# 3.3 Quadratic polynomial function  (Page 2/3)

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${y}_{\mathrm{max}}=-\frac{D}{4a}\phantom{\rule{1em}{0ex}}\text{at}\phantom{\rule{1em}{0ex}}x=-\frac{b}{2a}$

${y}_{\mathrm{min}}⇒-\infty$

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

Problem : Determine range of $f\left(x\right)=-3{x}^{2}+2x-4$

Solution : The determinant of corresponding quadratic equation is :

$D={b}^{2}-4ac=4-4X\left(-3\right)X\left(-4\right)=4-48=-44\phantom{\rule{1em}{0ex}}⇒D<0$

$a=-3\phantom{\rule{1em}{0ex}}⇒a<0$

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 :

$⇒{y}_{\mathrm{max}}=-\frac{D}{4a}=-\frac{-44}{4X-3}=-\frac{44}{12}=-\frac{11}{3}$

${y}_{\mathrm{min}}⇒-\infty$

Range = (-∞, -11/3)

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.

$\phantom{\rule{1em}{0ex}}D<0,a>0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)>0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$

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

$\phantom{\rule{1em}{0ex}}D<0,a<0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$

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.

$D=0,a>0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)\ge 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$

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

$D=0,a<0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)\le 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$

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).

$D>0,a>0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)>0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,\alpha \right)\cup \left(\beta ,\infty \right)\phantom{\rule{1em}{0ex}}⇒\text{Sign of function same as that of “a”}$

$D>0,a>0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)=0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left\{\alpha ,\beta \right\}\phantom{\rule{1em}{0ex}}$

$D>0,a>0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(\alpha ,\beta \right)\phantom{\rule{1em}{0ex}}⇒\text{Sign of function opposite to that of “a”}$

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,∞).

$D>0,a<0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(\alpha ,\beta \right)\phantom{\rule{1em}{0ex}}⇒\text{Sign of function same as that of “a”}$

$D>0,a<0\phantom{\rule{1em}{0ex}}⇒f\left(x\right)=0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left\{\alpha ,\beta \right\}\phantom{\rule{1em}{0ex}}$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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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
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
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

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