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Graph of quadratic function

The graph is parabola, which opens down.

y max = - D 4 a at x = - b 2 a

y min

Clearly, range of the function is (-∞, -D/4a].

Problem : Determine range of f x = - 3 x 2 + 2 x 4

Solution : The determinant of corresponding quadratic equation is :

D = b 2 4 a c = 4 4 X 3 X 4 = 4 48 = - 44 D < 0

a = - 3 a < 0

The graph of function is parabola opening down. Its vertex represents the maximum function value. The maximum and minimum values of function are given by :

y max = - D 4 a = - - 44 4 X - 3 = - 44 12 = - 11 3

y min

Range = (-∞, -11/3)

Nature of quadratic function

The discriminant of corresponding quadratic equation and coefficient of term “ x 2 ” of quadratic function together determine nature of quadratic function and hence its graph. Graphs of quadratic function is intuitive and helpful to remember results. As a matter of fact, we can interpret all properties of quadratic function, if we can draw its graph.

Case 1 : d<0

If D<0, then roots are complex conjugates. It means graph of function does not intersect x-axis. If a>0, then parabola opens up. The value of quadratic function is positive for all values of x i.e.

D < 0, a > 0 f x > 0 for x R

Graph of quadratic function

The discriminant is negative.

If a<0, then parabola opens down. The value of quadratic function is negative for all values of x i.e.

D < 0, a < 0 f x < 0 for x R

Sign rule : If D<0, then sign of function is same as that of “a” for all values of x in R.

Case 2 : d=0

If D=0, then roots are equal and is given by –b/2a. It means graph of function just touches x-axis. If a>0, then parabola opens up. The value of quadratic function is non-negative for all values of x i.e.

D = 0, a > 0 f x 0 for x R

Graph of quadratic function

The discriminant is zero.

If a<0, then parabola opens down. The value of quadratic function is non-positive for all values of x i.e.

D = 0, a < 0 f x 0 for x R

Sign rule : If D=0, then sign of function is same as that of “a” for all values of x in R except at x=-b/2a, at which f(x)=0. We do not associate sign with zero.

Case 3 : d>0

If D>0, then roots are unequal and are given by (–b±D)/2a. It means graph of function intersects x-axis at α and β (β>α). If a>0, then parabola opens up. The value of quadratic function is positive for all values of x in the interval (-∞,α) U (β,∞).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (α,β).

Graph of quadratic function

The discriminant is positive and a is positive.

D > 0, a > 0 f x > 0 for x - , α β , Sign of function same as that of “a”

D > 0, a > 0 f x = 0 for x { α , β }

D > 0, a > 0 f x < 0 for x α , β Sign of function opposite to that of “a”

If a<0, then parabola opens down. The value of quadratic function is positive for all values of x in the interval (α,β).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (-∞,α) U (β,∞).

Graph of quadratic function

The discriminant is positive and a is negative.

D > 0, a < 0 f x < 0 for x α , β Sign of function same as that of “a”

D > 0, a < 0 f x = 0 for x { α , β }

Questions & Answers

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?
Adin Reply
?
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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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?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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what is biological synthesis of nanoparticles
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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