4.1 Transformation of graphs using output

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We modify output of a function in a couple of ways through arithmetic operations like addition, subtraction, multiplication, division and negation. These operations are similar to the one that we use to modify independent variable. The general symbolic representation for modification to output of a function is represented as :

$af\left(x\right)+d;\phantom{\rule{1em}{0ex}}a,d\in R$

These changes are called external or post-composition modifications. These modifications compliment modifications by input, but in slightly different manner. In the case of modification to output, all effects take place in y-direction i.e. vertical direction as against horizontal transformation arising from modifications affected to input. Second, these transformations are in the direction of operation on output. For example, if we multiply output by a positive constant greater than 1, then graph of core function is stretched along y-axis. This means change in the output is reflected in the same direction in which operation takes place.

Addition and subtraction operation with function

In order to understand this type of transformation, we need to explore how output of the function changes as we add constant value to the output. If we add 1 unit to the function, then each value of function is incremented by 1 unit. It is a straight forward situation. In notation, we would say that the graph of “f(x) + 1” is same as the graph of f(x), which has been moved up by 1 unit. Alternatively, we can also describe this transformation by saying that vertical reference of measurement i.e. x-axis has moved down by 1 unit.

Similarly, if we subtract 1 unit from the function, then each value of function is decremented by 1 unit. In notation, we would say that the graph of “f(x) - 1” is same as the graph of f(x), which has been moved down by 1 unit. Alternatively, we can also describe this transformation by saying that vertical reference of measurement i.e. x-axis has moved up by 1 unit. We conclude :

The plot of y=f(x) + |a|; |a|>0 is the plot of y=f(x) shifted up by unit “a”.

The plot of y=f(x) - |a|; |a|>0 is the plot of y=f(x) shifted down by unit “a”.

We use these facts to draw plot of transformed function f(x±|a|) by shifting plot f(x) by unit “|a|” along y-axis. Each point forming the plot is shifted parallel to x-axis. In the figure below, the plot depicts modulus function y=|x|. It is shifted “1” unit up and the function representing shifted plot is y=|x|+1. Note that corner of plot at x=0 is also shifted by 1 unit along y-axis. Further, the plot is shifted “2” units down and the function representing shifted plot is |x|-2. In this case, corner of plot is shifted by 2 units down along y-axis.

Multiplication and division of function

Multiplication and division scales core graph in accordance with the operation. Scaling, however, is limited to vertical i.e. y-direction. This means modification due to either of these two arithmetic operations has no scaling impact in x-direction. If we multiply output of the function by a positive constant greater than 1, then graph of core function is stretched vertically by the factor, which is equal to the constant being multiplied. The magnification of graph i.e. stretching in y-direction is more noticeable in non-linear graphs like sine and cosine graphs, whose values are bounded in the interval [-1,1]. Let us consider function,

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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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?
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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Tarell
what is the actual application of fullerenes nowadays?
Damian
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Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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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
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
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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
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What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x