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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.

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
learn
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
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