# 1.1 Parametric equations  (Page 2/14)

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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 $x\left(t\right)$ and $y\left(t\right)$ generate a set of ordered pairs $\left(x,y\right).$ 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 $x\left(t\right)$ and $y\left(t\right).$

## Graphing a parametrically defined curve

Sketch the curves described by the following parametric equations:

1. $x\left(t\right)=t-1,\phantom{\rule{1em}{0ex}}y\left(t\right)=2t+4,\phantom{\rule{1em}{0ex}}-3\le t\le 2$
2. $x\left(t\right)={t}^{2}-3,\phantom{\rule{1em}{0ex}}y\left(t\right)=2t+1,\phantom{\rule{1em}{0ex}}-2\le t\le 3$
3. $x\left(t\right)=4\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,\phantom{\rule{1em}{0ex}}y\left(t\right)=4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\phantom{\rule{1em}{0ex}}0\le t\le 2\pi$
1. To create a graph of this curve, first set up a table of values. Since the independent variable in both $x\left(t\right)$ and $y\left(t\right)$ is t , let t appear in the first column. Then $x\left(t\right)$ and $y\left(t\right)$ will appear in the second and third columns of the table.
t $x\left(t\right)$ $y\left(t\right)$
−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. 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 $x\left(t\right)$ $y\left(t\right)$
−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 $t=-2\right)$ has coordinates $\left(1,-3\right),$ and the last point (corresponding to $t=3\right)$ has coordinates $\left(6,7\right).$ 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. Graph of the plane curve described by the parametric equations in part b.
3. In this case, use multiples of $\pi \text{/}6$ for t and create another table of values:
t $x\left(t\right)$ $y\left(t\right)$ t $x\left(t\right)$ $y\left(t\right)$
0 4 0 $\frac{7\pi }{6}$ $-2\sqrt{3}\approx -3.5$ 2
$\frac{\pi }{6}$ $2\sqrt{3}\approx 3.5$ $2$ $\frac{4\pi }{3}$ −2 $-2\sqrt{3}\approx -3.5$
$\frac{\pi }{3}$ $2$ $2\sqrt{3}\approx 3.5$ $\frac{3\pi }{2}$ 0 −4
$\frac{\pi }{2}$ 0 4 $\frac{5\pi }{3}$ 2 $-2\sqrt{3}\approx -3.5$
$\frac{2\pi }{3}$ −2 $2\sqrt{3}\approx 3.5$ $\frac{11\pi }{6}$ $2\sqrt{3}\approx 3.5$ 2
$\frac{5\pi }{6}$ $-2\sqrt{3}\approx -3.5$ 2 $2\pi$ 4 0
$\pi$ −4 0

The graph of this plane curve appears in the following graph. 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 $\left(4,0\right).$

Sketch the curve described by the parametric equations

$x\left(t\right)=3t+2,\phantom{\rule{1em}{0ex}}y\left(t\right)={t}^{2}-1,\phantom{\rule{1em}{0ex}}-3\le t\le 2.$ ## 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

$x\left(t\right)={t}^{2}-3,\phantom{\rule{1em}{0ex}}y\left(t\right)=2t+1,\phantom{\rule{1em}{0ex}}-2\le t\le 3.$

Solving the second equation for t gives

$t=\frac{y-1}{2}.$

This can be substituted into the first equation:

$x={\left(\frac{y-1}{2}\right)}^{2}-3=\frac{{y}^{2}-2y+1}{4}-3=\frac{{y}^{2}-2y-11}{4}.$

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 $\left(1,-3\right)$ and ended at $\left(6,7\right).$ These terminations were due to the restriction on the parameter t.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
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what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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