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Hi, I need 3 pages of my algebra workbook done by tomorrow 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 2 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.) 6 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 7 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? 8 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) 9 7. Let f(x)=(3x)2 and g(x)=9x2 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? ___________________________________________________________________________ 10 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 11 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 12 Definition (in your own words) _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ Facts/Characteristics _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ Examples Counter Examples 13 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 14 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 16 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 19 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 20 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 22 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 24 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 26 Concept Map: …