# 10.6 Parametric equations

 Page 1 / 6
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.

I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?