# 10.6 Parametric equations

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In this section, you will:
• Parameterize a curve.
• Eliminate the parameter.
• Find a rectangular equation for a curve defined parametrically.
• Find parametric equations for curves defined by rectangular equations.

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in [link] . At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.

In this section, we will consider sets of equations given by $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\left(t\right)\text{\hspace{0.17em}}$ where $t$ is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations : we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.

## Parameterizing a curve

When an object moves along a curve—or curvilinear path —in a given direction and in a given amount of time, the position of the object in the plane is given by the x- coordinate and the y- coordinate. However, both $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ vary over time and so are functions of time. For this reason, we add another variable, the parameter    , upon which both $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ are dependent functions. In the example in the section opener, the parameter is time, $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ The $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ position of the moon at time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ is represented as the function $\text{\hspace{0.17em}}x\left(t\right),\text{\hspace{0.17em}}$ and the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ position of the moon at time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ is represented as the function $\text{\hspace{0.17em}}y\left(t\right).\text{\hspace{0.17em}}$ Together, $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\left(t\right)\text{\hspace{0.17em}}$ are called parametric equations, and generate an ordered pair $\text{\hspace{0.17em}}\left(x\left(t\right),\text{\hspace{0.17em}}y\left(t\right)\right).\text{\hspace{0.17em}}$ Parametric equations primarily describe motion and direction.

When we parameterize a curve, we are translating a single equation in two variables, such as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y ,$ into an equivalent pair of equations in three variables, $\text{\hspace{0.17em}}x,y,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.

When we graph parametric equations, we can observe the individual behaviors of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and of $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ There are a number of shapes that cannot be represented in the form $\text{\hspace{0.17em}}y=f\left(x\right),\text{\hspace{0.17em}}$ meaning that they are not functions. For example, consider the graph of a circle, given as $\text{\hspace{0.17em}}{r}^{2}={x}^{2}+{y}^{2}.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ gives $\text{\hspace{0.17em}}y=±\sqrt{{r}^{2}-{x}^{2}},\text{\hspace{0.17em}}$ or two equations: $\text{\hspace{0.17em}}{y}_{1}=\sqrt{{r}^{2}-{x}^{2}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}_{2}=-\sqrt{{r}^{2}-{x}^{2}}.\text{\hspace{0.17em}}$ If we graph $\text{\hspace{0.17em}}{y}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}_{2}\text{\hspace{0.17em}}$ together, the graph will not pass the vertical line test, as shown in [link] . Thus, the equation for the graph of a circle is not a function.

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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