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Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of quadratic functions.

Functions of the form y = a ( x + p ) 2 + q

This form of the quadratic function is slightly more complex than the form studied in Grade 10, y = a x 2 + q . The general shape and position of the graph of the function of the form f ( x ) = a ( x + p ) 2 + q is shown in [link] .

Graph of f ( x ) = 1 2 ( x + 2 ) 2 - 1

Investigation : functions of the form y = a ( x + p ) 2 + q

  1. On the same set of axes, plot the following graphs:
    1. a ( x ) = ( x - 2 ) 2
    2. b ( x ) = ( x - 1 ) 2
    3. c ( x ) = x 2
    4. d ( x ) = ( x + 1 ) 2
    5. e ( x ) = ( x + 2 ) 2
    Use your results to deduce the effect of p .
  2. On the same set of axes, plot the following graphs:
    1. f ( x ) = ( x - 2 ) 2 + 1
    2. g ( x ) = ( x - 1 ) 2 + 1
    3. h ( x ) = x 2 + 1
    4. j ( x ) = ( x + 1 ) 2 + 1
    5. k ( x ) = ( x + 2 ) 2 + 1
    Use your results to deduce the effect of q .
  3. Following the general method of the above activities, choose your own values of p and q to plot 5 different graphs (on the same set of axes) of y = a ( x + p ) 2 + q to deduce the effect of a .

From your graphs, you should have found that a affects whether the graph makes a smile or a frown. If a < 0 , the graph makes a frown and if a > 0 then the graph makes a smile. This was shown in Grade 10.

You should have also found that the value of q affects whether the turning point of the graph is above the x -axis ( q < 0 ) or below the x -axis ( q > 0 ).

You should have also found that the value of p affects whether the turning point is to the left of the y -axis ( p > 0 ) or to the right of the y -axis ( p < 0 ).

These different properties are summarised in [link] . The axes of symmetry for each graph is shown as a dashed line.

Table summarising general shapes and positions of functions of the form y = a ( x + p ) 2 + q . The axes of symmetry are shown as dashed lines.
p < 0 p > 0
a > 0 a < 0 a > 0 a < 0
q 0
q 0

Phet simulation for graphing

Domain and range

For f ( x ) = a ( x + p ) 2 + q , the domain is { x : x R } because there is no value of x R for which f ( x ) is undefined.

The range of f ( x ) = a ( x + p ) 2 + q depends on whether the value for a is positive or negative. We will consider these two cases separately.

If a > 0 then we have:

( x + p ) 2 0 ( The square of an expression is always positive ) a ( x + p ) 2 0 ( Multiplication by a positive number maintains the nature of the inequality ) a ( x + p ) 2 + q q f ( x ) q

This tells us that for all values of x , f ( x ) is always greater than or equal to q . Therefore if a > 0 , the range of f ( x ) = a ( x + p ) 2 + q is { f ( x ) : f ( x ) [ q , ) } .

Similarly, it can be shown that if a < 0 that the range of f ( x ) = a ( x + p ) 2 + q is { f ( x ) : f ( x ) ( - , q ] } . This is left as an exercise.

For example, the domain of g ( x ) = ( x - 1 ) 2 + 2 is { x : x R } because there is no value of x R for which g ( x ) is undefined. The range of g ( x ) can be calculated as follows:

( x - p ) 2 0 ( x + p ) 2 + 2 2 g ( x ) 2

Therefore the range is { g ( x ) : g ( x ) [ 2 , ) } .

Domain and range

  1. Given the function f ( x ) = ( x - 4 ) 2 - 1 . Give the range of f ( x ) .
  2. What is the domain of the equation y = 2 x 2 - 5 x - 18 ?

Intercepts

For functions of the form, y = a ( x + p ) 2 + q , the details of calculating the intercepts with the x and y axes is given.

The y -intercept is calculated as follows:

y = a ( x + p ) 2 + q y i n t = a ( 0 + p ) 2 + q = a p 2 + q

If p = 0 , then y i n t = q .

For example, the y -intercept of g ( x ) = ( x - 1 ) 2 + 2 is given by setting x = 0 to get:

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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