<< 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

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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