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:
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.
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
$t=\mathrm{-2})$ has coordinates
$\left(1,\mathrm{-3}\right),$ and the last point (corresponding to
$t=3)$ 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.
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
$\frac{7\pi}{6}$
$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.5}$
2
$\frac{\pi}{6}$
$2\sqrt{3}\approx 3.5$
$2$
$\frac{4\pi}{3}$
−2
$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.5}$
$\frac{\pi}{3}$
$2$
$2\sqrt{3}\approx 3.5$
$\frac{3\pi}{2}$
0
−4
$\frac{\pi}{2}$
0
4
$\frac{5\pi}{3}$
2
$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.5}$
$\frac{2\pi}{3}$
−2
$2\sqrt{3}\approx 3.5$
$\frac{11\pi}{6}$
$2\sqrt{3}\approx 3.5$
2
$\frac{5\pi}{6}$
$\mathrm{-2}\sqrt{3}\approx \mathrm{-3.5}$
2
$2\pi $
4
0
$\pi $
−4
0
The graph of this plane curve appears in the following graph.
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
$\left(4,0\right).$
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
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
$\left(1,\mathrm{-3}\right)$ and ended at
$\left(6,7\right).$ These terminations were due to the restriction on the parameter
t.
Questions & Answers
where we get a research paper on Nano chemistry....?
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
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?
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Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?