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Key concepts

  • Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.
  • It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.
  • There is always more than one way to parameterize a curve.
  • Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.

For the following exercises, sketch the curves below by eliminating the parameter t . Give the orientation of the curve.

x = t 2 + 2 t , y = t + 1


A parabola open to the right with (−1, 0) being the point furthest the left with arrow going from the bottom through (−1, 0) and up.
orientation: bottom to top

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x = cos ( t ) , y = sin ( t ) , ( 0 , 2 π ]

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x = 2 t + 4 , y = t 1


A straight line passing through (0, −3) and (6, 0) with arrow pointing up and to the right.
orientation: left to right

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x = 3 t , y = 2 t 3 , 1.5 t 3

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For the following exercises, eliminate the parameter and sketch the graphs.

x = 2 t 2 , y = t 4 + 1

y = x 2 4 + 1
Half a parabola starting at the origin and passing through (2, 2) with arrow pointed up and to the right.

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For the following exercises, use technology (CAS or calculator) to sketch the parametric equations.

[T] x = t 2 + t , y = t 2 1

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[T] x = e t , y = e 2 t 1


A curve going through (1, 0) and (0, 3) with arrow pointing up and to the left.

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[T] x = 3 cos t , y = 4 sin t

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[T] x = sec t , y = cos t


A graph with asymptotes at the x and y axes. There is a portion of the graph in the third quadrant with arrow pointing down and to the right. There is a portion of the graph in the first quadrant with arrow pointing down and to the right.

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For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.

x = 6 sin ( 2 θ ) , y = 4 cos ( 2 θ )


An ellipse with minor axis vertical and of length 8 and major axis horizontal and of length 12 that is centered at the origin. The arrows go counterclockwise.

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x = cos θ , y = 2 sin ( 2 θ )

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x = 3 2 cos θ , y = −5 + 3 sin θ


An ellipse in the fourth quadrant with minor axis horizontal and of length 4 and major axis vertical and of length 6. The arrows go clockwise.

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x = 4 + 2 cos θ , y = −1 + sin θ

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x = sec t , y = tan t


A graph with asymptotes at y = x and y = −x. The first part of the graph occurs in the second and third quadrants with vertex at (−1, 0). The second part of the graph occurs in the first and fourth quadrants with vertex as (1, 0).
Asymptotes are y = x and y = x

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x = ln ( 2 t ) , y = t 2

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x = e −2 t , y = e 3 t

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x = 4 sec θ , y = 3 tan θ

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For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.

x = t 2 1 , y = t 2

x = 4 y 2 1 ; domain: x [ 1 , ) .

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x = 1 t + 1 , y = t 1 + t , t > −1

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x = 4 cos θ , y = 3 sin θ , t ( 0 , 2 π ]

x 2 16 + y 2 9 = 1 ; domain x [ −4 , 4 ] .

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x = 2 t 3 , y = 6 t 7

y = 3 x + 2 ; domain: all real numbers.

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x = 1 + cos t , y = 3 sin t

( x 1 ) 2 + ( y 3 ) 2 = 1 ; domain: x [ 0 , 2 ] .

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x = sec t , y = tan t , π t < 3 π 2

y = x 2 1 ; domain: x [ −1 , 1 ] .

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x = 2 cosh t , y = 4 sinh t

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x = cos ( 2 t ) , y = sin t

y 2 = 1 x 2 ; domain: x [ 2 , ) ( , −2 ] .

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x = 4 t + 3 , y = 16 t 2 9

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x = t 2 , y = 2 ln t , t 1

y = ln x ; domain: x ( 0 , ) .

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x = t 3 , y = 3 ln t , t 1

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x = t n , y = n ln t , t 1 , where n is a natural number

y = ln x ; domain: x ( 0 , ) .

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x = ln ( 5 t ) y = ln ( t 2 ) where 1 t e

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x = 2 sin ( 8 t ) y = 2 cos ( 8 t )

x 2 + y 2 = 4 ; domain: x [ −2 , 2 ] .

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x = tan t y = sec 2 t 1

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For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.

x = 3 t + 4 y = 5 t 2

line

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x = 2 t + 1 y = t 2 3

parabola

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x = 2 cos ( 3 t ) y = 2 sin ( 3 t )

circle

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x = 3 cos t y = 4 sin t

ellipse

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x = 2 cos ( 3 t ) y = 5 sin ( 3 t )

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x = 3 cosh ( 4 t ) y = 4 sinh ( 4 t )

hyperbola

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x = 2 cosh t y = 2 sinh t

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Show that x = h + r cos θ y = k + r sin θ represents the equation of a circle.

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Use the equations in the preceding problem to find a set of parametric equations for a circle whose radius is 5 and whose center is ( −2 , 3 ) .

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For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.

[T] x = θ + sin θ y = 1 cos θ

The equations represent a cycloid.
A graph starting at (−6, 0) increasing rapidly to a sharp point at (−3, 2) and then decreasing rapidly to the origin. The graph is symmetric about the y axis, so the graph increases rapidly to (3, 2) before decreasing rapidly to (6, 0).

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[T] x = 2 t 2 sin t y = 2 2 cos t

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[T] x = t 0.5 sin t y = 1 1.5 cos t


A graph starting at roughly (−6, 0) increasing to a rounded point and then decreasing to roughly (0, −0.5). The graph is symmetric about the y axis, so the graph increases to a rounded point before decreasing to roughly (6, 0).

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An airplane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 meters must drop an emergency package on a target on the ground. The trajectory of the package is given by x = 100 t , y = −4.9 t 2 + 4000 , t 0 where the origin is the point on the ground directly beneath the plane at the moment of release. How many horizontal meters before the target should the package be released in order to hit the target?

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The trajectory of a bullet is given by x = v 0 ( cos α ) t y = v 0 ( sin α ) t 1 2 g t 2 where v 0 = 500 m/s, g = 9.8 = 9.8 m/s 2 , and α = 30 degrees . When will the bullet hit the ground? How far from the gun will the bullet hit the ground?

22,092 meters at approximately 51 seconds.

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[T] Use technology to sketch the curve represented by x = sin ( 4 t ) , y = sin ( 3 t ) , 0 t 2 π .

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[T] Use technology to sketch x = 2 tan ( t ) , y = 3 sec ( t ) , π < t < π .


A graph with asymptotes roughly near y = x and y = −x. The first part of the graph is in the first and second quadrants with vertex near (0, 3). The second part of the graph is in the third and fourth quadrants with vertex near (0, −3).

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Sketch the curve known as an epitrochoid , which gives the path of a point on a circle of radius b as it rolls on the outside of a circle of radius a . The equations are

x = ( a + b ) cos t c · cos [ ( a + b ) t b ] y = ( a + b ) sin t c · sin [ ( a + b ) t b ] .
Let a = 1 , b = 2 , c = 1 .

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[T] Use technology to sketch the spiral curve given by x = t cos ( t ) , y = t sin ( t ) from −2 π t 2 π .


A graph starting at roughly (−6, −1) decreasing to a minimum in the third quadrant near (−1, −4.8) increasing through roughly (0, −4.7) and (3, 0) to a maximum near (1, 1.9) before decreasing through (0, 1.5) to the origin. The graph is symmetric about the y axis, so the graph increases through (0, 1.5) to a maximum in the second quadrant, decreases again through (0, −4.7), and then increases to (6, −1).

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[T] Use technology to graph the curve given by the parametric equations x = 2 cot ( t ) , y = 1 cos ( 2 t ) , π / 2 t π / 2 . This curve is known as the witch of Agnesi.

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[T] Sketch the curve given by parametric equations x = cosh ( t ) y = sinh ( t ) , where −2 t 2 .


A vaguely parabolic graph with vertex at the origin that is open to the right.

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Practice Key Terms 7

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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