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a f b x + c + d ; a , b , c R

In this case "a", "b", "c" and "d" can be either positive or negative depending on the particular transformation. A positive "d" means that graph is shifted up. On the other hand, we can specify constants to be positive in the following representation :

± a f ± b x ± c ± d ; a , b , c > 0

The form of representation appears to be cumbersome, but is more explicit in its intent. It delinks sign from the magnitude of constants. In this case, the signs preceding positive constants need to be interpreted for the nature of transformation. For example, a negative sign before c denotes right horizontal shift. It is, however, clear that both representations are essentially equivalent and their use depends on personal choice or context. This difference does not matter so long we understand the process of graphing.

Transformation of graph by input

Addition and subtraction to independent variable

In order to understand this type of transformation, we need to explore how output of the function changes as input to the function changes. Let us consider an example of functions f(x) and f(x+1). The integral values of inedependent variable are same as integral values on x-axis of coordinate system. Note that independent variable is plotted along x-axis as real number line. The integral x+1 values to the function f(x+1) - such that input values are same as that of f(x) - are shown on a separate line just below x-axis. The corresponding values are linked with arrow signs. Input to the function f(x+1) which is same as that of f(x) corresponds to x which is 1 unit smaller. It means graph of f(x+1) is same as graph of f(x), which has been shifted by 1 unit towards left. Else, we can say that the origin of plot (also x-axis) has shifted right by 1 unit.

Shifting of graph parallel to x-axis

Each element of graph is shifted left by same value.

Let us now consider an example of functions f(x) and f(x-2). Input to the function f(x-2) which is same as that of f(x) now appears 2 unit later on x-axis. It means graph of f(x-2) is same as graph of f(x), which has been shifted by 2 units towards right. Else, we can say that the origin of plot (also x-axis) has shifted left by 2 units.

Shifting of graph parallel to x-axis

Each element of graph is shifted right by same value.

The addition/subtraction transformation is depicted symbolically as :

y = f x y = f x ± | a | ; | a | > 0

If we add a positive constant to the argument of the function, then value of y at x=x in the new function y=f(x+|a|) is same as that of y=f(x) at x=x-|a|. For this reason, the graph of f(x+|a|) is same as the graph of y=f(x) shifted left by unit “a” in x-direction. Similarly, the graph of f(x-|a|) is same as the graph of y=f(x) shifted right by unit “a” in x-direction.

1 : The plot of y=f(x+|a|); is the plot of y=f(x) shifted left by unit “|a|”.

2 : The plot of y=f(x-|a|); is the plot of y=f(x) shifted right by unit “|a|”.

We use these facts to draw graph of transformed function f(x±a) by shifting graph of f(x) by unit “a” in x-direction. Each point forming the plot is shifted parallel to x-axis (see quadratic graph showm in the of figure below). The graph in the center of left figure depicts monomial function y = x 2 with vertex at origin. It is shifted right by “a” units (a>0) and the function representing shifted graph is y = x a 2 . Note that vertex of parabola is shifted from (0,0) to (a,0). Further, the graph is shifted left by “b” units (b>0) and the function representing shifted graph is y = x + b 2 . In this case, vertex of parabola is shifted from (0,0) to (-b,0).

Questions & Answers

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
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
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Berger describes sociologists as concerned with
Mueller Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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