Activities on exploring trigonometric graphs in context

Activities on exploring trigonometric graphs in context
Exploring Trigonometric
Graphs
Exploring Trigonometric Graphs
Index
Syllabus Extract: Relationship to
syllabus. . . . . . . . . . . . . . . . . . . . . 3
Leaving Certificate. . . . . . . . . . . . 3
Unit 1 . . . . . . . . . . . . . . . . . . . . . . 4
Unit 2 . . . . . . . . . . . . . . . . . . . . . . 10
Unit 3 . . . . . . . . . . . . . . . . . . . . . . 12
Questions Sheets - Trigonometric
Graphs in Context . . . . . . . . . . . . 13
Solutions . . . . . . . . . . . . . . . . . . . 16
Student Activity Sheet 1 Graphs of
sinx and cosx . . . . . . . . . . . . . . 34
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Exploring Trigonometric Graphs
Syllabus Extract: Relationship to syllabus.
Leaving Certificate.
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Exploring Trigonometric Graphs
Unit 1
Upon completion of this Unit:
Students will:
Understand terms such as
Maximum
Minimum
Period
Range
Amplitude
Horizontal midway line
Horizontal shape (stretch/shrink)
Vertical shape (stretch/shrink)
Transformations of the graphs of
and
In this unit students will explore functions of the type
a, b, c ∈ R
and examine how the values of “a”, “b” and “c” affect the curves.
Prior Knowledge
Students should be familiar with graphs of linear and quadratic
functions.
Students should be familiar with the graphs of
from Teaching and Learning
Plan 10 and the Student’s CD.
At the outset the students should be reminded of the features of the
graphs of the functions
and
by completing
Student Activity Sheet 1 [Page 37].
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Exploring Trigonometric Graphs
Materials required
For each student you will need:
Mini white board
Student Activity Sheet 1
and
Student Activity Sheet2a (Mathematical Language / Properties of
Trigonometric Graphs)
Student Activity Sheet 2b (Mathematical Language / Properties of
Trigonometric Graphs)
Student Activity Sheet 2c (Mathematical Language / Properties of
Trigonometric Graphs)
Student Activity Sheet 3 (Mathematical Language / Properties of
Trigonometric Graph)
Student Activity Sheet 5 (Mathematical Language / Properties of
Trigonometric Graphs)
Student Activity Sheet 6 (Transformations of Trigonometric
Graphs)
Student Activity Sheet 7 (Mathematical Language / Properties of
Trigonometric Graphs).
For each group of students you will need
Student’s CD
Card Set A1 (Trigonometric Functions )
Card Set A2 (Trigonometric Functions )
Card Set B (Trigonometric Functions Important Features)
Card Set C
Student Activity 4 (Mathematical Language / Properties of
Trigonometric Graphs).
Student Activities/Teacher’s support and actions
Use the interactive GeoGebra files at: www.projectmaths.ie
Working in groups [1]
Ask students to work in pairs, distribute one or both of
Card Set A1 (Trigonometric Functions g(x) = bsin x) [Page 38]
Card Set A2 (Trigonometric Functions g(x) = bcos x) [Page 39]
to each pair of student.
Ask each group to sort the cards into three sets, selection criteria
should be at the group’s discretion.
For example, they may group the functions where the coefficient of
sin x or cos x is 2.
Ask students to write a description of their selection criteria.
Ask students to write an equation of their own for each set (if
working with cards from one set only).
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Whole group activity using GeoGebra [1]
Share all the criteria students have come up with.
Hand out one of:
Student Activity Sheet 2a (Mathematical Language /Properties
of Trigonometric Graphs) [Page 40]
and
Student Activity Sheet 2b (Mathematical Language /Properties
of Trigonometric Graphs) [Page 41]
or
Student Activity Sheet 2c (Mathematical Language /Properties
of Trigonometric Graphs) (getting the students to use a mixture of
cards from Card Sets a1 and A2) [Page 42]
Now ask the students to use the interactive files
f(x)=a+bsin cx and f(x)=a+bcos cx at www.projectmaths.ie a
to justify their descriptions of the functions contained on Card Sets
A1/A2 and introduce students to the correct use of mathematical
language using the previous student activity.
Instructions: In the interactive files
f(x)=a+bsin cx and f(x)=a+bcos cx
Set Slider a to 0. Set Slider c to 1. Set Slider b to desired value.
Example:
A “b” value of 3 causes a stretch in the direction of the Y axis with
scale factor of 3.
If student need more practice with transformations and
trigonometric graphs use mini whiteboards to explore questions of
the type:
Give an equation to represent the function which results from:
stretching f(x)=sinx in the direction of the Y axis with a scale factor
of 4.
reflecting f(x)=cosx in the X axis.
Describe the transformation required to transform f(x)=sinx to
chlaochlú go dti f(x)=-2sinx
Define “amplitude”, “range”, “period” and link these to the idea of
stretch/shrink
Check the students’ understanding of how f(x)=a+bsin cx and
f(x)=a+bcos cx Check the students’ understanding of how give
information about amplitude, range and period by asking them to
find cards from Card Set A1 and/or Card Set A2 that fit certain
criteria. For example, the students might be asked to find equation(s)
which describe:
A graph having an amplitude of 2
Two graphs that have the same period
A graph that has been compressed along the X axis
A graph that has been stretched vertically
Two graphs that are images of each other by reflection.
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Working in groups [2]
Distribute to each pair of students
Card Set B (Trigonometric Functions-Important Features) [Page
43]
Ask each pair to sort the cards into two sets. Selection criteria should
be at their own discretion.
Ask students to write a description of the selection criteria.
Ask student to write an equation which would fit their selection
criteria.
Whole group activity using GeoGebra [2]
Share all the criteria students have come up with.
Distribute to each pair of students.
Student Activity Sheet 3 (Mathematical Language/Properties
of Trigonometric Graphs) [Page 44]
Now use the interactive files f(x)=a+bsin cx and f(x)=a+bcos cx
at www.projectmaths.ie to justify the descriptions and introduce
students to mathematical language.
Instructions: In the interactive files f(x)=a+bsin cx and
f(x)=a+bcos cx
Set Slider a to 0. Set Slider b to 1. Set Slider c to desired value.
Example
When “c” has a value of 3 this is a horizontal shrink with scale
factor 3.
If students need more practice with transformations and
trigonometric graphs use mini whiteboards to explore questions of
the type:
Give an equation to represent the function which results from:
Stretch f(x)=sin x horizontally by a scale factor of 4
Shrink f(x)=cos x horizontally by a scale factor of 0.5
Decreasing the period of f(x)=sin x by a factor of 2
Increasing the period of f(x)=sin x by a factor of…
What transformation converts:
f(x)=sin x to f(x)=sin2/5x,
f(x)=cos x to f(x)=cos5x?
Check student understanding of how the functions
f(x)=a+bsin cx and f(x)=a+bcos cx
give information about amplitude, range and period by asking them
to find cards from Card Set B that fit certain criteria.
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For example, the students might be asked to find equation(s) which
describe:
A graph having amplitude of 2.
Two graphs having the same period.
A graph that has been shrunk along the X axis.
A graph that has been vertically stretched.
A graph that has been horizontally stretched.
Two graphs that are images of each other by reflection.
Two graphs that have the same amplitude.
Working in groups [3]
Distribute to each pair of students:
Card Set C [Pages 45 – 47]
Student Activity Sheet 4 (Mathematical Language/Properties
of Trigonometric Graphs) [Page 48]
Student Activity Sheet 5 (Mathematical Language /Properties
of Trigonometric Graphs) [Page 49]
Ask each pair of students to find cards from Card Set C to match the
properties described on Student Activity Sheet 4.
When the students have found a suitable card they should place it in
the correct box on Student Activity Sheet 4 matching the property
described therein. They should aim to match as many boxes and cards
as possible.
Whole-group activity [3]
Choose one box from Student Activity Sheet 4 and write down all
the cards that were matched with it.
Discuss as a group if these cards are appropriate and agree reasons for
any decisions reached.
Repeat the above for the remaining boxes.
Check students understanding of how the equations f(x)=a+bsin cx
and f(x)=a+b coscx? give information about amplitude, range and
period by asking them to fill in the second section of Student Activity
Sheet 5.
Working in groups [4]
Distribute to each pair of students
Student Activity Sheet 6 (Transformation of Trigonometric
Graphs)[Page 50]
Student Activity Sheet 7 (Mathematical Language / Properties of
Trigonometric Graphs) [Page 51]
Discuss in groups how the graphs of f(x)=a+bsin cx and f(x)=a+b
coscx are affected by changing the value of “a”
If required reinforce the students’ prior learning by referring to the
properties of the graphs of linear and quadratic functions.
.
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Whole group activity using GeoGebra [4]
Share all the criteria student have come up with in the previous
activity Working in groups [4] and discuss the reasons for these
criteria.
Now use the interactive files f(x)=a+bsin cx and f(x)=a+b coscx
at www.projectmaths.ie to justify the descriptions and introduce
student to the appropriate use of mathematical language.
Instructions: In the interactive files f(x)=a+bsin cx and f(x)=a+b
coscx
Set Slider b to 1. Set Slider c to 1. Set Slider a to desired value.
For example
When “a” has a value of 3 this causes a vertical translation upwards
of length 3 units.
If student need more practice with transformations and
trigonometric graphs use mini whiteboards to answer questions of
the type:
Give an equation to represent the function which results from
translating: f(x)=sin x vertically upwards by 2 units.
Give the amplitude of the function which results from translating
f(x)=cos x vertically downwards by 3 units.
Give the range of the function which results from translating
f(x)=sin x vertically upwards by 4 units.
Give the period of the function which results from translating
f(x)=cos x vertically downwards by 2.5 units.
Give the transformation which has been used in transforming
f(x)=cos x to f(x)=7 + cos x
Check student understanding of how the equation gives information
about transformations amplitude, range and period by getting them
to complete Student Activity Sheet 7.
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Unit 2
Having completed this Unit:
Students will:
Understand how to identify the:
Period
Range
Amplitude
Horizontal midway line
Horizontal shape in the direction of the X axis
Vertical shape in the direction of the Y axis
Transformations of trigonometric functions from their graphs.
Furthermore, the students will be able to sketch trigonometric
graphs given their equations.
Prior Knowledge
Student should be familiar with the graphs f(x)=a+bsin cx and
f(x)=a+b cos cx from Unit 1.
Materials required
For each student you will need:
Mini white board
Student Activity Sheet 8 (Trigonometric Graphs)
For each group of student you will need:
Card Set D
Card Set E
Card Set F
Working in groups [5]
Distribute to each pair of students
Card Set D Trigonometric Graphs [Pages 52 – 53]
Student Activity Sheet 8 (Trigonometric Graphs) [Page 54]
Ask each group of students to fill in Student Activity Sheet 8 using
details from the graphs on Card Set D to do so.
Distribute to each pair of student Card Set E [Page 55]
Ask each group to match the graphs from Card Set D with the
trigonometric functions from Card Set E using the information
derived from the earlier group activity.
Check student understanding by asking them to explain why they
matched the cards as they did.
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Working in groups [6]
Distribute
Card Set D Trigonometric Graphs [Pages 52 – 53] to each pair of
student if not already handed out.
Ask each group of students to name and identify a number of points
which would allow them to draw a sketch of trigonometric graphs of
the form: f(x)=a+bsin cx and f(x)=a+b coscx
Student can use mini whiteboards to draw sketches as part of their
discussions.
Whole group activity [6]
Choose one group’s set of points.
Discuss as a group the merits or otherwise of this set of points.
Ask questions such as: Will this set of points give a unique graph?
Will this set of points give the graph for all possible trigonometric
functions of the form f(x)=a+bsin cx and f(x)=a+b coscx is this
the smallest set of points that will allow a sketch to be drawn?
Taking feedback from all the groups identify the five points required:
Maximum
Minimum
Three points of intersection with the horizontal midway line.
Working in groups [7]
Distribute to each pair of students
Card Set F [Page 56]
Check students understanding of how the equation gives the
information required to sketch a trigonometric function of the form
f(x)=a+bsin cx and f(x)=a+b cos cx by asking them to use mini
whiteboards to sketch the graphs from Card Set F.
Each group of students can check their sketches by using the
interactive files f(x)=a+bsin cx and f(x)=a+b cos cx
at www.projectmaths.ie
Note 1: if students do not have Card Set D or Student Activity
Sheet 8 in their possession they could use Card Set E for this
activity.
Note 2: Students could use the functions from Student Activity
Sheet 7 instead of using Card Set F, if this is considered desirable.
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Unit 3
Having completed this Unit:
Student will:
be able to formulate conjectures form patterns
understand the need to explain their findings and justify their
conclusions
have developed the skills to communicate mathematics verbally
and in written form using appropriate mathematical language
be able to apply their knowledge and skills to solve problems in
familiar and unfamiliar contexts
be able to analyse information presented verbally and translate
it into mathematical form devise, select and use appropriate
mathematical models, formulae or techniques to process
information and to draw relevant conclusions.
Prior Knowledge
Students should be familiar with the graphs of
f(x)=a+bsin cx and f(x)=a+b cos cx
from Units 1 and 2 of Exploring Trigonometric Graphs.
Materials required
For each students you will need:
Mini white board
Question Sheets (Trigonometric Graphs in Context)
Working in groups [7]
Ask student to work in pairs.
Distribute to each student Question Sheets (Trigonometric
Graphs in Context) [Page 15 - 17]
Ask each group to attempt Question 1.
Take feedback from groups using appropriate questioning to
establish and develop the students’ understanding.
Repeat the above for two additional questions.
Now ask each student to work independently on the remainder of
the questions on the Question Sheets(Trigonometric Graphs in
Context). Provide assistance to individuals as required and ensure
that the students’ learning is of a high quality and that their
understanding of the material is being developed and deepened.
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Questions Sheets Trigonometric Graphs in Context
Question 1
A wrecking ball attached to a crane swings back and
forth. The distance that the ball moves to the left
and to the right of its resting position with respect
to time is represented by the following graph.
a. What is the period of the crane’s motion. Explain your answer?
b. What is the equation of the horizontal midway line of the curve
and what does it represent?
c. What is the amplitude of the crane’s motion? Draw a diagram to
represent what the amplitude represents in terms of the motion
of the ball?
d. How many complete swings will the wrecking ball make in four
minutes?
e. What does point A represent?
Question 2
As a dolphin swims along by the side of a cruise
ship he jumps to a height of four metres above
the water surface and dives to a depth of four
meters below the surface. He does this in a
regular motion (simple harmonic motion). A
passenger using a stopwatch and starting it just as the dolphin is at
the surface determines that the dolphin completes one cycle every 8
seconds.
a. Describe the displacement of the dolphin relative to the water
surface using a sinusoid curve and sketch the graph of the
dolphin’s displacement relative to the water surface
b After three seconds how high above the surface is the dolphin?
c Is the dolphin above or below the surface of the water after 37
seconds?
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Question 3
The inside rim of a bicycle wheel whose diameter
is 25 inches, is 3 inches off the ground. An ant
is sitting on the inside rim of the wheel at the
point 3 inches off the ground. Sean starts riding
the bicycle at a steady rate. The wheel makes one
revolution every 1.6 seconds.
a. Find the equation of a sinusoid curve which
describes the motion of the ant and draw a graph of the
function.
b What height in centimetres from the ground will the ant be 25
seconds into the trip given that 1 inch = 2.54 cm.?
c Within the first 10 seconds how many times will the ant be at its
starting height?
Question 4
The number of people in thousands employed in a resort town is
represented by the function
Take t = 0 as last day of January
Take t = 1 as last day of February
Take t = 2 as last day of March
a. Draw a rough sketch showing the variation in the number of
people employed in the town for one complete period.
b When is the maximum number of people employed and what is
this maximum number?
c During which months of the year will the number of people
employed be 4,650 or greater?
d Does this model have any drawbacks and if yes identify one?
Question 5
A tsunami (tidal wave) is a fast moving wave caused
by an underwater earthquake. The water oscillates
about its normal level, with equal amplitudes above
and below this level The period is fifteen minutes.
Suppose that a tsunami with an amplitude of ten
metres approaches the pier at Honolulu, where the
normal depth of water is nine metres. Assuming that the depth of
water varies sinusoidally with time as the tsunami passes, predict
the depth of the water at the following times after the tsunami first
reaches the pier.
a Two minutes, four minutes and twelve minutes.
b According to your model what will be the minimum depth of the
water?
c How do you interpret this answer in terms of what will happen
in the real world?
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Question 6
A buoy in the ocean is bobbing up and down in harmonic motion.
At t = 0 seconds, the buoy is at its high point and returns to that high
point every 8 seconds. The buoy moves a distance of 1.44 meters
from its highest point to its lowest point.
a. Using a sinusoidal function create a mathematical model to
represent the depth of water under the buoy and sketch the curve.
b How high is the buoy at time = 29 seconds?
c Is it rising or falling at that time?
d How many times in 2 minutes will the buoy be at sea level?
Question 7
If the depth of water in a canal continually varies between a minimum
2m. below a specified buoy mark and a maximum of 2m above this
mark over a 24-hour period.
a Construct a formula involving a trigonometric function to describe
this situation.
The road to an island close to the shore is sometimes covered with
water. When the water rises to the level of the road, the road is closed.
On a particular day, the water at high tide is 5 m above the mean sea
level.
b Construct a formula involving a trigonometric function to model
this situation if high tide occurs every 12 hours and the tide
behaves in simple harmonic motion. Draw a sketch of the model.
c Find the height of the road above sea level if the road is closed for
3 hours on the day in question.
d If the road were raised so that it is impassable for only 2 hours 20
minutes, by how much was it raised?
Question 8
At a certain latitude the number (d) of hours of daylight in each day is
given by, d = A + B sin kt° where A and B are positive constants and t
is the time in days after the spring equinox.
Assuming the number of hours of daylight follows an annual cycle of
365 days;
a Find the value of k correct to three decimal places.
b If the shortest and longest days have 6 and 18 hours of daylight
respectively state the values of A and B.
c Find in hours and minutes the amount of daylight on New Year’s
day which is 80 days before the spring equinox.
d A town at this latitude holds a fair twice a year on days that have
exactly 10 hours of daylight. Find, in relation to the spring equinox,
which two days these are.
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Solutions
Question 1
A wrecking ball attached to a crane swings back and forth. The
distance that the ball moves to the left and to the right of its resting
position with respect to time is represented by the following graph.
a. What is the period of the crane’s motion. Explain your answer?
b. What is the equation of the horizontal midway line of the curve
and what does it represent?
c. What is the amplitude of the crane’s motion? Draw a diagram to
represent what the amplitude represents in terms of the motion
of the ball?
d. How many complete swings will the wrecking ball make in four
minutes?
e. What does point A represent?
a. The period is 8 seconds. It represents the time taken for the
wrecking ball to complete a full swing (i.e. from the extreme left
to the extreme right and back to the extreme left again).
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b. The equation of the horizontal midway line is d = 0. This
represents the resting position of the ball.
c. The amplitude is 4 metres. This distance represents the maximum
distance the wrecking ball swings to the left and right of its
resting position.
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d. 4 minutes = 240 seconds
period = 8 seconds
number of swings =
e. Point A represents the position of the wrecking ball when
collecting and recording of the data began
Question 2
As a dolphin swims along by the side
of a cruise ship he jumps to a height of
four metres above the water surface and
dives to a depth of four meters below
the surface. He does this in a regular
motion (simple harmonic motion).
A passenger using a stopwatch and
starting it just as the dolphin is at the surface determines that the
dolphin completes one cycle every 8 seconds.
a Describe the displacement of the dolphin relative to the water
surface using a sinusoid curve and sketch the graph of the
dolphin’s displacement relative to the water surface
b After three seconds how high above the surface is the dolphin?
c Is the dolphin above or below the surface of the water after 37
seconds?
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a.
a ± b sin cx
a ± b cos cx
a = horizontal midway line
a=0
b = aimplitude
b=4
period =
shape = function normal
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b)
After 3 seconds
h = 2.828
After 3 seconds the dolphin is 2.828m above the water
c)
After 37 seconds
h = -2.828
The dolphin is below the surface of the water
Question 3
The inside rim of a bicycle wheel whose diameter is 25 inches, is 3
inches off the ground. An ant is sitting on the inside rim of the wheel
at the point 3 inches off the ground. Sean starts riding the bicycle at
a steady rate. The wheel makes one revolution every 1.6 seconds.
a Find the equation of a sinusoid curve which describes the motion
of the ant and draw a graph of the function.
b What height in centimetres from the ground will the ant be 25
seconds into the trip given that 1 inch = 2.54 cm.?
c Within the first 10 seconds how many times will the ant be at its
starting height?
a.
Horizontal Midway Line
Horizontal Midway Line © Project Maths Development Team 2012
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Exploring Trigonometric Graphs
a ± b sin cx
a ± b cos cx
a = horizontal midway line
a = 12.5
b = aimplitude
b = 9.5
period
c = 1.25π
shape = cos function reflected
h = height off the ground
h = 12.5-9.5cxos1.25πt
b. 25 seconds into the trip
t = 25
h = 12.5-9.5 cos 1.2π (25)
h = 19.218 inches = (19.218) (2.54) = 48.81372cm
Tar éis 25 soicind tá an seangán 48.81372cm ón talamh.
c. period = 1.6 seconds
in 10 seconds
cycle
6.25 cycle
at starting point: 7 times
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Question 4
The number of people in thousands employed in a resort town is
represented by the function
Take t = 0 as last day of January
Take t = 1 as last day of February
Take t = 2 as last day of March
a Draw a rough sketch showing the variation in the number of
people employed in the town for one complete period.
b When is the maximum number of people employed and what is
this maximum number?
c During which months of the year will the number of people
employed be 4650 or greater?
d Does this model have any drawbacks and if yes identify one?
a.
a - b cos cx
3.8-1.7 cos
a = horizontal midway line = 3.8
b = aimplitude = 1.7
period
shape = cos function reflected
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a.
b.
Range = [3.8 - 1.7, 3.8 + 1.7]
Range = [2.1, 5.5]
Maximum number employed = 5,500 people on the last day of
July
c.
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d. A drawback of this model is that it assumes all months of the
year have the same number of days.
Question 5
A tsunami (tidal wave) is a fast moving wave caused by an
underwater earthquake. The water oscillates about its normal level,
with equal amplitudes above and below this level The period is
fifteen minutes. Suppose that a tsunami with an amplitude of ten
metres approaches the pier at Honolulu, where the normal depth
of water is nine metres. Assuming that the depth of water varies
sinusoidally with time as the tsunami passes, predict the depth of the
water at the following times after the tsunami first reaches the pier.
a Two minutes, four minutes and twelve minutes.
b According to your model what will be the minimum depth of the
water?
c How do you interpret this answer in terms of what will happen in
the real world?
a.
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b The minimum level of the water appears to be -1m.
c The water level can never be less than 0m. The water would
drain away from the pier for a short period of time.
Question 6
A buoy in the ocean is bobbing up and down in harmonic motion.
At t = 0 seconds, the buoy is at its high point and returns to that
high point every 8 seconds. The buoy moves a distance of 1.44
meters from its highest point to its lowest point.
a Using a sinusoidal function create a mathematical model to
represent the depth of water under the buoy and sketch the
curve.
b How high is the buoy at time = 29 seconds?
c Is it rising or falling at that time?
d How many times in 2 minutes will the buoy be at sea level?
a.
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normal
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b.
c.
d.
Question 7
If the depth of water in a canal continually varies between a
minimum 2m. below a specified buoy mark and a maximum of 2m
above this mark over a 24-hour period.
a. Construct a formula involving a trigonometric function to
describe this situation.
The road to an island close to the shore is sometimes covered with
water. When the water rises to the level of the road, the road is
closed. On a particular day, the water at high tide is 5 m above the
mean sea level.
b Construct a formula involving a trigonometric function to model
this situation if high tide occurs every 12 hours and the tide
behaves in simple harmonic motion. Draw a sketch of the model.
c Find the height of the road above sea level if the road is closed
for 3 hours on the day in question.
d If the road were raised so that it is impassable for only 2 hours
20 minutes, by how much was it raised?
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Exploring Trigonometric Graphs
a.
b.
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Exploring Trigonometric Graphs
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30
Exploring Trigonometric Graphs
c.
d.
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Exploring Trigonometric Graphs
Question 8
At a certain latitude the number (d) of hours of daylight in each day is
given by, d = A + B sin kt° where A and B are positive constants and
t is the time in days after the spring equinox. Assuming the number of
hours of daylight follows an annual cycle of 365 days;
a Find the value of k correct to three decimal places.
b If the shortest and longest days have 6 and 18 hours of daylight
respectively state the values of A and B.
c Find in hours and minutes the amount of daylight on New Year’s
day which is 80 days before the spring equinox.
d A town at this latitude holds a fair twice a year on days that have
exactly 10 hours of daylight. Find, in relation to the spring equinox,
which two days these are.
a.
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Exploring Trigonometric Graphs
b.
c.
d.
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Exploring Trigonometric Graphs
Student Activity Sheet 1 Graphs of sinx and
cosx
Complete the table below and then draw the graph of sinx and cosx
x
y = sinx
y = cosx
0°
30° 45°
60°
90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360°
0°
x
y = sin x
y = cos x
30° 45°
60°
90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360°
What is the horizontal midway line of each graph?
sin x____________________________________cos x_____________________________________
What is the maximum of each graph?
sin x____________________________________cos x_____________________________________
What is the minimum of each graph?
sin x____________________________________cos x_____________________________________
What is the range of each graph?
sin x____________________________________cos x_____________________________________
The amplitude of a graph is the greatest height the graph is above the horizontal midway line.
What is the amplitude of each graph?
sin x____________________________________cos x_____________________________________
When a graph repeats itself after a particular interval the interval is known as the period. What is
the period of each graph?
sin x____________________________________cos x_____________________________________
What difference if any is there between the curves of sin x and cos x?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Card Set A1 Trigonometric Functions g (x) = bsin x
A1
A2
g (x) = -3 sin x
A3
g (x) = 2 sin x
A4
g (x) = ½ sin x
A5
g (x) = 5 sin x
A6
g (x) = -4 sin x g (x) = ²⁄3 sin x
A7
A8
g (x) = -¼ sin x
A9
g (x) = sin x
A10
g (x) = -sin x
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g (x) = -¹⁄3 sin x
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Exploring Trigonometric Graphs
Card Set A2 Trigonometric Functions g (x) = bcos x
A11
A12
g (x) = -2 cos x
A13
g (x) = cos x
A14
g (x) = ¼ cos x
A15
g (x) = cos x
A16
g (x) = -5 cos x g (x) = ½ cos x
A17
A18
g (x) = -³⁄2 cos x g (x) = 6cos x
A19
A20
g (x) = -cos x g (x) = -¹⁄5 cos x
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Exploring Trigonometric Graphs
Student Activity Sheet 2a: Mathematical Language /
Properties of Trigonometric Graphs
Card
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
Curve
Amplitude
Range
g (x) = -3 sin x
g (x) = 2 sin x
g (x) = ½ sin x
g (x) = 5 sin x
g (x) = -4 sin x
g (x) = ²⁄3 sin x
g (x) = -¼ sin x
g (x) = sin x
g (x) = -sin x
g (x) = -¹⁄3 sin x
Period
Vertical
Transformation (3)
Vertical
Shape (3)
Horizontal
Shape (3)
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
From your observations above what is the effect of b on the curve?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Student Activity Sheet 2b: Mathematical Language /
Properties of Trigonometric Graphs
Card
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
Curve
Amplitude
Range
g (x) = -2 cos x
g (x) = cos x
g (x) = ¼ cos x
g (x) = cos x
g (x) = -5 cos x
g (x) = ½ cos x
g (x) = -³⁄2 cos x
g (x) = 6cos x
g (x) = -cos x
g (x) = -¹⁄5 cos x
Period
Vertical
Transformation (3)
Vertical
Shape (3)
Horizontal
Shape (3)
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
From your observations above what is the effect of b on the curve?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Student Activity Sheet 2c: Mathematical Language /
Properties of Trigonometric Graphs
Card
Curve
Amplitude
Range
Period
Vertical
Transformation (3)
Vertical
Shape (3)
Horizontal
Shape (3)
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
From your observations above what is the effect of b on the curve?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Card Set B: Trigonometric Functions Important features
B1
B2
g (x) = cos ½x
B3
g (x) = sin 2x
B4
g (x) = sin 3x
B5
g (x) = cos 4x
B6
g (x) = cos 3x
B7
g (x) = sin 4⁄5x
B8
g (x) = sin 4x
B9
g (x) = cos 2⁄3x
B10
g (x) = cos 2x
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g (x) = sin ¹⁄4x
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Exploring Trigonometric Graphs
Student Activity Sheet 3: Mathematical Language /
Properties of Trigonometric Graphs
Card
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
Curve
Amplitude
Range
g (x) = cos ½x
g (x) = sin 2x
g (x) = sin 3x
g (x) = cos 4x
g (x) = cos 3x
g (x) = sin 4⁄5x
g (x) = sin 4x
g (x) = cos 2⁄3x
g (x) = cos 2x
g (x) = sin ¹⁄4x
Period
Vertical
Transformation (3)
Vertical
Shape (3)
Horizontal
Shape (3)
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
Normal
Stretch
Stretch
Reflected
Shrink
Shrink
From your observations above what is the effect of c on the curve?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Card Set C
g (x) = 3sin ½x g (x) = 3sin ½x
g (x) = 3sin ½x g (x) = -5cos 3x
g (x) = -5cos 3x g (x) = -5cos 3x
g (x) = ½ sin 4x g (x) = ½ sin 4x
g (x) = ½ sin 4x g (x) = -2⁄3 cos ¹⁄4x
g (x) = -2⁄3 cos ¹⁄4x g (x) = -2⁄3 cos ¹⁄4x
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Exploring Trigonometric Graphs
Card Set C (continued)
g (x) = 5cos 2x g (x) = 5cos 2x
g (x) = 5cos 2x g (x) = 1⁄2 sin 2⁄3 x
g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x
g (x) = -4 cos 1⁄3x g (x) = -4 cos 1⁄3x
g (x) = -4 cos 1⁄3x g (x) = -2⁄5 cos 6x
g (x) = -2⁄5 cos 6x g (x) = -2⁄5 cos 6x
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Exploring Trigonometric Graphs
Card Set C (continued)
g (x) = 5cos 2x g (x) = 1⁄2 sin 2⁄3 x
g (x) = 1⁄2 sin 2⁄3 x g (x) = 1⁄2 sin 2⁄3 x
g (x) = -4 cos 1⁄3x g (x) = -4 cos 1⁄3x
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Exploring Trigonometric Graphs
Student Activity Sheet 4 Mathematical Language /
Properties of Trigonometric Graphs
Vertical Transformation
Normal
Reflected
Vertical Shape
Stretch
Shrink
Horizontal Shape
Shrink
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Stretch
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Exploring Trigonometric Graphs
Student Activity Sheet 5 Mathematical Language /
Properties of Trigonometric Graphs
Curve
g (x) =
g (x) =
g (x) =
g (x) =
g (x) =
g (x) =
g (x) =
g (x) =
g (x) =
© Project Maths Development Team 2012
Amplitude
Range
Period
6
∕3
2π
[-4, 4]
2
∕2
π
[-1∕2, 1∕2]
6π
[-2, 2]
2π
π
¹⁄5
3
3π
[-3, 3]
4π
¼
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Exploring Trigonometric Graphs
Student Activity Sheet 6: Transformation of
Trigonometric Graphs
Curve
Effect of the value a in
Horizontal Midway Line
Amplitude
Range
h(x) =1 + sin x
h(x) =2 + sin x
h(x) =3 + sin x
h(x) =-1 + sin x
h(x) =-2 + sin x
h(x) =-3 + sin x
h(x) =1 + cos x
h(x) =2 + cos x
h(x) =3 + cos x
h(x) =-1 + cos x
h(x) =-2 + cos x
h(x) =-3 + cos x
From your observations above what is the effect of a on the curve?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Exploring Trigonometric Graphs
Student Activity Sheet 7: Mathematical Language /
Properties of Trigonometric Graphs
Curve
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Vertical
Translation
Vertical Shape
Amplitude
Range
Period
g (x) = 2 + 3 sin 4
g (x) = 3 - 4 cos 2x
g (x) = -4 - 2 sin 3
g (x) = -1 + 5 sin 2
g (x) = 5 - cos x⁄2
g (x) = 3 sin 3x
g (x) = 1 + cos 4x
g (x) = ½ + 5 cos x
g (x) = 3 - 3 sin3x
g (x) = ¼ - ½ sin ¹∕3x
g (x) = 2 + 3 sin 4x
g (x) = 1 + 3 cosx
g (x) = 4 + 3 cos ½x
g (x) = 5 sinx +1
g (x) = -1 -sinx 2∕3x
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Exploring Trigonometric Graphs
Card Set D: Trigonometric Graphs
D1
D2
D3
D4
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Exploring Trigonometric Graphs
Card Set D: Trigonometric Graphs (continued)
D5
D6
D7
D8
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Exploring Trigonometric Graphs
Student Activity Sheet 8: Trigonometric Graphs
Graph Equation Horizontal
Midway line
Function Type And
Vertical Shape
Amplitude
Range
Number of
cycles in 2π
Period
D1
D2
D3
D4
D5
D6
D7
D8
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Exploring Trigonometric Graphs
Card Set E: Trigonometric Graphs
E1
E2
f (x) = ½ -2 sin 3x f (x) = 1 + sin ³⁄2 x
E3
E4
f (x) = -2 + 3 cos ¹⁄3x f (x) = 1 + 2 sin 3x
E5
E6
f (x) = 2 - ½ sin 5x f (x) = -1 + ½ cos 3x
E7
E8
f (x) = 3 -2 cos ½x f (x) = -2 - cos 4x
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Exploring Trigonometric Graphs
Card Set F
F1
F2
k (x) = 2 + 3 sin 4x k (x) = 5 - 4 cos 2x
F3
F4
k (x) = 1 + 3 cos ½ x k (x) = ½ - sin 3x
F5
F6
k (x) = 3 + 2 cos ¹⁄4 x k (x) = 1 - ½ sin x
F7
F8
k (x) = 2 + ¹⁄4 cos 3x
F9
k (x) = 4 sin 2x
F10
k (x) = -1 -2 sin 3x k (x) = -2 + 3 cos 4x
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