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This module is from Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.Objectives of this module: be able to construct the graph of a parabola.

## Overview

• Parabolas
• Constructing Graphs of Parabolas

## Parabolas

We will now study the graphs of quadratic equations in two variables with general form $\begin{array}{lllll}y=a{x}^{2}+bx+c,\hfill & \hfill & a\ne 0,\hfill & \hfill & a,b,c\text{\hspace{0.17em}}\text{are\hspace{0.17em}real\hspace{0.17em}numbers}\hfill \end{array}$

## Parabola

All such graphs have a similar shape. The graph of a quadratic equation of this type Parabola is called a parabola and it will assume one of the following shapes.

## Vertex

The high point or low point of a parabola is called the vertex of the parabola.

## Constructing graphs of parabolas

We will construct the graph of a parabola by choosing several $x$ -values, computing to find the corresponding $y$ -values, plotting these ordered pairs, then drawing a smooth curve through them.

## Sample set a

Graph $y={x}^{2}.$    Construct a table to exhibit several ordered pairs.

 $x$ $y={x}^{2}$ 0 0 1 1 2 4 3 9 $-1$ 1 $-2$ 4 $-3$ 9

This is the most basic parabola. Although other parabolas may be wider, narrower, moved up or down, moved to the left or right, or inverted, they will all have this same basic shape. We will need to plot as many ordered pairs as necessary to ensure this basic shape.

Graph $y={x}^{2}-2.$     Construct a table of ordered pairs.

 $x$ $y={x}^{2}-2$ 0 $-2$ 1 $-1$ 2 2 3 7 $-1$ $-1$ $-2$ 2 $-3$ 7

Notice that the graph of $y={x}^{2}-2$ is precisely the graph of $y={x}^{2}$ but translated 2 units down. Compare the equations $y={x}^{2}$ and $y={x}^{2}-2$ . Do you see what causes the 2 unit downward translation?

## Practice set a

Use the idea suggested in Sample Set A to sketch (quickly and perhaps not perfectly accurately) the graphs of

$\begin{array}{lllll}y={x}^{2}+1\hfill & \hfill & \text{and}\hfill & \hfill & y={x}^{2}-3\hfill \end{array}$

## Sample set b

Graph $y={\left(x+2\right)}^{2}.$

Do we expect the graph to be similar to the graph of $y={x}^{2}$ ? Make a table of ordered pairs.

 $x$ $y$ 0 4 1 9 $-1$ 1 $-2$ 0 $-3$ 1 $-4$ 4

Notice that the graph of $y={\left(x+2\right)}^{2}$ is precisely the graph of $y={x}^{2}$ but translated 2 units to the left. The +2 inside the parentheses moves $y={x}^{2}$ two units to the left. A negative value inside the parentheses makes a move to the right.

## Practice set b

Use the idea suggested in Sample Set B to sketch the graphs of

$\begin{array}{lllll}y={\left(x-3\right)}^{2}\hfill & \hfill & \text{and}\hfill & \hfill & y={\left(x+1\right)}^{2}\hfill \end{array}$

Graph $y={\left(x-2\right)}^{2}+1$

## Exercises

For the following problems, graph the quadratic equations.

$y={x}^{2}$

$y={x}^{2}$

$y=-{x}^{2}$

$y={\left(x-1\right)}^{2}$

$y={\left(x-1\right)}^{2}$

$y={\left(x-2\right)}^{2}$

$y={\left(x+3\right)}^{2}$

$y={\left(x+3\right)}^{2}$

$y={\left(x+1\right)}^{2}$

$y={x}^{2}-3$

$y={x}^{2}-3$

$y={x}^{2}-1$

$y={x}^{2}+2$

$y={x}^{2}+2$

$y={x}^{2}+\frac{1}{2}$

$y={x}^{2}-\frac{1}{2}$

$y={x}^{2}-\frac{1}{2}$

$y=-{x}^{2}+1$ (Compare with problem 2.)

$y=-{x}^{2}-1$ (Compare with problem 1.)

$y=-{x}^{2}-1$

$y={\left(x-1\right)}^{2}-1$

$y={\left(x+3\right)}^{2}+2$

$y={\left(x+3\right)}^{2}+2$

$y=-{\left(x+1\right)}^{2}$

$y=-{\left(x+3\right)}^{2}$

$y=-{\left(x+3\right)}^{2}$

$y=2{x}^{2}$

$y=3{x}^{2}$

$y=3{x}^{2}$

$y=\frac{1}{2}{x}^{2}$

$y=\frac{1}{3}{x}^{2}$

$y=\frac{1}{3}{x}^{2}$

For the following problems, try to guess the quadratic equation that corresponds to the given graph.

$y={\left(x-3\right)}^{2}$

$y=-{\left(x+3\right)}^{2}+2$

## Exercises for review

( [link] ) Simplify and write ${\left({x}^{-4}{y}^{5}\right)}^{-3}{\left({x}^{-6}{y}^{4}\right)}^{2}$ so that only positive exponents appear.

( [link] ) Factor ${y}^{2}-y-42.$

$\left(y+6\right)\left(y-7\right)$

( [link] ) Find the sum: $\frac{2}{a-3}+\frac{3}{a+3}+\frac{18}{{a}^{2}-9}.$

( [link] ) Simplify $\frac{2}{4+\sqrt{5}}.$

$\frac{8-2\sqrt{5}}{11}$

( [link] ) Four is added to an integer and that sum is doubled. When this result is multiplied by the original integer, the product is $-6.$ Find the integer.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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