<< Chapter < Page Chapter >> Page >

Notice in this definition that x and y are used in two ways. The first is as functions of the independent variable t. As t varies over the interval I , the functions x ( t ) and y ( t ) generate a set of ordered pairs ( x , y ) . This set of ordered pairs generates the graph of the parametric equations. In this second usage, to designate the ordered pairs, x and y are variables. It is important to distinguish the variables x and y from the functions x ( t ) and y ( t ) .

Graphing a parametrically defined curve

Sketch the curves described by the following parametric equations:

  1. x ( t ) = t 1 , y ( t ) = 2 t + 4 , −3 t 2
  2. x ( t ) = t 2 3 , y ( t ) = 2 t + 1 , −2 t 3
  3. x ( t ) = 4 cos t , y ( t ) = 4 sin t , 0 t 2 π
  1. To create a graph of this curve, first set up a table of values. Since the independent variable in both x ( t ) and y ( t ) is t , let t appear in the first column. Then x ( t ) and y ( t ) will appear in the second and third columns of the table.
    t x ( t ) y ( t )
    −3 −4 −2
    −2 −3 0
    −1 −2 2
    0 −1 4
    1 0 6
    2 1 8

    The second and third columns in this table provide a set of points to be plotted. The graph of these points appears in [link] . The arrows on the graph indicate the orientation    of the graph, that is, the direction that a point moves on the graph as t varies from −3 to 2.
    A straight line going from (−4, −2) through (−3, 0), (−2, 2), and (0, 6) to (1, 8) with arrow pointed up and to the right. The point (−4, −2) is marked t = −3, the point (−2, 2) is marked t = −1, and the point (1, 8) is marked t = 2. On the graph there are also written three equations: x(t) = t −1, y(t) = 2t + 4, and −3 ≤ t ≤ 2.
    Graph of the plane curve described by the parametric equations in part a.
  2. To create a graph of this curve, again set up a table of values.
    t x ( t ) y ( t )
    −2 1 −3
    −1 −2 −1
    0 −3 1
    1 −2 3
    2 1 5
    3 6 7

    The second and third columns in this table give a set of points to be plotted ( [link] ). The first point on the graph (corresponding to t = −2 ) has coordinates ( 1 , −3 ) , and the last point (corresponding to t = 3 ) has coordinates ( 6 , 7 ) . As t progresses from −2 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.
    A curved line going from (1, −3) through (−3, 1) to (6, 7) with arrow pointing in that order. The point (1, −3) is marked t = −2, the point (−3, 1) is marked t = 0, and the point (6, 7) is marked t = 3. On the graph there are also written three equations: x(t) = t2 − 3, y(t) = 2t + 1, and −2 ≤ t ≤ 3.
    Graph of the plane curve described by the parametric equations in part b.
  3. In this case, use multiples of π / 6 for t and create another table of values:
    t x ( t ) y ( t ) t x ( t ) y ( t )
    0 4 0 7 π 6 −2 3 −3.5 2
    π 6 2 3 3.5 2 4 π 3 −2 −2 3 −3.5
    π 3 2 2 3 3.5 3 π 2 0 −4
    π 2 0 4 5 π 3 2 −2 3 −3.5
    2 π 3 −2 2 3 3.5 11 π 6 2 3 3.5 2
    5 π 6 −2 3 −3.5 2 2 π 4 0
    π −4 0

    The graph of this plane curve appears in the following graph.
    A circle with radius 4 centered at the origin is graphed with arrow going counterclockwise. The point (4, 0) is marked t = 0, the point (0, 4) is marked t = π/2, the point (−4, 0) is marked t = π, and the point (0, −4) is marked t = 3π/2. On the graph there are also written three equations: x(t) = 4 cos(t), y(t) = 4 sin(t), and 0 ≤ t ≤ 2π.
    Graph of the plane curve described by the parametric equations in part c.

    This is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates ( 4 , 0 ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch the curve described by the parametric equations

x ( t ) = 3 t + 2 , y ( t ) = t 2 1 , −3 t 2 .


A curved line going from (−7, 8) through (−1, 0) and (2, −1) to (8, 3) with arrow going in that order. The point (−7, 8) is marked t = −3, the point (2, −1) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 − 1, and −3 ≤ t ≤ 2.

Got questions? Get instant answers now!

Eliminating the parameter

To better understand the graph of a curve represented parametrically, it is useful to rewrite the two equations as a single equation relating the variables x and y. Then we can apply any previous knowledge of equations of curves in the plane to identify the curve. For example, the equations describing the plane curve in [link] b. are

x ( t ) = t 2 3 , y ( t ) = 2 t + 1 , −2 t 3 .

Solving the second equation for t gives

t = y 1 2 .

This can be substituted into the first equation:

x = ( y 1 2 ) 2 3 = y 2 2 y + 1 4 3 = y 2 2 y 11 4 .

This equation describes x as a function of y. These steps give an example of eliminating the parameter . The graph of this function is a parabola opening to the right. Recall that the plane curve started at ( 1 , −3 ) and ended at ( 6 , 7 ) . These terminations were due to the restriction on the parameter t.

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
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask