# Great Lakes Christian College Unit 3 Average Rate of Change Exercises

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Im looking for a tutor who has foundation in the following topics trigomertic functions and characteristics functions, and can use the provided rubrics to solve the provided assignment.

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TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
Culminating Task: Making the World a Better Place
Part of the evaluation of this course will be in the form of a Culminating Task, worth 15% of
In this activity you will explore connections between the concepts learned in this course
and environmental issues. Specifically, you will create a mathematical model and visual
representations to provide evidence to convince investors that a way to improve an
environmental issue of your choice would be a worthy project in which to invest.
Overview
You have decided that you are going to make a pitch to the hosts of a reality TV show where
entrepreneurs request funding for projects, or products, designed to reduce the damage to
the environment caused by human activity.
Your pitch must combine a compelling reason why your initiative should be supported, with
evidence that justifies the need for action. You will research and collect secondary data,
use the data to create graphical and algebraic models, make predictions based on your
algebraic model and create a visual presentation using the media/ platform of your choice
(PowerPoint, WordPress, Prezi, Weebly, iMovie, Show Me, Blogger, Tumblr, YouTube, etc.).
The function used to create your algebraic model must be one of the types of functions
(polynomial of degree 3 or higher, exponential, logarithmic or trigonometric) covered in this
course.
Step 1:
Identify and describe an environmental issue and a proposed way to reduce impact and
improve outcomes. The proposal could be as simple as a campaign to convince people to
change daily habits that are harmful to the environment or as complex as developing a new
environmentally friendly product or process.
Step 2:
Research secondary data from a reliable source and create a mathematical model to predict
outcomes if no action to reduce harmful effects is taken.
Step 3:
Create algebraic and graphic representations of your model and provide a brief, but
convincing, description of the benefits of your proposal.
1
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
This assessment is out of 60 marks and is worth 15% of your final mark. You will submit
your Culminating Task upon completion of Unit 4 where you will be prompted at the end of
the final learning activity to submit your assessment for feedback and a grade by selecting
the “Assignments” link and following the submission directions.
Your proposal should contain the following elements:
Weight
(No. of marks)
A description of the environmental issue that needs to be addressed. 2
A description of the product or change in behaviour that could
reduce the damage caused by the environmental issue, with
justification.
5
A set of secondary data from a reliable source, in table form,
pertaining to the environmental issue. It must be appropriate for
modeling with one of the types of functions studied in this course:
polynomial of degree 3 or higher; exponential; logarithmic or
used to create the table and would facilitate graphing the data.
8
The APA reference for the data set.
2
5
A scatter plot of the data including appropriate titles, scales and
units, created with the graphing technology of your choice.
2
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
An algebraic model that fits the data. The model can feature one
of the functions mentioned above or could be a combination or
composite of two of those functions. The choice of model should be
based on visual inspection of data trends observed and does not
need to be supported by statistical analysis, which is beyond the
scope of this course.
5
An explanation of how the algebraic model was determined,
including justification for the choice of parent function (or combined
or composite function) and any transformations used to model the
data.
5
restrictions on the domain of your function and a description,
including justification, of potential limitations of the algebraic model
as it applies to correlation with the data and the ‘real world’ context.
5
A graph of the algebraic model, including appropriate titles, scales,
units, and restrictions on the domain, created with graphing
technology, and superimposed on the scatter plot.
5
At least one calculation using the algebraic model to make a
prediction, or for emphasis.
3
A brief sales pitch, using the graph as evidence, and calculations
involving the algebraic model to make predictions to convince the
reality TV show panel that your proposal deserves support.
10
Correct use of mathematical terminology and conventions
throughout.
5
3
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
The teacher will assess your work using the following rubric. Before submitting your
assessment, review the rubric to ensure that you are meeting the success criteria to the best
Categories
Knowledge and
Understanding
Level 4
Level 3
Level 2
Level 1
(80 – 100%)
A data table is
created with a
high degree of
effectiveness.
(70 – 79%)
A data table is
created with
considerable
effectiveness.
(60 – 69%)
A data table
is created
with some
effectiveness.
(50 – 59%)
A data table
is created
with limited
effectiveness.
Graphing
technology is
used to create
the required
graphs with a
high degree of
effectiveness.
Graphing
technology is
used to create
the required
graphs with
considerable
effectiveness.
Graphing
technology is
used to create
the required
graphs with some
effectiveness.
Graphing
technology
is used to
create the
required graphs
with limited
effectiveness.
Calculations
using the
algebraic model
are carried
out with a
high degree of
effectiveness.
Calculations
using the
algebraic model
are carried out
with considerable
effectiveness.
Calculations
using the
algebraic model
are carried
out with some
effectiveness.
Calculations
using the
algebraic model
are carried out
with limited
effectiveness.
4
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
Thinking
between the
environmental
issue, and
the product or
initiative being
proposed to
mitigate it, is
established with
a high degree of
effectiveness.
between the
environmental
issue, and
the product or
initiative being
proposed to
mitigate it, is
established with
considerable
effectiveness.
between the
environmental
issue, and
the product or
initiative being
proposed to
mitigate it, is
established
with some
effectiveness.
between the
environmental
issue, and
the product or
initiative being
proposed to
mitigate it, is
established
with limited
effectiveness.
The algebraic
model represents
the data with a
high degree of
effectiveness.
The algebraic
model represents
the data with
considerable
effectiveness.
The algebraic
model represents
the data
with some
effectiveness.
The algebraic
model represents
the data
with limited
effectiveness.
Restrictions on
the domain and
limitations of
the model suit
the context, and
they take into
consideration
the correlation
between the
algebraic
model and the
data, with a
high degree of
effectiveness.
Restrictions on
the domain and
limitations of
the model suit
the context, and
they take into
consideration
the correlation
between the
algebraic model
and the data,
with considerable
effectiveness.
Restrictions on
the domain and
limitations of
the model suit
the context, and
they take into
consideration
the correlation
between the
algebraic model
and the data,
with some
effectiveness.
Restrictions on
the domain and
limitations of
the model suit
the context, and
they take into
consideration
the correlation
between the
algebraic model
and the data,
with limited
effectiveness.
5
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
Application
Appropriate
secondary data
is selected with
a high degree of
effectiveness.
Appropriate
secondary data
is selected with
considerable
effectiveness.
Appropriate
secondary data
is selected
with some
effectiveness.
Appropriate
secondary data
is selected
with limited
effectiveness.
The process
used to
determine the
algebraic model
is described with
a high degree of
effectiveness.
The process
used to
determine the
algebraic model
is described with
considerable
effectiveness.
The process
used to
determine the
algebraic model
is described
with some
effectiveness.
The process
used to
determine the
algebraic model
is described
with limited
effectiveness.
Transformations
applied to the
chosen parent
function(s) are
appropriate and
are justified with
a high degree of
effectiveness.
Transformations
applied to the
chosen parent
function(s) are
appropriate and
are justified with
considerable
effectiveness.
Transformations
applied to the
chosen parent
function(s) are
appropriate and
are justified
with some
effectiveness.
Transformations
applied to the
chosen parent
function(s) are
appropriate and
are justified
with limited
effectiveness.
The algebraic
model is used
to support a
point or make
a prediction
within the
presentation with
a high degree of
effectiveness.
The algebraic
model is used
to support a
point or make a
prediction within
the presentation
with considerable
effectiveness.
The algebraic
model is used
to support a
point or make a
prediction within
the presentation
with some
effectiveness.
The algebraic
model is used
to support a
point or make a
prediction within
the presentation
with limited
effectiveness.
6
TVO ILC
MHF4U – Unit 2
Culminating Task: Making the World a Better Place
Communication
Mathematical
language
and notation,
throughout the
activity, are used
Mathematical
language
and notation,
throughout the
activity, are used
Mathematical
language
and notation,
throughout the
activity, are used
with a high
degree of
effectiveness.
with considerable with some
effectiveness.
effectiveness.
with limited
effectiveness.
The
environmental
issue is
described with a
high degree of
effectiveness.
The
environmental
issue is
described with
considerable
effectiveness.
The
environmental
issue is
described
with some
effectiveness.
The
environmental
issue is
described
with limited
effectiveness.
The reference
for the data
set is stated in
APA style with a
high degree of
effectiveness
The reference
for the data
set is stated in
APA style with
considerable
effectiveness.
The reference
for the data set
is stated in APA
style with some
effectiveness
The reference
for the data set
is stated in APA
style with limited
effectiveness.
Elements of the
activity that are
pertinent to the
presentation
are synthesized
to create a
convincing bid
for support with
a high degree of
effectiveness.
Elements of the
activity that are
pertinent to the
presentation
are synthesized
to create a
convincing
presentation
Elements of the
activity that are
pertinent to the
presentation
are synthesized
to create a
convincing
presentation
with considerable with some
effectiveness.
effectiveness.
Elements of the
activity that are
pertinent to the
presentation
are synthesized
to create a
convincing
presentation
with limited
effectiveness.
Mathematical
language
and notation,
throughout the
activity, are used
7
TVO ILC
MHF4U
Unit 3 Assessment
Unit 3 Assessment
1.
Determine the following: [K4]
a. Given a sector with an arc length of 4.5 cm and a radius of 3 cm, find the angle
subtended by the arc.
b. Given a sector with an angle of 13 radians and a radius of 7 m, find the arc length.
1
TVO ILC
MHF4U
Unit 3 Assessment
2. Fill in the table. Show your work and keep answers exact. [K4]
Degrees
_
​​  2π
45 ​​
​​
​2​ 30​​  ​
0
3. Determine the values of ​θ​if 0

4. Determine exact values of ​θ​if
≤ θ ≤ 2π​ given that ​cosx = − 0.3178​. Explain
​0 ≤ θ ≤ 2π​ given that ​tanθ =
_
−  ​ √  ​​
3 . Explain
5. Describe the transformations on the given cosine function. [C4]
f​​(x)​ = 0.5cos​
[0.5​(x− π)​]​− 12​
2
TVO ILC
MHF4U
Unit 3 Assessment
6. Find an equation for each function and explain your thought process.
a. A sinusoidal function with an amplitude of 15 units, a period of
radians to the right, axis at
y​ = ​33
, a phase shift of
[A4] [C2]
b. [A4] [C2]
3
TVO ILC
MHF4U
Unit 3 Assessment
7.
Determine the exact value of tan
. [T4]
8. Determine a value of x for each of the following. Give answers in simplest exact form.
[A6]
a.
_ ​ = sinx​
b. c
​ os ​11π
31
c.
​tan ​π_8 ​ = cotx​
9. Prove the following identity. Explain your thought process. [T3] [C2]
​cos​​  ​  x
​cot​​  2​  x =  ​ _
​1 − cos​​  2​  x ​
2
4
TVO ILC
MHF4U
Unit 3 Assessment
10.Solve the following trigonometric equation for ​0
[A4] [C2]

​cos​​  2​  x

≤ x ≤ 2π​. Explain your process.
1 = 0​
11.The depth of water in a harbour varies as a function of time. The maximum depth is
9 feet and the minimum depth is 1 foot. The depth can be modelled with a sinusoidal
function that has a period of 12 hours. If the depth is 5 feet at 12 midnight, and is
increasing,
a. C
reate an algebraic model to predict the depth of the water as a function of time.
b. The water must be at least 7 feet for Annie’s fishing boat to safely navigate the
harbour. She wants to enter the harbour during the afternoon.
i. Create a graph of this function using technology. [T2]
5
TVO ILC
MHF4U
Unit 3 Assessment
ii. What is the earliest time she can enter the harbour? [T2]
iii. How long can she safely stay in the harbour? [T2]
6
TVO ILC
MHF4U 4.8
Unit 4 Assignment
Unit 4 Assignment
1. Given that ​f(​ ​​x)​ ​​ ​​= { (1,3), ( 5,7), (9, 11), ( 13, -5)} and ​g​(​​x​)​​​​= { (-2,33), ( 1,-1), (5, 9)}.
Determine the following: [K3]
a.

​f​(​​x​)​​ + g​(​​x​)​​ =​
b.

​g​(​​x​)​​ − f​(​​x​)​​ =
c.

​f​(​​x​)​​ ∙ g​(​​x​)​​ =
2. Given that ​f​(x)​  = 2x− 3​and g​ ​(x)​ = 9 ​
x​​  2​− 5x+ 11​, determine the following:
a.
b.
c.
d.

​g​(​​x​)​​ − f​(​​x​)​​ =
[K2]

​f​(​​x​)​​ ∙ g​(​​x​)​​ =
[K2]

​  ​ =
[K1]
f​(x)​
_
g​(​​x​)​​

​g ∘ f​(​​x​)​​​​
[K2]
3. Use the given graphs to sketch the graph of ​f​(​​x​)​​ + g​(​​x​)​​​​. Explain your process. [A3] [C2]
​f​(​​x​)​​​​
​g​(​​x​)​​​
​f​(​​x​)​​ + g​(​​x​)​​​
1
TVO ILC
MHF4U 4.8
Unit 4 Assignment
4. State the domain and range of each function below. Explain your process.
Function
[A4] [C2]
Reasoning
_
f​(​ x)​ = 13 ​√ − 4x
+ 16 ​+ 19
1
​f​(x)​  =  ​  _
6x + 81​− 99​
5. Calculate the average rate of change for the function ​f(​ t)​ = 2 ​
x​​  3​+ 5​ over the interval
− 1 ≤ x ≤ 2​.
[K3]
6. a.Use an ​x− ​interval of 0.0001 to determine the instantaneous range of change of the
function ​f​(x)​= ​x​​  4​+ 8 ​x​​  3​ + 19 ​x​​  2​+ 12x​at x​ = − 2​.
Give answer to the nearest tenth.
b.
[A3]
Determine whether the function ​f​(x)​= ​x​​  4​+ 8 ​x​​  3​ + 19 ​x​​  2​+ 12x​ at ​x = − 2
has a maximum or minimum at the point where​x = − 2​. Justify your answer.
[A5] [C2]
7. Use algebra to determine whether each function is even, odd or neither.
a.

f​(x)​  = xcosx+ 1
b.
tanx

f​(x)​  =  ​_
​x​​  ​ ​​
[A6] [C2]
3
2
TVO ILC
MHF4U 4.8
Unit 4 Assignment
8. A marine biologist measures the presence of a pollutant in an ocean and concludes that
the concentration, ​C​, in parts per million (ppm), as a function of the population, ​P​, of the
people who visit the beach is given by ​C(​ P)​ = 1.38P+ 97.4​. The population of people
visiting the beach, in thousands, can be modeled by ​P(​ t)​  = 12 ​​(1.078)​​​ t​ where t is the
time in years since the first measurement.
a.
Determine an equation, in simplified form, for the concentration of pollutant as a
function of the number of years since the first measurement.
b.
[T2]
How long (to the nearest year) will it take for the concentration to reach 180 ppm?
[T3]
9. You are designing a water slide. Your first step is to create an algebraic model to
describe the slide’s profile, as shown.
a.
Use graphing technology to determine an equation for the algebraic model.
Consider the shape only; do not worry about scale. Describe how you arrived at
b.
Because of safety concerns, you want to make the slide less steep, and with smaller
bumps. Suggest changes to your algebraic model and an include a graph that
shows your original design and the one that has the improved safety features.
[T2] [C2]
3
Knowledge & Understanding: /16
Level 4
80-100%
Level 3
70-79%
Level 2
60-69%
Level 1
50-59%
Uses the
Uses the
Uses the
Uses the
relationship between relationship between relationship between relationship between
length and angle at length and angle at length and angle at length and angle at
the centre with a the centre with the centre with the centre with
high degree of considerable
some effectiveness limited effectiveness
effectiveness
effectiveness
Converts between
exact values in
degrees with a high
degree of
effectiveness
Converts between
exact values in
degrees with
considerable
effectiveness
Converts between
exact values in
degrees with some
effectiveness
Converts between
exact values in
degrees with limited
effectiveness
Solve linear
trigonometric
equations, for the
domain of real
values from 0 to 21
with a high degree
of effectiveness
Solve linear
trigonometric
equations, for the
domain of real
values from 0 to 2TT
with considerable
effectiveness
Solve linear
trigonometric
equations, for the
domain of real
values from 0 to 2TT
with some
effectiveness
Solve linear
trigonometric
equations, for the
domain of real
values from 0 to 2T
with limited
effectiveness
Thinking: /17
Level 4
80-100%
Level 3
70-79%
Level 2
60-69%
Level 1
50-59%
Uses compound
angle formulas to
determine exact
values of
trigonometric ratios
with a high degree
of effectiveness
Uses compound
angle formulas to
determine exact
values of
trigonometric ratios
with considerable
effectiveness
Uses compound
angle formulas to
determine exact
values of
trigonometric ratios
with some
effectiveness
Uses compound
angle formulas to
determine exact
values of
trigonometric ratios
with limited
effectiveness
Prove trigonometric
identities through
the application of
reasoning skills,
using a variety of
relationships with a
high degree of
effectiveness
Prove trigonometric
identities through
the application of
reasoning skills,
using a variety of
relationships with
considerable
effectiveness
Prove trigonometric
identities through
the application of
reasoning skills,
using a variety of
relationships with
some effectiveness
Prove trigonometric
identities through
the application of
reasoning skills,
using a variety of
relationships with
limited effectiveness
Solves problems
based on
applications
involving a
trigonometric
function using a
graph generated
with technology with
a high degree of
effectiveness
Solves problems
based on
applications
involving a
trigonometric
function using a
graph generated
with technology
considerable
effectiveness
Solves problems
based on
applications
involving a
trigonometric
function using a
graph generated
with technology
some effectiveness
Solves problems
based on
applications
involving a
trigonometric
function using a
graph generated
with technology
limited effectiveness
Application: /18
Level 4
80-100%
Level 3
70-79%
Level 2
60-69%
Level 1
50-59%
Represents a Represents a Represents a Represents a
sinusoidal function sinusoidal function sinusoidal function sinusoidal function
with an equation, with an equation, with an equation, with an equation,
given its graph or its given its graph or its given its graph or its given its graph or its
properties, with properties, with properties, with properties, with
angles expressed in angles expressed in angles expressed in angles expressed in
degree of
considerable
effectiveness
effectiveness
effectiveness
effectiveness
Solves equations
using equivalent
trigonometric
expressions with a
high degree of
effectiveness
Solves equations
using equivalent
trigonometric
expressions with
considerable
effectiveness
Solves equations
using equivalent
trigonometric
expressions with
some effectiveness
Solves equations
using equivalent
trigonometric
expressions with
limited effectiveness
trigonometric
equations for the
domain of real
values from 0 to 2TT
with a high degree
of effectiveness
trigonometric
equations for the
domain of real
values from 0 to 2TT
with considerable
effectiveness
trigonometric
equations for the
domain of real
values from 0 to 2TT
with some
effectiveness
trigonometric
equations for the
domain of real
values from 0 to 2TT
with limited
effectiveness
Communication: /18
Level 4
80-100%
Level 3
70-79%
Level 2
60-69%
Level 1
50-59%
Expresses and
organizes
mathematical
thinking with a high
degree of
effectiveness
Expresses and
organizes
mathematical
thinking with
considerable
effectiveness
Expresses and
organizes
mathematical
thinking with some
effectiveness
Expresses and
organizes
mathematical
thinking with limited
effectiveness
Uses mathematical
conventions,
vocabulary, and
terminology with a
high degree of
effectiveness
Uses mathematical
conventions,
vocabulary, and
terminology with
considerable
effectiveness
Uses mathematical
conventions,
vocabulary, and
terminology with
some effectiveness
Uses mathematical
conventions,
vocabulary, and
terminology with
limited effectiveness

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