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y = 4 sin x

Scaling of graph

Sine graph is stretched and shrunk.

The amplitude of function "4sinx" is 4 times that of core graph "sinx". In the same fashion, a division by a positive constant greater than 1 results in shrinking of core graph by the factor, which is equal to constant being multiplies. Let us consider division of function :

y = 1 2 sin x

The amplitude of the graph "sinx" changes from 1 to 1/2 in the graph of "1/2 sinx".

Negation of function

What would happen if we negate output of a function? Answer is easy. All positive values will turn negative and all negative values will turn positive. It means that graph of core function which is being negated will be swapped across x-axis in the transformation. The graph of “f(-x)”, therefore, is mirror image in x-axis. In other words, we would need to flip the graph f(x) across x-axis to draw graph “-f(x)”.

Changing sign of the graph

The transformed graph is image of the core graph across x-axis.

Problem : Draw graph of y = log e 1 x .

Solution : We simplify given function as :

y = log e 1 x = log e 1 log e x = log e x

Here, core function is f(x) = log e (x) . Clearly, given function is transformed function of type y=-f(x). We obtain its graph by taking mirror image of the graph of y=f(x) about x-axis. We obtain its graph by taking mirror image of graph of core function about x-axis.

Changing sign of the graph

The transformed graph is image of the core graph about x-axis.

Combined output operations

Certain functions are derived from core function as a result of multiple arithmetic operations on the output of core function. Consider an example :

f x = - 2 sin x - 1

We can consider this as a function composition which is based on sine function f(x) = sinx as core function. Here, sequence of operations on the function is important. Difference in interpreting input and output composition is that input composition is evaluated such that defining input transitions are valid. This results in a order of evaluation which gives precedence to addition/subtraction over multiplication/division. This evaluation order is clearly opposite to normal composition order of arithmetic operations in which multiplication/division is given precedence over addition/subtraction. We, therefore, say that decomposition of function for input operation is opposite to that of composition order. In the case of output operation, however, composition order of arithmetic operations is maintained during decomposition. It is logical also. After all, we are operating on a value – not something that goes into function to generate values in accordance with function rule as is the case with independent variable. It is, therefore, expected that we carry out arithmetic operations on the function just the way we evaluate algebraic expressions. In the nutshell, we shall give precedence to multiplication/division over addition/subtraction. In the example abvoe, we subtract "-1" to "-2sinx" - not to core function "sinx".

Keeping above in mind, the correct sequence of operation for graphing is :

(i) 2f(x) i.e. multiply function f(x) by 2 i.e. stretch the graph vertically by 2.

(ii) -2f(x) i.e. negate function f(x) i.e. flip the graph across x-axis.

(iii) -2f(x) – 1 i.e. subtract 1 from -2f(x) i.e. shift the graph down by 1 units.

Changing sign of the graph

The transformed graph is image of the base graph about x-axis.

Combined input and output operations

The combined input and output operation is symbolically represented as :

a f b x + c + d ; a , b , c , d R Carrying out output operation before input operation does not make sense. There will be two different outputs which are not connected to each other. Hence, logical order is that we first carry out input operations then follow it with output operations.

Problem : Draw y 1 = log e x 2

Solution : We rewrite the function :

y = log e x 2 + 1

In order to plot this function, we plot the graph of core function y = log e x . Note that when y=0,

y = log e x = 0 x = e 0 = 1

In this case, plot intersects x-axis at x=1. Now, the plot of y = log e x 2 is plot of y = log e x shifted right by 2 units. Note that when y=0,

y = log e x 2 = 0 x 2 = e 0 = 1 x = 3

The plot of y = log e x 2 + 1 is plot of y = log e x 1 shifted up by 1 unit.

Shifting of logarithmic graph parallel to y-axis

Each element of graph is shifted by same value.

There is yet another alternative to obtain graph of transformed function by shifting axes themselves instead of plot. In the case of shifting either in x or y direction, the operation of shifting graph is equivalent to shifting of axis. Therefore, transformation involving shifting can be affected by shifting axes in opposite directions to that required for the graph. In the example case, we need to move y-axis by 2 units towards left and move x-axis by 1 unit downwards.

Shifting of graph

Each element of graph is shifted by same value in either direction.

Problem : Draw the plot y = cos 2 x .

Solution : We know that :

y = cos 2 x = 1 + cos 2 x 2 = 1 2 + cos 2 x 2

Here, core graph is y = cos x . Multiplying independent variable by 2 shrinks core graph horizontally. As a result its period is reduced from 2π to π as shown in the graph. Division of cos2x by 2 is division operation on function. This operation shrinks the graph cos2x by 2 vertically. Note that amplitude of graph is reduced to 1/2 due to this operation. In the figure, lower graph corresponds to (cos2x)/2. Once we draw graph of (cos2x)/2, we draw given function y = cos 2 x by shifting the graph of (cos2x)/2 by 1/2 units up.

Graph of squared cosine

Each element of graph is shifted by same value.


Author wishes to thank Ms. Aditi Singh, New Delhi for her editorial suggestions.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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