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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.

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
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
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
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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