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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 :

a f x + d ; a , d 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.

Shifting of graph parallel to y-axis

Each element of graph is shifted by same value.

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.

Shifting of graph parallel to y-axis

Each element of graph is shifted by same value.

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,

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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s. Reply
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are you nano engineer ?
s.
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.
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
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.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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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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