<< Chapter < Page Chapter >> Page >

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

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
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
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
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?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Other chapter Q/A we can ask
Moahammedashifali Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?

Ask