8.6 Parametric equations  (Page 5/6)

What is a system of parametric equations?

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.

What is a benefit of writing a system of parametric equations as a Cartesian equation?

What is a benefit of using parametric equations?

Why are there many sets of parametric equations to represent on Cartesian function?

$\{\begin{array}{l}x\left(t\right)=5-t\hfill \\ y\left(t\right)=8-2t\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=6-3t\hfill \\ y\left(t\right)=10-t\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=2t+1\hfill \\ y\left(t\right)=3\sqrt{t}\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=3t-1\hfill \\ y\left(t\right)=2t^2\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=2e^t\hfill \\ y\left(t\right)=1-5t\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=e^{-2t}\hfill \\ y\left(t\right)=2e^{-t}\hfill \end{array}$

$\{\begin{array}{l}x(t)=4\text{log}(t)\hfill \\ y(t)=3+2t\hfill \end{array}$

$\{\begin{array}{l}x(t)=\text{log}(2t)\hfill \\ y(t)=\sqrt{t-1}\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=t^3-t\hfill \\ y\left(t\right)=2t\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=t-t^4\hfill \\ y\left(t\right)=t+2\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=e^{2t}\hfill \\ y\left(t\right)=e^{6t}\hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=t^5\hfill \\ y\left(t\right)=t^{10}\hfill \end{array}$

$\{\begin{array}{l}x(t)=4\text{cos}t\hfill \\ y(t)=5\sin t \hfill \end{array}$

$\{\begin{array}{l}x\left(t\right)=3\sin t\hfill \\ y\left(t\right)=6\cos t\hfill \end{array}$

$\{\begin{array}{l}x(t)=2{\text{cos}}^{2}t\hfill \\ y(t)=-\sin t \hfill \end{array}$

$\{\begin{array}{l}x(t)=\cos t+4\\ y(t)=2{\sin }^{2}t\end{array}$

$\{\begin{array}{l}x(t)=t-1\\ y(t)=t^2\end{array}$

$\{\begin{array}{l}x(t)=-t\\ y(t)=t^3+1\end{array}$

$\{\begin{array}{l}x(t)=2t-1\\ y(t)=t^3-2\end{array}$

$\{\begin{array}{l}x(t)=2t-1\\ y(t)=t+4\end{array}$

$\{\begin{array}{l}x(t)=4-t\\ y(t)=3t+2\end{array}$

$\{\begin{array}{l}x(t)=2t-1\\ y(t)=5t\end{array}$

$\{\begin{array}{l}x(t)=4t-1\\ y(t)=4t+2\end{array}$

$y\left(x\right)=3x^2+3$

$y\left(x\right)=2\sin x+1$

$x\left(y\right)=3\log \left(y\right)+y$

$x\left(y\right)=\sqrt{y}+2y$

$\frac{x^2}{4}+\frac{y^2}{9}=1$

$\frac{x^2}{16}+\frac{y^2}{36}=1$

$x^2+y^2=16$

$x^2+y^2=10$

Parameterize the line from $(3,0)$ to $(\mathrm{-2},\mathrm{-5})$ so that the line is at $(3,0)$ at $t=0,$ and at $(\mathrm{-2},\mathrm{-5})$ at $t=1.$

Parameterize the line from $(\mathrm{-1},0)$ to $(3,\mathrm{-2})$ so that the line is at $(\mathrm{-1},0)$ at $t=0,$ and at $(3,\mathrm{-2})$ at $t=1.$

Parameterize the line from $(\mathrm{-1},5)$ to $(2,3)$ so that the line is at $(\mathrm{-1},5)$ at $t=0,$ and at $(2,3)$ at $t=1.$

Parameterize the line from $(4,1)$ to $(6,\mathrm{-2})$ so that the line is at $(4,1)$ at $t=0,$ and at $(6,\mathrm{-2})$ at $t=1.$

$\{\begin{array}{l}{x}_1(t)=3t\hfill \\ y_1(t)=2t-1\hfill \end{array}\text{ and }\{\begin{array}{l}{x}_2(t)=t+3\hfill \\ y_2(t)=4t-4\hfill \end{array}$

$\{\begin{array}{l}{x}_1(t)=t^2\hfill \\ y_1(t)=2t-1\hfill \end{array}\text{ and }\{\begin{array}{l}{x}_2(t)=-t+6\hfill \\ y_2(t)=t+1\hfill \end{array}$

$\{\begin{array}{l}{x}_1\left(t\right)=3t^2-3t+7\hfill \\ y_1\left(t\right)=2t+3\hfill \end{array}$

$\{\begin{array}{l}{x}_1\left(t\right)=t^2-4\hfill \\ y_1\left(t\right)=2t^2-1\hfill \end{array}$

$\{\begin{array}{l}{x}_1\left(t\right)=t^4\hfill \\ y_1\left(t\right)=t^3+4\hfill \end{array}$

Find two different sets of parametric equations for $y={\left(x+1\right)}^2.$

Find two different sets of parametric equations for $y=3x-2.$

Find two different sets of parametric equations for $y=x^2-4x+4.$