# 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.

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
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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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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is Bucky paper clear?
CYNTHIA
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Harper
Do you know which machine is used to that process?
s.
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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
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China
Cied
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many many of nanotubes
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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what is nano technology
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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Damian
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Azam
Hello
Uday
I'm interested in Nanotube
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