<< Chapter < Page Chapter >> Page >

Functions of the form y = tan ( k θ )

In the equation, y = tan ( k θ ) , k is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = tan ( 2 θ ) .

The graph of tan ( 2 θ ) (solid line) and the graph of g ( θ ) = tan ( θ ) (dotted line). The asymptotes are shown as dashed lines.

Functions of the form y = tan ( k θ )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = tan 0 , 5 θ
  2. b ( θ ) = tan 1 θ
  3. c ( θ ) = tan 1 , 5 θ
  4. d ( θ ) = tan 2 θ
  5. e ( θ ) = tan 2 , 5 θ

Use your results to deduce the effect of k .

You should have found that, once again, the value of k affects the periodicity (i.e. frequency) of the graph. As k increases, the graph is more tightly packed. As k decreases, the graph is more spread out. The period of the tan graph is given by 180 k .

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = tan ( k θ ) .
k > 0 k < 0

Domain and range

For f ( θ ) = tan ( k θ ) , the domain of one branch is { θ : θ ( - 90 k , 90 k ) } because the function is undefined for θ = - 90 k and θ = 90 k .

The range of f ( θ ) = tan ( k θ ) is { f ( θ ) : f ( θ ) ( - , ) } .

Intercepts

For functions of the form, y = tan ( k θ ) , the details of calculating the intercepts with the x and y axis are given.

There are many x -intercepts; each one is halfway between the asymptotes.

The y -intercept is calculated as follows:

y = tan ( k θ ) y i n t = tan ( 0 ) = 0

Asymptotes

The graph of tan k θ has asymptotes because as k θ approaches 90 , tan k θ approaches infinity. In other words, there is no defined value of the function at the asymptote values.

Functions of the form y = sin ( θ + p )

In the equation, y = sin ( θ + p ) , p is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = sin ( θ + 30 ) .

Graph of f ( θ ) = sin ( θ + 30 ) (solid line) and the graph of g ( θ ) = sin ( θ ) (dotted line).

Functions of the form y = sin ( θ + p )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = sin ( θ - 90 )
  2. b ( θ ) = sin ( θ - 60 )
  3. c ( θ ) = sin θ
  4. d ( θ ) = sin ( θ + 90 )
  5. e ( θ ) = sin ( θ + 180 )

Use your results to deduce the effect of p .

You should have found that the value of p affects the position of the graph along the y -axis (i.e. the y -intercept) and the position of the graph along the x -axis (i.e. the phase shift ). The p value shifts the graph horizontally. If p is positive, the graph shifts left and if p is negative tha graph shifts right.

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = sin ( θ + p ) . The curve y = sin ( θ ) is plotted with a dotted line.
p > 0 p < 0

Domain and range

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

The range of f ( θ ) = sin ( θ + p ) is { f ( θ ) : f ( θ ) [ - 1 , 1 ] } .

Intercepts

For functions of the form, y = sin ( θ + p ) , the details of calculating the intercept with the y axis are given.

The y -intercept is calculated as follows: set θ = 0

y = sin ( θ + p ) y i n t = sin ( 0 + p ) = sin ( p )

Functions of the form y = cos ( θ + p )

In the equation, y = cos ( θ + p ) , p is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = cos ( θ + 30 ) .

Graph of f ( θ ) = cos ( θ + 30 ) (solid line) and the graph of g ( θ ) = cos ( θ ) (dotted line).

Functions of the form y = cos ( θ + p )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = cos ( θ - 90 )
  2. b ( θ ) = cos ( θ - 60 )
  3. c ( θ ) = cos θ
  4. d ( θ ) = cos ( θ + 90 )
  5. e ( θ ) = cos ( θ + 180 )

Use your results to deduce the effect of p .

You should have found that the value of p affects the y -intercept and phase shift of the graph. As in the case of the sine graph, positive values of p shift the cosine graph left while negative p values shift the graph right.

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = cos ( θ + p ) . The curve y = cos θ is plotted with a dotted line.
p > 0 p < 0

Domain and range

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

The range of f ( θ ) = cos ( θ + p ) is { f ( θ ) : f ( θ ) [ - 1 , 1 ] } .

Intercepts

For functions of the form, y = cos ( θ + p ) , the details of calculating the intercept with the y axis are given.

The y -intercept is calculated as follows: set θ = 0

y = cos ( θ + p ) y i n t = cos ( 0 + p ) = cos ( p )

Functions of the form y = tan ( θ + p )

In the equation, y = tan ( θ + p ) , p is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = tan ( θ + 30 ) .

The graph of tan ( θ + 30 ) (solid lines) and the graph of g ( θ ) = tan ( θ ) (dotted lines).

Functions of the form y = tan ( θ + p )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = tan ( θ - 90 )
  2. b ( θ ) = tan ( θ - 60 )
  3. c ( θ ) = tan θ
  4. d ( θ ) = tan ( θ + 60 )
  5. e ( θ ) = tan ( θ + 180 )

Use your results to deduce the effect of p .

You should have found that the value of p once again affects the y -intercept and phase shift of the graph. There is a horizontal shift to the left if p is positive and to the right if p is negative.

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = tan ( θ + p ) . The curve y = tan ( θ ) is plotted with a dotted line.
k > 0 k < 0

Domain and range

For f ( θ ) = tan ( θ + p ) , the domain for one branch is { θ : θ ( - 90 - p , 90 - p } because the function is undefined for θ = - 90 - p and θ = 90 - p .

The range of f ( θ ) = tan ( θ + p ) is { f ( θ ) : f ( θ ) ( - , ) } .

Intercepts

For functions of the form, y = tan ( θ + p ) , the details of calculating the intercepts with the y axis are given.

The y -intercept is calculated as follows: set θ = 0

y = tan ( θ + p ) y i n t = tan ( p )

Asymptotes

The graph of tan ( θ + p ) has asymptotes because as θ + p approaches 90 , tan ( θ + p ) approaches infinity. Thus, there is no defined value of the function at the asymptote values.

Functions of various form

Using your knowledge of the effects of p and k draw a rough sketch of the following graphs without a table of values.

  1. y = sin 3 x
  2. y = - cos 2 x
  3. y = tan 1 2 x
  4. y = sin ( x - 45 )
  5. y = cos ( x + 45 )
  6. y = tan ( x - 45 )
  7. y = 2 sin 2 x
  8. y = sin ( x + 30 ) + 1

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
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
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket 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
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
Berger describes sociologists as concerned with
Mueller 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