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
Is there any normative that regulates the use of silver nanoparticles?
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
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?