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

 Page 2 / 4

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

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
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
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x