# 8.6 Parametric equations  (Page 5/6)

 Page 5 / 6

## Verbal

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?

## Algebraic

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.

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

$y=-2+2x$

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

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

$y=3\sqrt{\frac{x-1}{2}}$

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

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

$x=2{e}^{\frac{1-y}{5}}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y=1-5ln\left(\frac{x}{2}\right)$

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

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

$x=4\mathrm{log}\left(\frac{y-3}{2}\right)$

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

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

$x={\left(\frac{y}{2}\right)}^{3}-\frac{y}{2}$

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

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

$y={x}^{3}$

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

${\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{5}\right)}^{2}=1$

$\left\{\begin{array}{l}x\left(t\right)=3\mathrm{sin}\text{\hspace{0.17em}}t\hfill \\ y\left(t\right)=6\mathrm{cos}\text{\hspace{0.17em}}t\hfill \end{array}$

${y}^{2}=1-\frac{1}{2}x$

$\left\{\begin{array}{l}x\left(t\right)=\mathrm{cos}\text{\hspace{0.17em}}t+4\\ y\left(t\right)=2{\mathrm{sin}}^{2}t\end{array}$

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

$y={x}^{2}+2x+1$

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

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

$y={\left(\frac{x+1}{2}\right)}^{3}-2$

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an $x\text{-}y$ table.

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

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

$y=-3x+14$

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

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

$y=x+3$

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting $x\left(t\right)=t$ or by setting $\text{\hspace{0.17em}}y\left(t\right)=t.$

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

$y\left(x\right)=2\mathrm{sin}\text{\hspace{0.17em}}x+1$

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

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

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

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

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 $\text{\hspace{0.17em}}y\left(t\right)=b\mathrm{sin}\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ Identify the curve.

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

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

$\left\{\begin{array}{l}x\left(t\right)=4\mathrm{cos}\text{\hspace{0.17em}}t\hfill \\ y\left(t\right)=6\mathrm{sin}\text{\hspace{0.17em}}t\hfill \end{array};\text{\hspace{0.17em}}$ Ellipse

${x}^{2}+{y}^{2}=16$

${x}^{2}+{y}^{2}=10$

$\left\{\begin{array}{l}x\left(t\right)=\sqrt{10}\mathrm{cos}t\hfill \\ y\left(t\right)=\sqrt{10}\mathrm{sin}t\hfill \end{array};\text{\hspace{0.17em}}$ Circle

Parameterize the line from $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(-2,-5\right)\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}\left(-2,-5\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$

Parameterize the line from $\text{\hspace{0.17em}}\left(-1,0\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}\left(-1,0\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}\left(3,-2\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$

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

Parameterize the line from $\text{\hspace{0.17em}}\left(-1,5\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(2,3\right)$ so that the line is at $\text{\hspace{0.17em}}\left(-1,5\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$

Parameterize the line from $\text{\hspace{0.17em}}\left(4,1\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}\left(6,-2\right)\text{\hspace{0.17em}}$ so that the line is at $\text{\hspace{0.17em}}\left(4,1\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=0,\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}\left(6,-2\right)\text{\hspace{0.17em}}$ at $\text{\hspace{0.17em}}t=1.$

$\left\{\begin{array}{l}x\left(t\right)=4+2t\hfill \\ y\left(t\right)=1-3t\hfill \end{array}$

## Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.

yes, at $t=2$

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

$\left\{\begin{array}{l}{x}_{1}\left(t\right)=3{t}^{2}-3t+7\hfill \\ {y}_{1}\left(t\right)=2t+3\hfill \end{array}$

$t$ $x$ $y$
–1
0
1

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

$t$ $x$ $y$
1
2
3
$t$ $x$ $y$
1 -3 1
2 0 7
3 5 17

$\left\{\begin{array}{l}{x}_{1}\left(t\right)={t}^{4}\hfill \\ {y}_{1}\left(t\right)={t}^{3}+4\hfill \end{array}$

$t$ $x$ $y$
-1
0
1
2

## Extensions

Find two different sets of parametric equations for $\text{\hspace{0.17em}}y={\left(x+1\right)}^{2}.$

Find two different sets of parametric equations for $\text{\hspace{0.17em}}y=3x-2.$

Find two different sets of parametric equations for $\text{\hspace{0.17em}}y={x}^{2}-4x+4.$

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this