$ Ellipse
${x}^{2}+{y}^{2}=16$
${x}^{2}+{y}^{2}=10$
$\{\begin{array}{l}x\left(t\right)=\sqrt{10}\mathrm{cos}t\hfill \\ y\left(t\right)=\sqrt{10}\mathrm{sin}t\hfill \end{array};\text{\hspace{0.17em}}$ Circle
Parameterize the line from $\text{\hspace{0.17em}}(3,0)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}(\mathrm{-2},\mathrm{-5})\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}(3,0)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}(\mathrm{-2},\mathrm{-5})\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$
Parameterize the line from $\text{\hspace{0.17em}}(\mathrm{-1},0)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}(3,\mathrm{-2})\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}(\mathrm{-1},0)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}(3,\mathrm{-2})\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$
$\{\begin{array}{l}x\left(t\right)=-1+4t\hfill \\ y\left(t\right)=-2t\hfill \end{array}$
Parameterize the line from $\text{\hspace{0.17em}}(\mathrm{-1},5)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}(2,3)$ so that the line is at $\text{\hspace{0.17em}}(\mathrm{-1},5)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}(2,3)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$
Parameterize the line from $\text{\hspace{0.17em}}(4,1)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}(6,\mathrm{-2})\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}(4,1)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}(6,\mathrm{-2})\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$
$\{\begin{array}{l}x\left(t\right)=4+2t\hfill \\ y\left(t\right)=1-3t\hfill \end{array}$
For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
$\{\begin{array}{l}{x}_{1}(t)=3t\hfill \\ {y}_{1}(t)=2t-1\hfill \end{array}\text{and}\{\begin{array}{l}{x}_{2}(t)=t+3\hfill \\ {y}_{2}(t)=4t-4\hfill \end{array}$
$\{\begin{array}{l}{x}_{1}(t)={t}^{2}\hfill \\ {y}_{1}(t)=2t-1\hfill \end{array}\text{and}\{\begin{array}{l}{x}_{2}(t)=-t+6\hfill \\ {y}_{2}(t)=t+1\hfill \end{array}$
yes, at $t=2$
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.
$\{\begin{array}{l}{x}_{1}\left(t\right)=3{t}^{2}-3t+7\hfill \\ {y}_{1}\left(t\right)=2t+3\hfill \end{array}$
$t$ | $x$ | $y$ |
---|---|---|
–1 | ||
0 | ||
1 |
$\{\begin{array}{l}{x}_{1}\left(t\right)={t}^{2}-4\hfill \\ {y}_{1}\left(t\right)=2{t}^{2}-1\hfill \end{array}$
$t$ | $x$ | $y$ |
---|---|---|
1 | ||
2 | ||
3 |
$t$ | $x$ | $y$ |
---|---|---|
1 | -3 | 1 |
2 | 0 | 7 |
3 | 5 | 17 |
$\{\begin{array}{l}{x}_{1}\left(t\right)={t}^{4}\hfill \\ {y}_{1}\left(t\right)={t}^{3}+4\hfill \end{array}$
$t$ | $x$ | $y$ |
---|---|---|
-1 | ||
0 | ||
1 | ||
2 |
Find two different sets of parametric equations for $\text{\hspace{0.17em}}y={\left(x+1\right)}^{2}.$
answers may vary: $\text{\hspace{0.17em}}\{\begin{array}{l}x\left(t\right)=t-1\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}\text{and}\{\begin{array}{l}x\left(t\right)=t+1\hfill \\ y\left(t\right)={\left(t+2\right)}^{2}\hfill \end{array}$
Find two different sets of parametric equations for $\text{\hspace{0.17em}}y=3x-2.$
Find two different sets of parametric equations for $\text{\hspace{0.17em}}y={x}^{2}-4x+4.$
answers may vary: , $\text{\hspace{0.17em}}\{\begin{array}{l}x\left(t\right)=t\hfill \\ y\left(t\right)={t}^{2}-4t+4\hfill \end{array}\text{and}\{\begin{array}{l}x\left(t\right)=t+2\hfill \\ y\left(t\right)={t}^{2}\hfill \end{array}$
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