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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.

Acknowledgment

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

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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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