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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 exponential functions.

Functions of the form y = a b ( x + p ) + q For b > 0

This form of the exponential function is slightly more complex than the form studied in Grade 10.

General shape and position of the graph of a function of the form f ( x ) = a b ( x + p ) + q .

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

  1. On the same set of axes, with 5 x 3 and 35 y 35 , plot the following graphs:
    1. f ( x ) = - 2 · 2 ( x + 1 ) + 1
    2. g ( x ) = - 1 · 2 ( x + 1 ) + 1
    3. h ( x ) = 0 · 2 ( x + 1 ) + 1
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 2 · 2 ( x + 1 ) + 1
    Use your results to understand what happens when you change the value of a . You should find that the value of a affects whether the graph curves upwards ( a > 0 ) or curves downwards ( a < 0 ). You should also find that a larger value of a (when a is positive) stretches the graph upwards. However, when a is negative, a lower value of a (such as -2 instead of -1) stretches the graph downwards. Finally, note that when a = 0 the graph is simply a horizontal line. This is why we set a 0 in the original definition of these functions.
  2. On the same set of axes, with 3 x 3 and 5 y 20 , plot the following graphs:
    1. f ( x ) = 1 · 2 ( x + 1 ) - 2
    2. g ( x ) = 1 · 2 ( x + 1 ) - 1
    3. h ( x ) = 1 · 2 ( x + 1 ) + 0
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 1 · 2 ( x + 1 ) + 2
    Use your results to understand what happens when you change the value of q . You should find that when q is increased, the whole graph is translated (moved) upwards. When q is decreased (poosibly even made negative), the graph is translated downwards.
  3. On the same set of axes, with 5 x 3 and 35 y 35 , plot the following graphs:
    1. f ( x ) = - 2 · 2 ( x + 1 ) + 1
    2. g ( x ) = - 1 · 2 ( x + 1 ) + 1
    3. h ( x ) = 0 · 2 ( x + 1 ) + 1
    4. j ( x ) = 1 · 2 ( x + 1 ) + 1
    5. k ( x ) = 2 · 2 ( x + 1 ) + 1
    Use your results to understand what happens when you change the value of a . You should find that the value of a affects whether the graph curves upwards ( a > 0 ) or curves downwards ( a < 0 ). You should also find that a larger value of a (when a is positive) stretches the graph upwards. However, when a is negative, a lower value of a (such as -2 instead of -1) stretches the graph downwards. Finally, note that when a = 0 the graph is simply a horizontal line. This is why we set a 0 in the original definition of these functions.
  4. Following the general method of the above activities, choose your own values of a and q to plot 5 graphs of y = a b ( x + p ) + q on the same set of axes (choose your own limits for x and y carefully). Make sure that you use the same values of a , b and q for each graph, and different values of p . Use your results to understand the effect of changing the value of p .

These different properties are summarised in [link] .

Table summarising general shapes and positions of functions of the form y = a b ( x + p ) + q .
p < 0 p > 0
a > 0 a < 0 a > 0 a < 0
q > 0
q < 0

Domain and range

For y = a b ( x + p ) + q , the function is defined for all real values of x . Therefore, the domain is { x : x R } .

The range of y = a b ( x + p ) + q is dependent on the sign of a .

If a > 0 then:

b ( x + p ) > 0 a · b ( x + p ) > 0 a · b ( x + p ) + q > q f ( x ) > q

Therefore, if a > 0 , then the range is { f ( x ) : f ( x ) [ q , ) } . In other words f ( x ) can be any real number greater than q .

If a < 0 then:

b ( x + p ) > 0 a · b ( x + p ) < 0 a · b ( x + p ) + q < q f ( x ) < q

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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.
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Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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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