# 4.2 Transformation of graphs by modulus function  (Page 2/4)

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From the point of construction of the graph of y=f(|x|), we need to modify the graph of y=f(x) as :

1 : remove left half of the graph

2 : take the mirror image of right half of the graph in y-axis

This completes the construction for y=f(|x|).

Problem : Draw graph of $y=\mathrm{sin}|x|$ .

Solution : First we draw graph of sinx. In order to obtain the graph of y=sin|x|, we remove left half of the graph and take the mirror image of right half of the graph of in y-axis.

Problem : Draw graph of $y={e}^{|x+1|}$ .

Solution : We first draw graph of $y={e}^{x}$ . Then, we shift the graph left by 1 unit to obtain the graph of ${e}^{x+1}$ . At $x=0,y={e}^{0+1}=e$ . In order to obtain the graph of $y={e}^{|x+1|}$ , we remove left part of the graph and take the mirror image of right half of the graph of $y={e}^{x+1}$ in y-axis.

In order to obtain the graph of $y={e}^{|x+1|}$ , we remove left part of the graph and take the mirror image of right half of the graph of $y={e}^{x+1}$ in y-axis.

Problem : Draw graph of $y={x}^{2}-2|x|-3$

Solution : The given expression $f\left(x\right)={x}^{2}-2|x|-3$ is obtained by taking modulus of the independent variable of the corresponding quadratic polynomial in x as given here, $f\left(x\right)={x}^{2}-2x-3$ . Hence, we first draw $f\left(x\right)={x}^{2}-2x-3$ . The corresponding quadratic equation $f\left(x\right)={x}^{2}-2x-3=0$ has real roots -1 and 3. The co-efficient of “ ${x}^{2}$ ” is positive. Hence, its plot is a parabola which opens upward and intersects x-axis at x=-1 and x=3.

In order to draw the graph of $f\left(x\right)={|x|}^{2}-2|x|-3={x}^{2}-2|x|-3$ , we remove left half of the graph and take the mirror image of right half of the core graph of quadratic function in y-axis.

Problem : Draw graph of function defined as :

$⇒y=\frac{1}{|x|+1}$

Solution : It is clear that we can obtain given function by applying modulus operator to the independent variable of function given here :

$⇒y=\frac{1}{x+1}$

This function, in tern, can be obtained by applying shifting modification to the argument of the function given as :

$⇒y=\frac{1}{x}$

We, therefore, first draw $f\left(x\right)=1/x$ . Then we draw $g\left(x\right)=f\left(x+1\right)=1/\left(x+1\right)$ by shifting the graph left by 1 unit. Finally, we draw $h\left(x\right)=g\left(|x|\right)=1/\left(|x|+1\right)$ by removing left half of the graph and taking mirror image of right half of the graph in y-axis. .

## Modulus function applied to the function

The form of transformation is depicted as :

$y=f\left(x\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}y=|f\left(x\right)|$

It can be seen that modulus operator here modifies the value of the function itself. In other words, it is like changing output of the function in accordance with nature of modulus function. The output of the function is now either zero or positive number. This has the implication that part of the graph y=f(x) corresponding to negative function values is not present in the graph of y=|f(x)|. Rather, negative function value of f(x) is converted to positive function value. This change in the sign of function takes place without changing magnitude of the value. It implies that we can obtain function values, which correspond to negative function value in y=f(x) by taking image of negative function values across x-axis. This is image in x-axis.

#### Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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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
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 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?
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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