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FLORIDA INTERNATIONAL UNIVERSITY

MATH GYM STUDENT WORKBOOK

Fall 2021

1

Table of Contents

WELCOME NOTE……………………………………………………………………………………………………………………2

TOPICS FOR FALL 2021 ……………………………………………………………………………………………………..3

MAC 1105 MATH GYM FALL 2021 WEEK 1 AND WEEK 2……………………………………………………6

MAC 1105 MATH GYM FALL 2021 WEEK 3………………………………………………………………………13

MAC 1105 MATH GYM FALL 2021 WEEK 4………………………………………………………………………16

MAC 1105 MATH GYM FALL 2021 WEEK 5………………………………………………………………………19

MAC 1105 MATH GYM FALL 2021 WEEK 6………………………………………………………………………23

MAC 1105 MATH GYM FALL 2021 WEEK 7………………………………………………………………………28

MAC 1105 MATH GYM FALL 2021 WEEK 8………………………………………………………………………31

MAC 1105 MATH GYM FALL 2021 WEEK 9………………………………………………………………………35

MAC 1105 MATH GYM FALL 2021 WEEK 10……………………………………………………………………38

MAC 1105 MATH GYM FALL 2021 WEEK 11……………………………………………………………………40

MAC 1105 MATH GYM FALL 2021 WEEK 12……………………………………………………………………42

MAC 1105 MATH GYM FALL 2021 WEEK 13……………………………………………………………………46

MAC 1105 MATH GYM FALL 2021 WEEK 14……………………………………………………………………49

MAC 1105 MATH GYM FALL 2021 WEEK 15……………………………………………………………………51

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WELCOME NOTE
Welcome to Math Gym. This element of your College Algebra course is designed to provide you with the

opportunity to gain more in-depth understanding of the concepts involved in the course, as well as

provide you with the chance to work collaboratively with your peers and engage with Mathematical

processes. While the Lab gives you the chance to practice skills of algebraic manipulation and test your

conceptual understanding frequently by combining multiple concepts into one problem, the math gym

problems will ask you to think deeply about the concepts and use your own words to explain that

thinking. Also, throughout the workbook you will work on questions you may see in future classes.

How Math Gym operates:

• Once you signed up for a Math Gym, you will continue to meet each week at that Math gym

• You are to complete the Math Gym worksheet for that respective week prior to attending each Math

Gym. If you cannot answer a question, write what about the question is difficult, be specific.

• For Virtual Gyms: You must upload the complete worksheet for that respective week via google

classroom. You LA will provide the link to your respective google classroom

You must have a working webcam and microphone to enter your Math Gym. Your cam must remain on

at all times, while you are in the math gym session

• You and your classmates will share your work, defend your answers and pose questions to each other

and your LA

• Concept maps and/or challenge questions will be graded for clarity and correctness. A grade of 0%

represents minimal effort and/or inconsistent or incoherent work. A grade of 50% represents work that

shows effort to fully answer the question being asked but lacks the mathematical accuracy or cohesion.

A grade of 100% represents an answer that attempts to fully address the intent of the problem and is

mathematically coherent

• If you miss a Math Gym for an excused absence, you need to speak to your professor (not LA) to ensure

that the excuse is accepted.

• It is expected that the work in Math Gym be done in groups. Your LA may have to move you in order to

maximize the effectiveness of the learning environment

• Bring your own questions to Math Gym. Ask “why” a lot. Be on time and attend every week!

The Math Gym questions are designed to get at the meaning of the Math. If something does not make

sense, or you are doing a step just because that is a step that you saw someone else do (teacher, LA,

peer, help me solve this) then ask for help. Math makes sense-always.

All of us here in the Mastery Math Lab wish you all the best for this semester and hope to be able to

help you on your journey of understanding.

3

TOPICS FOR FALL 2021

Fall 2021
Week

Starting
Topics (Sections – Blitzer)

Week 1 8/23

Pre-Class Assignment: Functions

In-Class: Functions (2.1)
Functional Notation (2.1)
Domain and Range (2.1)

Week 2 8/30

Pre-Class Assignment: Multiple representations of functions

In-Class: Graphs of Functions (2.1)
Properties of Functions (2.2)

Week 3 9/6

Scheduling period for Test 1:

Pre-Class Assignment: Library of Functions

In-Class: Graphing Techniques (2.5)

Week 4 9/13

Test 1 in Math Lab (2.1, 2.2, and 2.5)

Pre-Class Assignment: Introduction to piece wise functions and Average rate of
change.

In-Class: Piecewise Functions (2.2)
Average rate of change (2.5)

Week 5 9/20

Pre-Class Assignment: Find Sum, Difference, Product of Functions

In-Class: Quotient of Functions (2.6)
Composition of Functions (2.6)
Difference Quotient (2.2)

Week 6 9/27

Pre-Class Assignment: Graph of a Quadratic Function Intro

In-Class: Quadratic functions and Their Graphs (3.1), Mathematical Models (3.1)
to Graphing Techniques (2.5)

Week 7 10/4

Scheduling period for Test 2

4

Fall 2021
Week

Starting
Topics (Sections – Blitzer)

Pre-Class Assignment: Domain of Rational Functions

In-Class: Rational Functions: Domain, Asymptotes, and Graph (3.5)

Week 8 10/11

Test 2 in Lab (2.2, 2.5, 2.6, 3.1, 3.5)

Pre-Class Assignment: Solving from Graph

In-Class: Solving Polynomial and Rational Inequalities (3.6)

Week 9 10/18

Pre-Class Assignment: Basics of one-to-one

In-Class: One-to-one Functions (2.7)
Inverse Functions (2.7)

Week 10 10/25

Scheduling period for Test 3

Pre-Class Assignment: Exponential Exercise

In-Class: Exponential Functions (4.1), Basic Exponential Equations (4.4)

Week 11 11/1

Last day to Drop is Monday, 11/2 at 11:59pm

Test 3 in Lab (2.7, 3.6, 4.1, 4.4)

Pre-Class Assignment: Finding the inverse of the exponential function

In-Class: Logarithmic Functions (4.2), Domain, Natural Log, Graphs,

Week 12 11/8

Pre-Class Assignment: Rules of exponents and properties of logs

In-Class: Properties of Logarithms (4.3), Solving Exponential and Logarithmic
Equations (4.4)

Week 13 11/15

Pre-Class Assignment:
Pythagorean Theorem

In-Class: Exponential Modeling (4.5)
Midpoint and Distance Formulas (2.8)
Circles (2.8)

5

Fall 2021
Week

Starting
Topics (Sections – Blitzer)

Week 14 11/22

Scheduling period for Test 4

Pre-Class Assignment: Systems of Linear Equations

In-Class: Systems of Non-Linear Equations (8.4)
Solving Quadratics over Imaginary Numbers (1.5)

Week 15 11/29

Test 4 in Lab (4.2, 4.3, 4.4, 4.5,2.8 )

Review for Final Exam

Week- 16 12/6 Final Exam: Comprehensive (Scheduled in the lab. Same way as tests.)

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MAC 1105 MATH GYM FALL 2021 WEEK 2

1. Sometimes we make mistakes out of carelessness or moving too quickly, but sometimes it is
because we are not really sure what we are doing and are simply trying to “match” a similar looking
example. This means that we do not understand the mathematical meaning in the problem.

Write down an example of a mistake you made on the homework /quizzes:

What do you need to know/understand so that you will not make this mistake again?
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________

Goals for week 1 and week 2

Check if you understand:

□ What it means to be a function

□ The 3 main ways a function may be represented; verbally,

graphically and algebraically

□ What the domain of a function means

□ What the range of a function means

□ The different properties of a function

□ What it means to be the graph of a function

Check if you are able to:

□ Find the domain of a function

□ The range of a function

□ How to graph a function

□ Identify intervals of decreasing, increasing or constant on the

graph of a function

□ Identify relative maxima or minima on the graph of a function

□ Identify odd or even functions and their respective symmetries

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2. Your younger cousin saw you working on “My Labs Plus” and saw the word “function”. Curious they

ask you, “What is a function?” explain to them, in detail, what is a function.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

After your explanation your cousin says, “Wow you have a lot of questions on functions! Why

are functions so important anyway?” Explain the importance of functions to your cousin.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

3. What does it mean for when someone asks where a function is not defined? Give an example

______________________________________________________________________________

______________________________________________________________________________

Explain what the domain of a function is

______________________________________________________________________________

______________________________________________________________________________

4. Someone claims that the fuel efficiency (miles per gallon of a car) is an example of a function.

a) Make a reasonable argument why fuel efficiency is a function

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

b) Make a reasonable argument why fuel efficiency is not a function

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

c) Which side of the argument do you agree with?

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5. The graph of a function is a picture representation of that function. All the x values (input values) on

the graph also known as the _____________ give all corresponding y values (output values) on the

graph also known as the ____________.

How can we use this idea to find out if a given point is on the graph of a given function?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

6. a) Graph the function 𝑓(𝑥) = 2𝑥 − 1

What type of function is 𝑓(𝑥) = 2𝑥 − 1

__________________________________________________

b) Show on the graph as well as algebraically that the following points belong to 𝑓(𝑥). If a

coordinate is missing, show how you can find the missing coordinate.

(1, 4) (x, -7) (0, y)

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7. Let f(x)=(3x)2 and g(x)=9×2

a) Find f (2) and g (2)

b) Find f (-2) and g (-2)

c) Are f and g equivalent functions? Why or why not?

___________________________________________________________________________
___________________________________________________________________________

d) Let f(k) =k+6 and g (k) =k+6. Are f and g equivalent functions? Why or why not?

___________________________________________________________________________

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Facts/Characteristics: Expressions may consist of multiple terms. We can add or subtract
expressions by combining like terms. We can also multiply and divide expressions using properties of

exponents or factoring to simplify completely.

Concept Map During every class meeting your professor will cover key concepts that are important for
your course. It is critical that you identify these concepts and actively work toward understanding their
connections to other previous mathematical topics and ideas. A concept map is a great way to make and
organize these connections, and is very useful when you want to review for an exam.
Every week before your math gym, you will be required to create a concept map based on the topics
already covered in your College Algebra class during that same week. You may select any of the key
concepts covered that week to produce your map; some weeks, however, there will be only one key
concept covered. You may use the schedule of topics that is included for you here in the packet (the
same topics that are in your syllabus) as a guide to the key concepts that will be covered every week. For
the first few weeks we will provide you the concepts that were taught, and you can use these to design
your concept map. Going forward, you will need to know how to recognize and locate concepts on your
own. If you are struggling identifying concepts, talk with your professor or any of the LAs in the lab.
During math gym, compare your maps with your math gym classmates and correct the map when you
find any misconceptions. Write your work in the provided boxes as neatly as possible (pencil works
best). Note you will not receive credit if your work is not presented in a clear manner.
Here is an example of a concept map from a week one topic. Note that the key concept is at the center
of the map:

Expressions

Definition (in your own words): A single term or more than one term containing variables
or constants or operations between values. There is no equal sign.

Examples:
𝑥2

7𝑧 − 25

𝑘3 − √2

Non-Examples:

𝑥 − 13 = 4

ℎ2 + ℎ = 0

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Using the provided list, create a concept map for two of the topics taught in weeks 1 and 2.

List: Functions, Functional Notation, Domain, Range, and Graphs. Don’t forget to compare your maps

and make corrections in order to receive full credit.

Definition (in your own words)

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Facts/Characteristics

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Examples Counter Examples

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Definition (in your own words)

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Facts/Characteristics

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Examples Counter Examples

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MAC 1105 MATH GYM FALL 2021 Week 3

1. If the graph of the function f(x)= cos(x) looks like

Use the coordinate system below to graph g(x) = cos (x+ π/2).

Goals for week 3

Check if you understand:

□ What it means to transform a graph

Check if you are able to:

□ Recall the library of functions and their respective graphs

□ Identify functions by their respective graph

□ Transform points of a graph

□ Transform entire graphs

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One of your classmates is confused and says “but I have never seen cos(x) before” …you say “it does not

matter; you already know how to do this because we just…” Complete this statement to help your

classmate understand why they already know how to get the graph of cos(x+ π/2) using

transformations.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

2. Draw a parabola with at least 3 transformations from the parent function. Write the function for
your parabola. Note that your parabola should contain distinct points rather than be a sketch or
approximation. (At least 3 points)

Your function:

How do you know that your function matches your graph?

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

15

Concept map. List of topics: Library of Functions, and Transformations on Functions.

Definition (in your own words)

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Facts/Characteristics

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Examples Counter Examples

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MAC 1105 MATH GYM FALL 2021 Week 4

1. Given the following 𝑓(𝑥) = {
−|𝑥| + 1 𝑤ℎ𝑒𝑛 𝑥 < −2
6 𝑤ℎ𝑒𝑛 − 2 ≤ 𝑥 ≤ 3
(𝑥 − 3)2 + 2 𝑤ℎ𝑒𝑛 3 < 𝑥

Is 𝑓(𝑥) a function? Explain how you know.

______________________________________________________________________________

______________________________________________________________________________

For 𝑓(𝑥):

What are the intercepts?

What is the domain?

What is the range?

Goals for week 4

Check if you understand:

□ What it means to be a piecewise function

□ How to use a piecewise function

Check if you are able to:

□ Create a piecewise function

□ Find the domain and range of a piecewise function

□ Compute the average rate of change of a function on an interval

17

Graph f(x):

18

Concept Map: There was one key concept introduced this week, what was it? Create a map for that

concept.

The Key concept was: __________________________________________________________

Definition (in your own words)

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Facts/Characteristics

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Examples Counter Examples

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MAC 1105 MATH GYM FALL 2021 Week 5

1. Given the table of values, find the outputs of the given compositions for the given inputs.

X -3 -2 -1 0 1 2 3

f(x) 11 9 7 5 3 1 -1

g(x) 8 -3 0 1 0 -3 -8

f◦g (1) = _______ g◦f (3)=_______
f◦g (2) =_______ f◦f(3)=_______
f◦g (-1) =_______ g◦g (1)=_______

Goals for week 5

Check if you understand:

□ What it means to be a composite function

Check if you are able to:

□ Find the sum, difference, product and quotient of functions

□ Form a composite function

□ Find the domain and range of a composite function

□ Find and simplify the difference quotient of a function

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2. Given 𝑓(𝑥) = −2𝑥 2 − 3𝑥 + 1
a. Find and simplify the difference quotient

Initial Evaluation: Write what you expect your final answer to look like:

Write down each step

Explain why your step gets you closer to

an answer

Final Evaluation: Did your final answer match what you expected in your initial

evaluation?

21

Concept Map: Choose a key concept for this week (Operations on Functions, Composite Functions,

Difference Quotient), and create a concept map for that key concept.

The Key concept was: __________________________________________________________

Definition (in your own words)

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Facts/Characteristics

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

Examples Counter Examples

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Extra credit Challenge Question:

The resistance of blood flow in a blood vessel (R) is inversely proportional to the fourth power of the

radius (r) of the respective blood vessel

a. Build a function based on the given information.

b. What would be the domain of the function you created? (Remember to think in terms of

this question). Give answer in interval notation

c. In terms of this question, describe in words what the dependent variable of the function you

created is _________________________________________________________

d. In terms of this question, describe in words what the independent variable of the function

you created is __________________________________________________

e. Based on the function you created, describe why it is or isn’t possible to have a blood vessel

that has zero (0) resistances of blood flow

________________________________________________________________________

________________________________________________________________________

f. Graph the function you created.

23

MAC 1105 MATH GYM FALL 2021 Week 6

1. The Revenue, in dollars, is equal to the unit selling price, p, of the product, times the number x

of units sold. Suppose that p and x are related by: 𝑝(𝑥) = −
1

4
𝑥 + 3.

a. What does the function p(x) represents?
___________________________________________________________________________
___________________________________________________________________________

Goals for week 6

Check if you understand:

□ The characteristics of a quadratic function

□ The characteristics of a parabola

Check if you are able to:

□ Graph a parabola

□ Determine the minimum and maximum of a quadratic function

□ Solve problems involving the minimum and maximum of a

quadratics function

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b. Express the Revenue as a function of the number x of units sold and as a function of price.

What do you expect your final answers to look like?

Why?

Revenue as a function of the number of units
sold:

Why did you do it that way

Revenue as a function of price:

Why did you do it that way

Someone in your math gym says “I do not know what is meant when they write ‘is a function

of”, several others agree. What do you understand is meant by this phrase?

25

2. Given that 𝑓(𝑥) = 2𝑥2 + 4𝑥 − 1
a. Find the domain of 𝑓(𝑥)

b. range of 𝑓(𝑥)

c. x-intercepts

d. y-intercept

e. Represent the function 𝑓(𝑥) in vertex form

f. Once in standard form, identify the transformations of 𝑓(𝑥) in the correct order.

g. Graph 𝑓(𝑥) = 2𝑥2 + 4𝑥 − 1

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Concept Map: …

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