1 M121Q: College Algebra Structured Notes Workbook for Fall 2025 Sections 1 – 25 2 5 24x x− = X=± 1/4 1 1 )3( 2 )( 2      − +− = x x if if x x xf 2 3 Acknowledgements There are many people who deserve thanks for making this structured note packet / workbook possible. In particular, I am indebted to Megan Wickstrom for her wisdom, listening-ear, encouragement and support. It is such a joy to have her as a course supervisor for M121Q: College Algebra at Montana State University (MSU)! Several others contributed in significant ways to making this packet and the transition to OER a reality for M121Q: College Algebra. • Tom Hayes, Mary Ann Sodja and Jocelyn Reid were the first to create a structured note packet / workbook for use with College Algebra instruction at MSU many years ago. The packet has undergone complete transformation(s) since then, but the structure and philosophy are largely inherited from them – perhaps even a few items from their work remain here and there in the packet. • Brian Rossman and Christina Trunnell and the Montana State University (MSU) Library OER Initiative provided funding support to transition the workbook from connection to a traditionally published textbook to being supported by publicly available materials. Jacqueline Frank (Accessibility Librarian) provided guidance on making the packet accessible and the library reviewed the packet for accessibility. Furthermore, the MSU Library OER Initiative provided a means of publishing the workbook under a Creative Commons license. • David Lippman and the MyOpenMath community deserve many thanks for providing an excellent free online mathematics homework platform and problems that can be used with M121Q: College Algebra at MSU. • Many people / entities deserve thanks for creating and making publicly available online materials that serve as reference materials for this note packet. Definitions used in the packet frequently mirror those in reference materials and examples often utilize problems that are “sisters” to ones in reference materials. In order of frequency of use as reference material: o OpenStax: Intermediate Algebra, 2e, Lynn Marecek and Andrea Honeycutt Mathis o OpenStax: College Algebra, 2e, Jay Abramson o Scottsdale Community College Mathematics Division OER Materials Page: College Algebra: An Investigation of Functions, 4e, primary authors David Lippman, Melonie Rasmussen and Jay Abramson o OpenStax: Elementary Algebra, 2e, Lynn Marecek, Mary-Anne Anthony Smith and Andrea Honeycutt Mathis o Khan Academy | Math o OpenStax: College Algebra 2e with Corequisite Support, Jay Abramson and Sharon North o OpenTextBookStore: Precalculus / College Algebra 3e, Carl Stitz and Jeff Zeager Heidi Staebler 4 To Students Please bring this note packet / workbook to class with you every day. We will use it for instruction and it will provide you with a well-organized set of class notes. Sections that are labeled (R) should be review material for you. They are topics that you should already be skilled in OR be able to quickly get back your skills by doing some review. We will teach some of these sections in class. Other sections will not be taught in class, but lecture videos that match the note pages will be provided with required sectional online homework assignments. In addition, a small number of the sections are considered optional – to be utilized if you personally need to review the topics. Lecture videos for optional sections will be provided with associated optional online homework assignments. Course Expectations This course requires a significant amount of time outside of class. The general expectation for a university-level course is that you put in 2 to 3 hours of work outside of class for every 50-minute class session. That means, it is normal / expected for students to work on M121Q: College Algebra 6 – 9 hours per week OUTSIDE OF CLASS. You may need to spend more time on this course if you struggle with algebra. Specific Expectations: • Attend EVERY class session and ACTIVELY participate. Mathematics is not a spectator sport. The only way to learn mathematics is to DO mathematics. • Do ALL online homework assignments and complete them ON TIME. • GET HELP when you need it. There are many support resources available. Please make use of them. • STUDY for quizzes, not just for exams. In particular, WORK THROUGH the problems in weekly quiz preparation handouts. • Prior to an exam, WORK THROUGH the set of exam review problems that will be provided to you in Brightspace D2L. That is probably the best way to prepare for exams in this course. • REGULARLY check announcements within Brightspace D2L – at least every other day. Important information will be posted there. • BE IN GOOD COMMUNICATION with your instructor. Let them know when you are sick or are experiencing significant issues that are preventing class attendance. Seek out their help and advice. They truly care about you and want you to be successful at MSU and in this course. Best Wishes for a Good Semester! 5 Table of Contents Preliminaries Chapter Cover Page . . . . . p. 7 Section P.1: Solving Linear Equations (R) . . . . . p. 9 Section P.2: Radicals and Power Equations (R) . . . . . p. 13 Section P.3: The Rectangular Coordinate System and Graphs of Equations (R) . . . . . p. 17 Section P.4: Slopes and Lines (R) . . . . . p. 20 Chapter 1 Cover Page . . . . . p. 23 Section 1.1: Relations and Functions . . . . . p. 25 Section 1.2: Function Notation and Net Change . . . . . p. 31 Section 1.3: Interval Notation and Basic / Toolkit Functions . . . . . p. 34 Section 1.4: Getting Information from a Graph . . . . . p. 37 Section 1.5: Domains of Functions Presented Algebraically. . . . . p. 39 Section 1.6: Piecewise Defined Functions. . . . . p. 42 Section 1.7: Rates of Change and Behavior of Graphs. . . . . p. 48 Section 1.8: Models and Applications. . . . . p. 53 Chapter 2 Cover Page . . . . . p. 59 Section 2.1: Introduction to Linear Functions . . . . . p. 61 Section 2.2: Using Point-Slope Form to Create Linear Models . . . . . p. 65 Chapter 3 Cover Page . . . . . p. 69 Section 3.1: Factoring Foundations (R) . . . . . p. 71 Section 3.2: Solving Quadratic Equations (R) . . . . . p. 76 Section 3.3: Transformations of x2 . . . . . p. 81 Section 3.4: Three Forms of a Quadratic Function . . . . . p. 85 Section 3.5: Modeling with Quadratic Functions . . . . . p. 89 Section 3.6: X-intercepts of a Quadratic Function. . . . . p. 96 Section 3.7: Polynomial Functions . . . . . p. 102 6 Chapter 4 Cover Page . . . . . p. 107 Section 4.1: Rational Expressions . . . . . p. 109 Section 4.2: Rational Equations . . . . . p. 113 Section 4.3: Rational Functions. . . . . p. 116 Section 4.4: Arithmetic of Functions . . . . . p. 124 Section 4.5: Function Composition . . . . . p. 128 Section 4.6: Function Notation – Beyond the Basics . . . . . p. 131 Chapter 5 Cover Page . . . . . p. 137 Section 5.1: Exponent Properties (R) . . . . . p. 139 Section 5.2: Radicals and Rational Exponents . . . . . 142 Section 5.3: Using Like Bases to Solve Exponential Equations . . . . . p. 145 Section 5.4: Exponential Growth and Decay . . . . . p. 149 Section 5.5: Finding Equations of Exponential Functions . . . . . p. 155 Chapter 6 Cover Page . . . . . p. 159 Section 6.1: Inverse Functions . . . . . p. 161 Section 6.2: Logarithmic Functions . . . . . p. 168 Section 6.3: Logarithm Properties . . . . . p. 173 Section 6.4: Using Logarithms to Solve Exponential Equations. . . . . p. 180 Section 6.5: Base e Exponential Models. . . . . p. 182 Bibliography . . . . . p. 187 7 Preliminaries Chapter Sections P.1 – P.4 8 P.1 – Solving Linear Equations 9 Section P.1: Solving Linear Equations (R) Part 1: Solving Linear Equations with Whole Numbers & Decimals Reference Materials: Open Stax: Elementary Algebra 2e by Maracek, Anthony-Smith & Honeycutt Mathis Chapter 2 Section 2.5: Solve Equations with Fractions or Decimals Objectives: • Review how to solve linear equations that involve whole numbers and decimals Important Vocabulary Words: Expression, Equation, Solve Basic Linear Equations in One Variable What is the major difference between an equation and an expression? Expression Example: Equation Example: What does it mean to solve an equation (in one variable)? Example 1: Solve the equation for x. 6 ( 2) 3( 4) 8x x x− + = − − https://openstax.org/books/elementary-algebra-2e/pages/2-5-solve-equations-with-fractions-or-decimals P.1 – Solving Linear Equations 10 Example 2: Solve the equation for x. 4 0.05(3 2) 0.2(6 5)x x x+ − = + P.1 – Solving Linear Equations 11 Part 2: Solving Linear Equations with Fractions Reference Materials: Open Stax: Elementary Algebra 2e by Maracek, Anthony-Smith & Honeycutt Mathis Chapter 1 Section 1.6: Add and Subtract Fractions Chapter 2 Section 2.5: Solve Equations with Fractions or Decimals Objectives: • Review how to solve linear equations that involve fractions Important Vocabulary Words: Expression, Equation, Solve, “Eliminate the Fractions” Strategy, Least Common Multiple (LCM) Solving Linear Equations with Fractions – Simplify and Eliminate the Fractions Least Common Multiple (LCM) of a Set of Numbers: Simplify By Adding / Subtracting Fractions to Combine Terms: Determine the LCM of the denominators and multiply each term by “1” in the form n n to create a common denominator = LCM of the denominators. Then add / subtract to combine terms. Eliminate the Fractions Strategy: Multiply both sides of the equation by the LCM of the denominators and use the LCM to cancel out the denominators. Example 1: Solve for x. 1 1 ( 2) 4 5 3 x x x− + = https://openstax.org/books/elementary-algebra-2e/pages/1-6-add-and-subtract-fractions https://openstax.org/books/elementary-algebra-2e/pages/2-5-solve-equations-with-fractions-or-decimals P.1 – Solving Linear Equations 12 Example 2: Solve for x. 3 4 1 2 6 10 x x− + − = You Try! Example 3: Solve for x. 6 3 2 3 4 12 x x x− + − = P.2 – Radicals and Power Equations 13 Section P.2: Radicals and Power Equations (R) Part 1: Evaluating Radicals Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 8 Section 8.1: Simplify Expressions with Roots Khan Academy Resources. All Khan Academy content is available for free at www.khanacademy.org . • YouTube Video: Introduction to Square Roots • YouTube Video: Introduction to Cube Roots Objectives: Review / learn how to work with simple radical expressions Important Vocabulary Words: Radical, (Principle) Root, Absolute Value Introduction to Radicals What is the square root of 9, i.e. 9 ? What is the cube root of 8, i.e. 3 8 ? *Note: For even roots, if you are given a radical you must assume that we want only the positive (principal) root. Can we take even roots of negative numbers? (√−16) What about odd roots of negative numbers? (√−8 3 ) Why? https://openstax.org/books/intermediate-algebra-2e/pages/8-1-simplify-expressions-with-roots http://www.khanacademy.org/ https://youtu.be/mbc3_e5lWw0 https://youtu.be/87_qIofPwhg P.2 – Radicals and Power Equations 14 Example 1: Evaluate / simplify each of the following. A. 64 =______ B. =3 125 _____ C. =4 81 _____ Another way to think: Working with a Variable: Recall Absolute Value: if 0 if a is 0 or a positive number if 0 if a is a negative number a a a a a   =  −   Examples: 5 5= 23.7 23.7= 0 0= But: 8 ( 8) 8− = − − = 46.72 ( 46.72) 46.72− = − − = Rule: When n is even: When n is odd: n nx x= n nx x= Example 2: Simplify each of the following. A. =2x ______ B. =3 3x _______ P.2 – Radicals and Power Equations 15 Question: Is 4 9 4 9+ = + ? Example 3: For 0,x  simplify each of the following or state that it cannot be simplified. Recall: If 0x  , then x x= . A. 2 9x + B. 3 3 8x − IMPORTANT!! You Try! Example 4: For 0x  , evaluate / simplify each of the following or state that it cannot be simplified. A. 3 27− = B. 225x = C. 249 x+ = P.2 – Radicals and Power Equations 16 Part 2: Solving Power Equations Reference Materials: Math Class Rocks Resources: https://mathclassrocks.com • YouTube Video: Solve Equations Using Square and Cube Roots Objectives: Review / learn how to solve power equations Example 1: Solve each of the following equations. A. 362 =x B. 3 125x = − C. 814 =x D. 492 −=x Rule: Solve cxn = , where c is a real number. https://mathclassrocks.com/ https://www.youtube.com/watch?v=w1c9DPm8MQU P.3 – The Rectangular Coordinate System and Graphs of Equations 17 Section P.3: The Rectangular Coordinate System and Graphs of Equations (R) Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.1: Graph Linear Equations in Two Variables Objectives: • Be able to plot (x, y) coordinate pairs • Be able to use a T-chart to plot graphs of equations in x and y • Understand what x- and y-intercepts are Important Vocabulary Words: Solution, Rectangular (Cartesian) Coordinate System, Graph, x-axis, y-axis, x-intercept, y-intercept Plotting Solutions to an Equation in the Rectangular (Cartesian) Coordinate System We can represent an ordered pair ( , )x y using the coordinate (Cartesian) plane. Example 1: Plot and label the following ordered pairs in the rectangular (Cartesian) coordinate plane. A. (4, 2) B. ( 1,3)− C. (1, 5)− D. ( 3, 4)− − Definition: Solution of an Equation Involving x and y An ordered pair ( , )x y is a solution to an equation involving x and y if the equation is a true statement when the x- and y- values of the ordered pair are substituted into the equation. https://openstax.org/books/intermediate-algebra-2e/pages/3-1-graph-linear-equations-in-two-variables P.3 – The Rectangular Coordinate System and Graphs of Equations 18 Example 2: Find some solutions to 3 2y x= − . List solutions as ordered pairs in the form ),( yx . Graphing Equations Definition: The graph of an equation in x and y is the set of all points ( , )x y that satisfy the equation. Graph of a Line The graph of an equation of the form y mx b= + ,where m and b are real numbers, is a line. • Every point on the line is a solution to the equation. • Every solution to the equation is a point on the line. Basic Equation Graphing Method: 1. Make a T-chart or table of solutions to the equation by choosing values of x and using the equation to calculate corresponding values of y. 2. Plot enough solutions ( , )x y to know what the overall graph looks like and then connect the points with a smooth curve or line. Example 3: Use the basic equation graphing method to draw the graph of 2 4y x= − + . x y 0 1 2 3 4 P.3 – The Rectangular Coordinate System and Graphs of Equations 19 Definitions: X-intercept & Y-intercept The x-intercept of a graph is the point ( ,0)x where the graph crosses the x-axis. The y-intercept of a graph is the point (0, )y where the graph crosses the y-axis. Example 4: What are the x- and y-intercepts for the graph of 2 4y x= − + ? x-intercept: y-intercept: Example 5: Use the basic equation graphing method to draw the graph of 22 2y x= − and determine the x- and y-intercepts of the graph. P.4 – Slopes and Lines 20 Section P.4: Slopes and Lines (R) Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.2: Slope of a Line & Section 3.3 Find the Equation of a Line Objectives: • Develop the slope formula and the slope-intercept form of the equation of a line • Use the slope formula and slope-intercept form of the equation of a line to determine the equations of lines and to graph lines when an equation is provided. Important Vocabulary Words: Slope, Slope-Intercept Form Formula for Slope Given Two Points on a Line: Slope = m = 2 1 2 1 rise run y y x x − = − Example 1: Determine the slopes of the lines shown below. A. B. https://openstax.org/books/intermediate-algebra-2e/pages/3-2-slope-of-a-line https://openstax.org/books/intermediate-algebra-2e/pages/3-3-find-the-equation-of-a-line P.4 – Slopes and Lines 21 Example 2: Determine the slopes of the following lines. A. B. Summary: Line Direction ↔Slope Positive Slope ↔ Negative Slope ↔ Horizontal Line: Vertical Line: Slope-Intercept Form of the Equation of a Line: Slope-Intercept Form of the Equation of a Line y mx b= + m = slope of the line (0, )b is the y-intercept of the line P.4 – Slopes and Lines 22 Example 3: Determine the equation of the line whose graph is shown. Example 4: Determine the equation of the line whose graph passes through (3, 1)− and (2, 4) . CHAPTER 1 COVER PAGE 23 Chapter 1 Functions Sections 1.1 – 1.8 CHAPTER 1 COVER PAGE 24 1.1 – Relations and Functions 25 Section 1.1: Relations and Functions Part 1: Relations Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.5: Relations and Functions and Section 3.6 Graphs of Functions Objectives: • Become familiar with various means of visualizing data and how they represent a relation • Learn and be able to apply the terms domain and range • Be able to write a relation as a set of ordered pairs Important Vocabulary Words: Relation, Domain, Range Example 1: A developer contracted with a realtor to survey buyers with a strong interest in purchasing a condo in the location and price per square foot that the developer wishes to build. The developer wanted to know the number of bedrooms and the number of covered parking spaces potential buyers would want. Results from the first five buyers surveyed are shown below. Number of bedrooms 1 2 2 3 4 Number of parking spaces 0 0 1 2 2  (1,0), (2,0), (2,1), (3,2), (4,2) 0 1 2 3 0 1 2 3 4 5 Number of Parking Spaces Desired Number of Bedrooms Desired Buyer Survey Results Number of Bedrooms Number of Parking Spaces 1 2 3 4 0 1 2 https://openstax.org/books/intermediate-algebra/pages/3-5-relations-and-functions https://openstax.org/books/intermediate-algebra/pages/3-6-graphs-of-functions 1.1 – Relations and Functions 26 Definitions: Relation: Any set of ordered pairs. Input ↔ first value in the ordered pair Output ↔ second value in the ordered pair Domain: The set of all input values. Range: The set of all output values For Example 1: Domain: Range: You Try! Example 2: The graph below illustrates the area in square feet of each style of unit in the proposed condominium development based on the purchase price for it in thousands of dollars. A. How many square feet are in a unit that is priced at $600,000? 900 950 1000 1050 1100 1150 1200 1250 1300 1350 475 500 525 550 575 600 625 A re a o f U n it in S q u ar e Fe et Purchase Price in Thousands of Dollars Unit Size Based on Purchase Price 1.1 – Relations and Functions 27 B. How much would a unit with 1100 square feet cost? C. Provide the set of ordered pairs for the relation that is shown in the graph. B. State the domain and range of the relation. Write your answers using set notation. Domain: Range: Part 2: Introduction to Functions Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.5: Relations and Functions Objectives: • Be able to determine if a relation is a function based on the definition of function • Be able to use the vertical line test to determine if a graph represents a function Important Vocabulary Words: Function, Vertical Line Test, Function Notation Compare Part 1 Example 1 and Example 2. Besides the contexts and the number values, what is different about the two relations? https://openstax.org/books/intermediate-algebra/pages/3-5-relations-and-functions 1.1 – Relations and Functions 28 Definition of Function A function is a relation that assigns to each element in its domain exactly one element in its range. In other words, each “input value” corresponds to exactly one “output value.” Example 1: Function? Yes / No Example 2: { (1, 4), (2, 5), (2, 6), (3, 7) } Function? Yes / No Example 3: Function? Yes / No Input 2 4 6 8 Output 3 5 3 9 You Try! Example 4: Determine if the following relations represent functions. Use the definition of function to support your answers. A. B. x y 1 3 2 7 3 1 4 4 Fall Semester Year Required Minimum Math SAT Score For M121Q 2015 540 2016 570 2017 570 2018 570 2019 570 2021 560 Number of People in Household Number of Bedrooms in Home 2 1 4 4 3 2 4 3 2 2 5 4 1.1 – Relations and Functions 29 You Try! Example 5: Determine if the following graphs represent functions. Use the definition of function to support your answers. A. B. Vertical Line Test: If _______________ vertical line intersects a graph ________________________________, then the graph DOES represent a function. However, If ___________ vertical line intersects a graph __________________________________, then the graph does NOT represent a function. Example 6: Use the Vertical Line Test to determine if the following graphs represent functions. A. B. 1.1 – Relations and Functions 30 Pull It All Together You Try! Problems for Section 1.1 Part 2 1. A relation is a function if __________________________________________________________. 2. Determine whether each relation / graph represents a function. Respond with "Yes" or "No." A. { (-2,3), (-1,3), (0,3), (1,4), (2,5) } _______ B. { (-2,-1), (-2,0), (0,1), (1,2), (2, 3) } _______ C. ________ D. ________ 1.2 – Function Notation and Net Change 31 Section 1.2: Function Notation and Net Change Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.5: Relations and Functions Objectives: • Be able to use and interpret function notation. • Understand the phrases: independent variable, dependent variable, ___ is a function of ___ • Be able to determine net change and interpret its meaning in context Important Vocabulary Words: Function Notation, independent variable, dependent variable, ____ is a function of _____, Net Change New Way to Think About Functions A function f is a rule that assigns exactly one output to each input. Equation Form Function Notation Form 42 += xy ↔ 4)( 2 += xxf Find )2(f . What does )2(f represent? Example 1: Let xxxg 3)( 2 +−= . Evaluate each of the following. A. )1(g B. )2(−g C. ( )g a You Try! Example 2: Let 2( ) 2( 3)g x x= − . A. What is the name of the function? _______ B. What letter represents the input? ________ C. What represents the output? D. Determine (5)g and ( )g k . https://openstax.org/books/intermediate-algebra/pages/3-5-relations-and-functions 1.2 – Function Notation and Net Change 32 Dependent & Independent Variables The input variable is referred to as the independent variable and the output variable is referred to as the dependent variable. ↔ What comes out of a function depends upon what goes into it. ____ is a function of ___ We say: The output variable “is a function of” the input variable Example 3: According to Cosmic Pizza’s online menu, the cost, C, of an 18-inch pizza with t toppings is given by: ( ) 22.5 2.5C t t= + . (Cosmic Pizza, 2022) A. Describe the input and output variables. Input Variable = ______ = __________________________________________ Output Variable = ______ = _________________________________________ We say: _______ is a function of _______ B. Determine (3)C and write a sentence, using everyday English, that interprets the meaning of your answer in the given context. C. Determine (5) (3)C C− and write a sentence, using everyday English, that interprets the meaning of your answer in the given context. https://www.cosmicpizza.net/menu 1.2 – Function Notation and Net Change 33 Net Change in a Function The net change in a function f from input a to input b = ( ) ( )f b f a− ↔ represents the increase (+) or decrease (-) in the function outputs You Try! Example 4: Assuming normal air density, the amount of power generated by a wind turbine depends upon the type of turbine and the windspeed. For windspeeds between 4 m/s and 12 m/s, the power P, in kilowatts (kW), generated by a particular type of wind turbine can be approximated by 3( ) 0.955P w w= , where w is the windspeed in meters per second (m/s). Informed by a Sciencing website (K. Lee, 2020) and Wind farm Bop website (F. Miceli, 2012) A. Fill in the blanks: Independent Variable: _________ Dependent Variable: _______ We say: _______ is a function of _______ Input Variable: _______ = _________________________________________________ Word Description Output Variable: _______ = _________________________________________________ Word Description B. Determine the net change in P from 6w = to 10w = . Include appropriate units with your answer. C. Write a complete sentence, using everyday English, that interprets the meaning of your answer to B in the given context. https://sciencing.com/much-power-wind-turbine-generate-6917667.html https://www.windfarmbop.com/power-curve-what-is-and-how-to-measure-it/ 1.3 – Interval Notation and Basic / Toolkit Functions 34 Section 1.3: Interval Notation and Basic / Toolkit Functions Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 2 Section 2.5: Solve Linear Inequalities Chapter 3 Section 3.6: Graphs of Functions Objectives: • Be able to represent an interval using interval notation • Be able to graph functions by plotting points • Know shapes & key points, domain and range for 6 basic / toolkit types of functions Important Vocabulary Words: Interval Notation, Domain, Range, Basic / Toolkit Functions Interval Notation: Inequality Number Line Graph 1 Number Line Graph 2 1 3x  Interval Notation: 2x  Interval Notation: 1x  Interval Notation: You Try! Example 1: Express each of the following using interval notation. A. 4x  B. 3 5x−   C. 1x  − Interval Notation: Interval Notation: Interval Notation: https://openstax.org/books/intermediate-algebra/pages/2-5-solve-linear-inequalities https://openstax.org/books/intermediate-algebra/pages/3-6-graphs-of-functions 1.3 – Interval Notation and Basic / Toolkit Functions 35 Basic Graphing Strategy When graphing functions, x-coordinates ↔ input values and y-coordinates ↔ output values. Hence, we talk about the graph of “ ( )y f x= .” A basic technique is to create a T-chart. Choose x-values and use the algebraic expression of the function to calculate corresponding y-values. The graph is the set of all such ( , )x y points. Plot enough points to be able to determine the overall shape of the graph. Six Basic / Toolkit Functions Example 2: Graph the following functions by plotting points. Use interval notation to state the domain (x values) and range (y values) of each function. Learn the shapes of all six of the basic / toolkit function types. A. Constant functions: #)( =xf B. Linear Functions: ( )f x mx b= + Example: 3)( =xf Example: ( ) 2 1f x x= − C. The square function: 2)( xxf = D. The square root function xxf =)( 1.3 – Interval Notation and Basic / Toolkit Functions 36 E. The cube function: 3)( xxf = F. The absolute value function xxf =)( Graphing Simple Transformations of Basic / Toolkit Functions Strategy: Set up a “T-chart” to get several points to plot and also use what you know about the shapes of the six basic / toolkit functions. You Try! Example 3: Sketch the graph of 2( ) ( 2)f x x= − − and determine its domain and range. 1.4 – Getting Information from a Graph 37 Section 1.4: Getting Information from a Graph Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.6: Graphs of Functions Objectives: Based on the graph of a function, be able to: • Determine the domain and range • Evaluate the function for a given input value and solve equations of the form ( ) #f x = • Determine x- and y-intercepts Important Vocabulary Words: Interval Notation, Domain, Range, X-intercept, Y-intercept Introductory Example The graph of ( )y d x= is shown, where ( )d x is the snow depth, in inches, on the x th day of January. A. What is the domain of d ? B. What is the range of d ? C. How deep was the snow on January 9? ↔ Determine (9)d . D. When were there 16 inches of snow on the ground? ↔ Find the value(s) of x for which ( ) 16d x = . E. When was the snow depth 7 inches or less? ↔ Find the value(s) of x for which ( ) 7d x  . https://openstax.org/books/intermediate-algebra/pages/3-6-graphs-of-functions 1.4 – Getting Information from a Graph 38 Piecewise-defined Function Examples: Example 1: The graph of ( )y f x= is shown. Use the graph to determine the following. Use interval notation and coordinate pair notation whenever appropriate. Domain of f: _______________________________ Range of f: ________________________________ (2)f = __________ Value(s) of x for which 0)( =xf : _________ Values of x for which ( ) 0f x  : _________________________ x-intercept(s): __________________ y-intercept: ___________ You Try! Example 2: The graph of ( )y g x= is shown. Use the graph to determine the following. Use interval notation and coordinate pair notation whenever appropriate. Domain of g: _______________________________ Range of g: ________________________________ ( 1)g − = ___________ ( ) 1g x = − for x = ______________ Value(s) of x for which ( ) 0g x  : __________________ Value(s) of x for which ( ) 2g x  : __________________ x-intercept(s): __________________ y-intercept: ___________ 1.5 – Domains of Functions Presented Algebraically 39 Section 1.5: Domains of Functions Presented Algebraically Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 3 Section 3.2: Domain and Range Objectives: • Be able to determine the domain of a given function (assuming real number output values) and write it using interval notation Important Vocabulary Words: Domain, Interval Notation, Polynomial Function, Rational Function, Root / Radical Function Introductory Example Let ( ) 4 x f x x = − . Find each of the following: A. (5)f =_______ B. (0)f = _______ C. (4)f = ______________ D. ( 9)f − = ________________ F. Domain of 𝑓 so that it outputs only real numbers: To determine the domain of a function that is presented algebraically so that it outputs only real numbers, there are two issues to consider: 1. 2. https://openstax.org/books/college-algebra-2e/pages/3-2-domain-and-range 1.5 – Domains of Functions Presented Algebraically 40 How do we determine the domain of a polynomial function? Examples: 3 2( ) 4 3 5 12f x x x x= − + − Domain: 53)( 2 −= xxg , 102 − x Domain: How do we determine the domain of a rational (fractional) function? BIG IDEA: Example: 3 15 ( ) 4 x f x x + = − Domain: How do we determine the domain of a root/radical function so that it outputs only real numbers? BIG IDEA: n numbernegative is “okay” when n is ODD, BUT, is “not a real number” when n is EVEN. → For ODD roots, one does NOT need to be concerned about restricting x in order to avoid a negative number under the root symbol. → For EVEN roots, one MUST consider the necessity of restricting x in order to avoid a negative number under the root symbol. Example 1: Determine the domains of the following functions so that they output only real numbers. Write answers using interval notation. A. xxf −= 7)( B. 3( ) 2 8g x x= + Domain: Domain: 1.5 – Domains of Functions Presented Algebraically 41 Example 2: Determine the domains of the following functions so that they output only real numbers. Write answers using interval notation. A. 5 ( ) 2 x h x x + = − B. 3 ( ) 1 k x x = − Domain: Domain: You Try! Example 3: Determine the domains of the following functions so that they output only real numbers. Write answers using interval notation. A. 5 ( ) 3 21 x f x x − = + B. 3 4 ( ) 3 x x g x − = Domain: Domain: C. 3( ) 5 2h x x= + D. ( ) 12 2i x x= − Domain: Domain: E. 3 ( ) 7 x j x x − = − F. 7 ( ) 3 x k x x − = − Domain: Domain: 1.6 – Piecewise Defined Functions 42 Section 1.6: Piecewise Defined Functions Part 1: Introduction to Piecewise Functions Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 3 Section 3.2: Domain and Range Objectives: • Understand the notation for piecewise defined functions and be able to evaluate such a function when given an input value • Be able to use piecewise defined functions to answer application questions Important Vocabulary Words: Piecewise Defined Functions Example 1: 1 1 1 )( 2      + = x x if if x x xh Find each of the following: ( 3)h − (0)h (3)h (1)h https://openstax.org/books/college-algebra-2e/pages/3-2-domain-and-range 1.6 – Piecewise Defined Functions 43 You Try! Example 2: The monthly water bill B in dollars for a residence in a city is given by ( )B x , where x is the number of metered units of water used during the month. (1 metered unit = 100 cubic feet of water.) Informed by Salt Lake City Utilities website (Salt Lake City, 2023) 13.5 2 0 10 ( ) 6 2.75 10 x x B x x x +   =  +  if if A. Find (5)B and (20)B . Include appropriate units with your answers. B. How many cubic feet of water was used if the monthly water bill is $30? Set up and solve an equation to answer the question. https://www.slc.gov/utilities/pay-my-bill/current-rates/ 1.6 – Piecewise Defined Functions 44 Strategy to Determine an Algebraic Expression for a Piecewise Function in an Application 1. Determine what the input and output variables represent. 2. Determine “boundary” input value(s) and also pick an input value from each of regions created by the boundary value(s). Write down EVERY arithmetic step used to determine corresponding output values. 3. To create algebraic expressions for each region, mimic the arithmetic steps used in #2. I.e. replace the input numbers with “x” and follow the same arithmetic processes. Example 3: Suppose a state taxed income as follows: • 5% tax for taxable income up to $30,000 • 7% tax on additional taxable income over $30,000 Then the amount of tax a person owes, T, is a piecewise function of a person’s taxable income x. A. Definitions (words): Input: x = ______________________________________________ Output: ( )T x = __________________________________________ B. Determine (20,000)T , (30,000)T and (40,000)T . Write down EVERY mathematical calculation! C. Express T as a piecewise defined function: _______________________ ________________ ( ) _______________________ ________________ if T x if  =   1.6 – Piecewise Defined Functions 45 Part 2: Graphing Piecewise Functions Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 3 Section 3.2: Domain and Range Objectives: • Be able to graph piecewise defined functions by plotting points and by using knowledge about the graphs of the six basic / toolkit functions Graphing Piecewise Defined Functions 1. Getting Set Up: Determine the “boundaries” for the rules and indicate them on graph. (NOTE: These are not actually part of the graph, but they serve as guides.) Label the regions determined by the boundaries and indicate which rule is to be graphed in each of the regions. 2. Plot Graph within Each Region: For each region, look at the corresponding rule and determine “shape” information about what a graph that corresponds to that rule should look like. (Is it a line? If so, what is the slope? Is it a one of the basic / toolkit functions or a simple transformation of one of those functions? Etc.) For each region, create a T-chart of points to plot and use what you know about the basic shape to help sketch the graph in that region. BE CAREFUL! Do not let graphs extend beyond their regions. 3. Points at Boundaries: Use the rules to plot points on the boundaries of the regions to which they correspond. Based on whether or not a boundary is actually part of the region for a rule, use a “closed” or “open” dot. Make sure your graphs “connect” to the points on the boundary line(s). Example 1: Sketch the graph of 1 2 2 2 ( ) 2 63 x if x f x if xx  −   =    − https://openstax.org/books/college-algebra-2e/pages/3-2-domain-and-range 1.6 – Piecewise Defined Functions 46 You Try! Example 2: Sketch the graph of 3 0 ( ) 2 1 0 if x f x x if x −  =  − +  You Try! Example 3: Sketch the graph of 5 5 2 ( ) 1 2 x if x f x x if x + −   − =  −  − 1.6 – Piecewise Defined Functions 47 You Try! Example 4: Sketch the graph of 3 2 2 ( ) 3 25 2 if x f x if xx x −   =  − 1.7 – Rates of Change and Behavior of Graphs 48 Section 1.7: Rates of Change and Behavior of Graphs Introductory Example – A Continuation The graph of ( )y d x= is shown, where ( )d x is the snow depth, in inches, on the x th day of January. A. When was the snow depth increasing, decreasing, constant? ↔ Find the interval(s) on which d is increasing, decreasing, constant. Increasing: Decreasing: Constant: B. What was the maximum snow depth? On which day was the snow the deepest? ↔ Find the maximum value of d. ↔ Find the location of the maximum value of d. C. On average, did the snow depth increase more quickly between Day 3 and Day 4 or between Day 8 and Day 12? How can you tell? 1.7 – Rates of Change and Behavior of Graphs 49 Part 1: Average Rate of Change Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 3 Section 3.3: Rates of Change and Behavior of Graphs Objectives: • Understand the definition of average rate of change and how it relates to the graph of a function • Be able to apply average rate of change to a variety of function settings • Be able to write a sentence, using everyday English, that interprets the meaning of average rate of change in a given context Important Vocabulary Words: Average Rate of Change, Secant Line Definition of Average Rate of Change The average rate of change in a function f between two input values x1 and x2 describes how quickly, on average, the function increases or decreases over the interval 1 2[ , ]x x . It is the total change of the function values (output values) divided by the change in the input values. Average rate of change Example 1: Let xxxg 2)( 2 −= . A. Determine the average rate of change in g between 1x = and 3x = . 2 1 2 1 1 1 2 2 1 1 2 2 ( ) ( ) ( , ) ( , ) ( ) ( ) f x f xy x x x x y x y y f x y f x − = =  − = = = Slope of the secant line between and where and https://openstax.org/books/college-algebra-2e/pages/3-3-rates-of-change-and-behavior-of-graphs 1.7 – Rates of Change and Behavior of Graphs 50 24.7 15.4 17.5 15.1 16.4 0 5 10 15 20 25 30 0 2 4 6 8 10Lo ss es in B ill io n s o f D o lla rs Number of Years Since 2012 Annual Identity Fraud Losses B. You Try! Determine the average rate of change in g over the given intervals. Use function notation in your work. i.) Between 1x = − and 0x = ii.) Between 1x = − and 3x = C. Why is the answer to A positive while the answer to B part i) is negative? Example 2: The graph below shows the annual amount of loss, L, in billions of dollars, in the U.S. due to identity fraud between 2012 and 2021, where x is the number of years since 2012. A. Determine the average rate of change in 𝐿 between 0x = and 6x = . Include appropriate units with your answer. B. Write a complete sentence that interprets your answer from A in the given context. Informed by Bureau of Justice Statistics Bulletins for 2012 (E. Harrell & L. Langton, 2013), 2014 (E. Harrell, 2015), 2016 (E. Harrell, 2019), 2018 (E. Harrell, 2021), 2021 (E. Harrell & A. Thompson, 2023) https://bjs.ojp.gov/content/pub/pdf/vit12.pdf https://bjs.ojp.gov/content/pub/pdf/vit14.pdf https://bjs.ojp.gov/content/pub/pdf/vit16.pdf https://bjs.ojp.gov/library/publications/victims-identity-theft-2018 https://bjs.ojp.gov/library/publications/victims-identity-theft-2021 1.7 – Rates of Change and Behavior of Graphs 51 C. Determine the net change in 𝐿 between 0x = and 6x = . Include appropriate units with your answer. D. Write a complete sentence that interprets your answer from C in the given context. Part 2: Behavior of Graphs Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 3 Section 3.3: Rates of Change and Behavior of Graphs Objectives: • Be able to determine where a function is increasing, decreasing and constant from its graph • Be able to determine local minimum and local maximum values and where they occur Important Vocabulary Words: Increasing, Decreasing, Constant, Local Minimum, Local Maximum Increasing, Decreasing, Constant A function is increasing over an interval ( , )a b if 1 2( ) ( )f x f x whenever 1 2x x for any two values 1x and 2x in (a,b). A function is decreasing over an interval ( , )a b if 1 2( ) ( )f x f x whenever 1 2x x for any two values 1x and 2x in (a,b). Example 1: The graph of ( )y f x= is shown. Use the graph to determine where f is increasing, decreasing and constant. Write your answers using interval notation. Increasing: __________________________ Decreasing: _________________________ Constant: _________________________ https://openstax.org/books/college-algebra-2e/pages/3-3-rates-of-change-and-behavior-of-graphs 1.7 – Rates of Change and Behavior of Graphs 52 You Try! Example 2: The graph of ( )y g x= is shown. Use the graph to determine where g is increasing and decreasing. Write your answers using interval notation. Increasing: __________________________ Decreasing: _________________________ Example 3: For the graph of g shown in Example 2, determine the local maximum value(s) and local minimum values(s) and where they occur. _______ is a local maximum value and it occurs at ________________ _______ is a local __________________ value and it occurs at ________________ _______ is a local __________________ value and it occurs at ________________ Local Maximum and Local Minimum Values ( )y f c= is a local maximum value occurring at x c= if there is an interval ( , )a b containing c such that ( ) ( )f c f x for all values x in ( , )a b . ( )y f c= is a local minimum value occurring at x c= if there is an interval ( , )a b containing c such that ( ) ( )f c f x for all values x in ( , )a b . You Try! Example 4: The graph of ( )y h x= is shown. Determine the following: Local maximum value(s) of h: ____________ Value(s) of x at which they occur: ____________ Local minimum value(s) of h: ____________ Value(s) of x at which they occur: ____________ 1.8 – Models and Applications 53 Section 1.8: Models and Applications Part 1: Working with Formulas Reference Materials: Open Stax: College Algebra 2e with Corequisite Support by Jay Abramson and Sharon North Chapter 2 Section 2.3: Models and Applications Objectives: • Be able to solve a formula for a specified variable Important Vocabulary Words: Formula What is a Formula? A formula is an equation involving multiple variables / quantities which illustrates relationships between the quantities that always hold true. Example 1: The formula 2mv F r = calculates the centripetal force F felt by an object of mass m traveling at a velocity v on a circular path with resistance r. (Example: The force pushing a car off the road on a curve, where resistance corresponds to friction between the tires and the road.) Solve the formula for v. https://openstax.org/books/college-algebra-corequisite-support-2e/pages/2-3-models-and-applications 1.8 – Models and Applications 54 Example 2: A fully enclosed rectangular box is shown. A. Find a formula for the surface area of the box. B. Solve the formula above for H. You Try! Example 3: Simple Interest Formula: PrtPA += A = amount after t years t = time in years P = amount invested r = annual interest rate as a decimal A. Solve the formula for r. B. Solve the formula for P. 1.8 – Models and Applications 55 Part 2: Geometric Applications Reference Materials: Open Stax: College Algebra 2e with Corequisite Support by Jay Abramson and Sharon North Chapter 2 Section 2.3: Models and Applications Scottsdale Community College – Math Blog: College Algebra, An Investigation of Functions, primary authors David Lippman, Melonie Rasmussen and Jay Abramson Chapter 3 Section 3.2: Quadratic Functions Example 1 - page 173 of the pdf file (printed page 165) Objectives: • Be able to create a mathematical model for a geometric scenario and use the model to answer questions about the scenario. Important Vocabulary Words: Mathematical Model Definition of Mathematical Model A mathematical model is a mathematical representation (such as an equation or function) of a real-world situation. Useful Modeling Strategies: 1. Choose variables to use. 2. Draw and label a diagram. 3. Use simple cases to explore the situation numerically. 4. Describe relationships verbally first, and then represent them algebraically. 5. Check the model with simple cases to make sure it makes sense. Example 1: A company wants to package its beverages in a rectangular carton with a square base They want the carton to be two and a half times as tall as it is wide. A. Determine an expression for the volume of the carton as a function of the width, x, of its base. B. What dimensions should the company use for the carton if they want it to hold 1280 milliliters? NOTE: 1 milliliter = 1 cm3 https://openstax.org/books/college-algebra-corequisite-support-2e/pages/2-3-models-and-applications https://sccmath.wordpress.com/wp-content/uploads/2017/07/college-algebra-proof-4-with-links-ed4-fall-2017.pdf 1.8 – Models and Applications 56 Example 2: A farmer has 180 feet of fencing and needs to enclose a rectangular area divided by a fence into two pens for his hogs. The farmer intends to build the pen up against the side of a long barn so that he will not have to use fencing along that side. His goal is to create as big of an enclosure as he can with the amount of fencing that he has. A. Start by experimenting with a number to get a sense of how things work. Determine the total area of the enclosure when x is 10 feet. B. Label the total length of the enclosure as L. Then use x and L to create a fencing “constraint” equation involving 180. Constraint Equation: C. Determine an expression for the total area of the two hog pens as a function of only x, where x is the width of the hog pens as shown in the diagram. HINT: Solve the constraint equation for L. Area = 𝐴(𝑥) = 1.8 – Models and Applications 57 Use Graphing Technology for D: (You will learn a “pencil & paper” algebraic strategy in Chapter 3.) D. Based upon the graph of 𝐴(𝑥), what width should the farmer use for the rectangular enclosure in order to provide maximum area for the hogs within it? You Try! Example 3: The length of a rectangular room is 5 feet less than twice its width w. A. Determine an expression for the perimeter P of the room as a function of its width w. B. What is the width of the room if it has a perimeter of 80 feet? C. What is the area of the room if it has a perimeter of 80 feet? 1.8 – Models and Applications 58 CHAPTER 2 COVER PAGE 59 Chapter 2 Linear Functions Sections 2.1 – 2.2 CHAPTER 2 COVER PAGE 60 2.1 – Introduction to Linear Functions 61 Section 2.1: Introduction to Linear Functions Part 1: Definition of Linear Function Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 4 Section 4.1: Linear Functions Objectives: • Know the definition of a linear function and be able to apply it when information is presented in table form, in graph form and in word form. Important Vocabulary Words: Linear Function, Form of a Linear Function, Rate of Change, Slope, Initial Value, Y-intercept, Units Definition of Linear Function A function f is a linear function if the graph of f is a line. This occurs when f has constant average rate of change; that is, its graph has constant slope. Form of a Linear Function The form of a linear function matches the slope-intercept form of a line: ( )f x mx b= + m = constant rate of change = slope b = initial or starting value (when 0x = ) ↔ y-intercept: (0, )b Example 1: To make money for school, Sue sells snow cones during the summer. She started a work shift with $10 in her cash box . With each customer, Sue kept track of the total number of snow cones sold and the total amount in the cash box as shown in the table below. Total Number of Snow Cones Sold 0 3 7 9 Amount in Cash Box $10 $18.25 $29.25 $34.75 The amount of money in the cash box, A, is a function of the total number of snow cones sold, n. Is ( )A n a linear function? A. Draw and label a graph of the data. Does it appear to be linear? https://openstax.org/books/college-algebra-2e/pages/4-1-linear-functions 2.1 – Introduction to Linear Functions 62 B. Sometimes points appear to lie on a line, but actually do not. To truly determine if ( )A n is a linear function, calculate the average rates of change ↔ slopes to see if they are constant. Include units in your work and your answers. C. Is ( )A n a linear function? If so, write an algebraic expression for ( )A n . Circle One: Yes No ( )A n = __________________________ You Try! Example 2: The graph of a linear function f is shown below. Determine an algebraic expression for ( )f x . =)(xf ________________________ Example 3: At sea level the boiling point of water is 100 °C. But as one climbs in elevation, the boiling point of water decreases by approximately 0.0033 ° per meter in elevation. Find a linear function ( )B x that approximates the boiling point of water at an elevation of x meters. Informed by online altitude / pressure / boiling point calculator (B. Szyk, 2024) https://www.omnicalculator.com/chemistry/boiling-point-altitude 2.1 – Introduction to Linear Functions 63 Part 2: Interpreting Slope and Y-intercept in an Applied Scenario Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 4 Section 4.1: Linear Functions Objectives: • Be able to write complete sentences, using everyday English, that interpret the meanings of the slope and the y-intercept for a given linear model in an applied setting. Important Vocabulary Words: Rate of Change, Slope, Initial Value, Y-intercept, Units Example 1: The speed of sound, S, changes with the air temperature, t. For temperatures between 0°C (32°F) and 50°C (122°F), the relationship between the two can be approximated by the following: ( ) 13 741S t t= + where ( )S t is the speed of sound in miles per hour and t is the temperature in degrees Celsius. Informed by a speed of sound calculator on a National Weather Service website (T. Brice and T. Hall) A. Draw a rough sketch of the graph of ( )y S t= . i. Label the y-intercept with the appropriate number. ii. Label the horizontal and vertical axes with appropriate words and units. iii. Indicate 0 on the horizontal axis. iv. Draw a slope triangle: Label the “run” as 1 and the “rise” with the unit increase from the slope. In addition, label the rise and run with the appropriate units. B. Write a complete sentence, using everyday English, which interprets the meaning of 741 in the given context. https://openstax.org/books/college-algebra-2e/pages/4-1-linear-functions https://www.weather.gov/epz/wxcalc_speedofsound 2.1 – Introduction to Linear Functions 64 C. Write a complete sentence, using everyday English, which interprets the meaning of 13 in the given context. 2.2 – Using Point-Slope Form to Create Linear Models 65 Section 2.2: Using Point-Slope Form to Create Linear Models Part 1: Point-Slope Form of a Line Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 3 Section 3.3: Find the Equation of a Line Objectives: • Know and understand the point-slope form of the equation a line • Be able to use the point-slope form to create the equation of a line through two given points Important Vocabulary Words: Slope-Intercept Form of a Line, Point-Slope Form of a Line Slope Between Two Points on a Line 2 1 2 1 y yy x x x − = = = = − rise change in Slope m run change in where ),( 11 yx and ),( 22 yx are two points on the line Slope-Intercept Form of the Equation of a Line y mx b= + where ),0( b is the y-intercept m = slope of the line Development of the Point-Slope Form of the Equation of a Line Point-Slope Form of the Equation of a Line where ),( 11 yx is a point on the line and m = slope of the line https://openstax.org/books/intermediate-algebra-2e/pages/3-3-find-the-equation-of-a-line 2.2 – Using Point-Slope Form to Create Linear Models 66 Example 1: A. Use point-slope form to find the equation of the line that has slope = 3 and that passes through the point (4,10) . B. Rewrite the equation in slope-intercept form. You Try! Example 2: Use the point-slope form of the equation of a line to determine the equation of the line that passes through (5, -2) and (-1, 13). Then convert the equation to slope-intercept form. 2.2 – Using Point-Slope Form to Create Linear Models 67 Part 2: Creating a Linear Model Based on Two Data Points Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 4 Section 4.1: Linear Functions Objectives: • Be able to create a linear function model when given two data points in an applied scenario • Be able to write complete sentences, using everyday English, that interpret the meanings of the slope and the y-intercept for a given linear model in an applied setting. Important Vocabulary Words: Point-Slope Form of the Equation of a Line Example 1: Cigarette smoking used to be much more popular than it is now. In 1964 the U.S. Surgeon General issued a first report on the dangers of smoking. Then, in January 1971 smoking ads became banned on television and radio. As a result of these efforts and others, cigarette smoking has been in steady decline in the United States. 37.1% of US adults smoked cigarettes in 1974. By 2004 that figured had dropped to 20.9% of adults. (See references at end of problem) A. Find a linear function ( )S t that models the percent of US adults who smoked cigarettes t years after 1964. If needed, round decimal numbers to two decimal places. Linear Model: _____________________________________ https://openstax.org/books/college-algebra-2e/pages/4-1-linear-functions 2.2 – Using Point-Slope Form to Create Linear Models 68 B. Draw and label a rough sketch of the graph of ( )y S t= . In particular, label the y-intercept and draw and label a “slope triangle.” C. For the linear model found in Part A, write complete sentences, using everyday English, that interpret the meanings of the b and the m values in the given context. Meaning of b: Meaning of m: D. According to the model, what percent of U.S. adults smoked in 2014? See how your answer compares with *data provided by the American Lung Association! E. If the model continued to be valid, in which year does it predict that only 10% of US adults smoke cigarettes? Informed by an online PBS News article (Ruble, 2014), an online article in the New Yorker (Whiteside, 1970) and *data found on American Lung Association Website (American Lung Association) https://www.pbs.org/newshour/health/first-surgeon-general-report-on-smokings-health-effects-marks-50-year-anniversary https://www.newyorker.com/magazine/1970/12/19/the-fight-to-ban-smoking-ads https://www.lung.org/research/trends-in-lung-disease/tobacco-trends-brief/data-tables/ad-cig-smoke-rate-sex-race-age CHAPTER 3 COVER PAGE 69 Chapter 3 Quadratic and Polynomial Functions Sections 3.1 – 3.7 CHAPTER 3 COVER PAGE 70 3.1 – Factoring Foundations 71 Section 3.1: Factoring Foundations (R) Part 1: Factoring Out the Greatest Common Factor Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 6 Section 6.1: Greatest Common Factor and Factor by Grouping Objectives: • Find the greatest common factor for two of more monomial expressions • Factor the greatest common factor from a polynomial Important Vocabulary Words: Factors, Product, Greatest Common Factor, Factor Factors, Product and Greatest Common Factor 2 6 12 = ↔ 2 and 6 are factors of 12 and 12 is referred to as the product of 2 and 6 Similarly, by the Distributive Property of Multiplication Across Addition or Subtraction: 23 ( 5) 3 15x x x x+ = + ↔ 3x and 5x + are factors of 23 15x x+ 23 15x x+ is the product of 3x and 5x + When comparing two or more numbers, the greatest common factor of the numbers is the largest number that is a factor of all of the numbers. For example, 6 is the greatest common factor of 12, 24 and 30. The numbers 2, 3 and 6 are all common factors of 12, 24 and 30, but 6 is the largest common factor. This idea can be extended to algebraic expressions. Definition of Greatest Common Factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all of the expressions. Example 1: Find the greatest common factor of 210x , 318x and 44x . https://openstax.org/books/intermediate-algebra-2e/pages/6-1-greatest-common-factor-and-factor-by-grouping 3.1 – Factoring Foundations 72 Factoring a Polynomial Expression By Un-distributing The Greatest Common Factor BIG IDEA: Factoring a polynomial expressing means “reversing” the distributive property of multiplication across addition or subtraction. Applying the Distributive Property: 23 ( 5) 3 15x x x x+ → + Factoring – Reverse the Distributive Property: 23 15 3 ( 5)x x x x+ → + Process for Factoring By Un-distributing the GCF 1. Find the GCF of all of the terms of the polynomial. 2. Rewrite each term as a product using the GCF. 3. Un-distribute, i.e. the “reverse” Distributive Property to factor the expression. 4. Check by using the Distributive Property to multiply the two factors. Example 2: Factor each of the following polynomial expressions by factoring out the greatest common factor. A. 4 38 28x x+ B. 3 250 30 60x x x+ + You Try! Example 3: Factor each of the following polynomial expressions by factoring out the greatest common factor. A. 3 22 14 26x x x+ − B. 4 25 35x x− 3.1 – Factoring Foundations 73 Part 2: Factoring Trinomials With x2 As the Leading Term Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 6 Section 6.2: Factor Trinomials Objectives: • Factor trinomials of the form 2ax bx c+ + by using trial and error Important Vocabulary Words: Prime BIG IDEA: We will utilize patterns that occur when you multiply factors of the form ( )x a+ or ( )x a− where a is a positive whole number. Factoring Trinomials of the Form x2 + bx + c and x2 - bx + c Example 1: Use the Distributive Property to multiply ( 3)( 5)x x+ + , ( 3)( 5)x x− − and notice a useful pattern for factoring the resulting products. Example 2: Factor the following: A. 2 11 18x x+ + B. 2 9 20x x− + https://openstax.org/books/intermediate-algebra-2e/pages/6-2-factor-trinomials 3.1 – Factoring Foundations 74 Factoring Trinomials of the Form x2 - bx + c and x2 - bx - c Example 3: Use the Distributive Property to multiply ( 3)( 5)x x− + , ( 3)( 5)x x+ − and notice a useful pattern for factoring the resulting products. Example 4: Factor the following: A. 2 7 30x x− − B. 2 6 16x x+ − C. 2 6 8x x+ − Put It All Together You Try! Example 5: Factor the following or state that it is prime. A. 2 10 21x x− + B. 2 3x x+ − C. 2 3 18x x+ − 3.1 – Factoring Foundations 75 Part 3: Factoring A Difference of Squares Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 6 Section 6.3: Factor Special Products Objectives: • Factor expressions of the form 2 2a b− • Understand that expressions of the form 2 2a b+ generally cannot be factored Important Vocabulary Words: Difference of Squares, Prime Example 1: Multiply each of the following. What do you notice? A. (2 3)(2 3)x x+ + B. (2 3)(2 3)x x− + C. (2 3)(2 3)x x− − Factoring a Difference of Squares If a and b are real numbers, then 2 2 ( )( )a b a b a b− = + − Example 2: Factor the following: A. 2 36x − B. 216 1x − C. 2 4x + You Try! Example 3: Factor the following or state that it is prime. A. 2 81x − B. 29 25x + C. 29 25x − https://openstax.org/books/intermediate-algebra-2e/pages/6-3-factor-special-products 3.2 – Solving Quadratic Equations 76 Section 3.2: Solving Quadratic Equations (R) What is a quadratic equation? A quadratic equation is an equation in which the variable is squared and no exponent larger than 2 is used. The following are examples of quadratic equations: 22 32 0x − = 2 12 0x x+ − = 23 13 10x x− = 2( 2) 9 0n + − = The last example can be more-clearly seen as being quadratic when it is simplified. 2 2 2( 2) 9 0 ( 2)( 2) 9 0 2 2 4 9 0 4 5 0n n n n n n n n+ − = → + + − = → + + + − = → + − = Part 1: Factoring Method for Solving Quadratic Equations Reference Materials: Open Stax: Elementary Algebra 2e by Maracek, Anthony-Smith & Honeycutt Mathis Chapter 7 Section 7.6: Quadratic Equations Objectives: • Be able to solve quadratic equations by using factoring and the Zero Product Property Important Vocabulary Words: Zero Product Property Zero Product Property If 0a b = , then 0a = or 0b = (or both) Example 1: Solve the following equations. A. ( 2)( 5) 0x x+ − = B. 2 7 12 0x x− + = https://openstax.org/books/elementary-algebra/pages/7-6-quadratic-equations 3.2 – Solving Quadratic Equations 77 Example 2: Solve the following equations. A. 23 15 0x x+ = B. 2 2 35x x+ = You Try! Example 4: Solve the following equations. A. 2 6 0x x− = B. 2 7 6 0x x+ + = C. 2 7 18x x− = It is possible to use this technique to solve more-complex quadratic equations, such as the example below. However, most students find the quadratic formula method (Part 3) to be easier to use when solving equations of the form 2 0ax bx c+ + = and 1a  . Solve: 23 13 10 0x x− − = → (3 2)( 5) 0x x+ − = → or 3 2 0 2 2 3 2 3 2 3 3 2 3 x x x x + = − − = − − = −= 5 0 5 5 5 x x − = + + = 3.2 – Solving Quadratic Equations 78 Part 2: Square Root Method for Solving Quadratic Equations Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 9 Section 9.1: Solve Quadratic Equations Using the Square Root Property Objectives: • Be able to solve quadratic equations by the square root property Important Vocabulary Words: Square Root Property Square Root Property Solutions to 2x a= are x a= and x a= − ↔ x a=  Square Root Method For Solving Quadratic Equations 1. Completely isolate the squared part of the equation. 2. Square root both sides and apply the Square Root Property → Do not forget to use ± ! There are two solutions. Example 1: Solve the following equations. A. 22 10 0x − = B. 23 12 0x + = C. 25( 1) 10 70x − − = Something Important to Notice: In all of the above problems x is only found in the squared part of the equation. When that is not the case, other strategies for solving quadratic equations are easier to use. For example, the square root method is not the best method for solving 23( 4) 2 0x x+ + = because there is an x outside of 2( 4)x + . https://openstax.org/books/intermediate-algebra/pages/9-1-solve-quadratic-equations-using-the-square-root-property 3.2 – Solving Quadratic Equations 79 You Try! Example 2: Solve the following equations using the square root method OR the factoring method. A. 23 75 0x − = B. 2 4 0x x− = C. 24( 3) 2 30x + + = Part 3: Using the Quadratic Formula to Solve Quadratic Equations Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 9 Section 9.3: Solve Quadratic Equations Using the Quadratic Formula Objectives: • Know the quadratic formula • Be able to use the quadratic formula to solve quadratic equations Important Vocabulary Words: Quadratic Formula Quadratic Formula The solutions to 2 0ax bx c+ + = for 0a  are given by: 2 4 2 b b ac x a −  − = You are expected to memorize the quadratic formula. There are songs / rhymes / stories available to aid memorization. YouTube is a good place to look for them. https://openstax.org/books/intermediate-algebra/pages/9-3-solve-quadratic-equations-using-the-quadratic-formula 3.2 – Solving Quadratic Equations 80 Example 1: Solve the following equation. 23 13 10 0x x− − = Example 2: Solve the following equation. 22 5 1x x+ = You Try! Example 3: Solve the following equation. 25 2 7x x− = 3.3 – Transformations of x^2 81 Section 3.3: Transformations of x2 Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 9 Section 9.7: Graph Quadratic Functions Using Transformations Objectives: • Be able to use the ideas of vertical and horizontal shifts, vertical reflection and vertical stretch / compression to graph quadratic functions of the form 2( ) ( )f x a x h k= + + • Be able to create an algebraic expression for a quadratic function from its graph based upon the vertex and a point on the graph that is one horizontal unit away from the vertex. Important Vocabulary Words: Parabola, Vertical Shift, Horizontal Shift, Vertex, Axis of Symmetry, Vertical Reflection, Vertical Stretch / Compression Factor The Overall Idea One way to think about quadratic functions is as transformations of the basic / toolkit function 2y x= . You graphed some transformations of basic / toolkit functions in the Section 1.3 homework assignment and likely noticed some patterns. In this section you will use a Desmos activity site to work through an exploration of those patterns. Desmos Exploration 1. Turn on the first quadratic function, f(x) 2x k= + . Then, vary the k slider to see how the parameter k alters the graph. Write down your observations below. Be sure to discuss the difference between when k is negative and when k is positive. 2. Turn off f(x) and turn on g(x) 2( )x h= + . Vary the h slider to see how the parameter h alters the graph. Write down your observations below. Be sure to discuss the difference between when h is positive and when h is negative. Does the graph behave the way you had expected?! https://openstax.org/books/intermediate-algebra-2e/pages/9-7-graph-quadratic-functions-using-transformations https://www.desmos.com/calculator/vynxhv4twy 3.3 – Transformations of x^2 82 3. Turn off g(x) and turn on i(x) 2ax= AND x = {0 0: a < 0: y-intercept: Factored Form of a Quadratic Function: ))(()( nxmxaxf −−= ; where m, n are real numbers. If a > 0: a < 0: x-intercepts: Standard Form of a Quadratic Function: 2( ) ( )f x a x h k= − + ; where a, h, k are real numbers. If a > 0: a < 0: Vertex: Focus on Standard and General Form You Try! Example 2: Let 2( ) 2( 1) 8f x x= − − + . A. Determine the coordinates of the vertex. B. The graph of )(xfy = is a parabola. Does the parabola open up or down? C. Determine the y-intercept of the graph. D. Express the function in general form. 3.4 – Three Forms of a Quadratic Function 87 Determining X-intercepts and Y-intercepts To find the x-intercept(s) of the graph of f : Solve for x: ( ) 0f x = To find the y-intercept of the graph of f : Determine: (0)y f= E. Determine the x-intercepts of the graph of ( )y f x= . There are three ways to do this! First Way: Second Way: Third Way: 3.4 – Three Forms of a Quadratic Function 88 Quadratic Formula: The solutions for 2 0ax bx c+ + = are 2 4 2 b b ac x a −  − = . You MUST know this! F. Sketch the graph of ( )y f x= . 3.5 – Modeling with Quadratic Functions 89 Section 3.5: Modeling with Quadratic Functions Part 1: General Form - A Method for Finding the Vertex Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 5 Section 5.1: Quadratic Functions Objectives: • Be able to determine the vertex for a quadratic function presented in general form • Be able to answer maximum / minimum value questions for a quadratic function in general form Important Vocabulary Words: General Form, Vertex, Maximum Value, Minimum Value Development of a Formula / Procedure to Find the Vertex BIG IDEA: The x-coordinate of the vertex of a quadratic function occurs exactly half-way between the x-intercepts of the graph of that function. Graph of 2( )f x ax bx c= + + Formula / Procedure to Find the Vertex of a Quadratic Function in General Form: If cbxaxxf ++= 2)( , then the vertex of the graph of ( )y f x= is at: , 2 2 b b f a a  − −         . https://openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions 3.5 – Modeling with Quadratic Functions 90 Application of the Formula / Procedure Example 1: Let 2 ( ) 2 7 3 x f x x= − + + . A. What is the vertex of the graph of f ? B. Does f have a maximum or minimum value? State the maximum or minimum value of f and the value of x at which it occurs. You Try! Example 2: A volleyball flies into the air after a player successfully reacts to a strong serve. The height above the ground, in feet, of the volleyball t seconds after the player received it is given by 2( ) 16 48 1h t t t= − + + . What is the maximum height attained by the ball, and when did it occur? A. Draw a rough sketch of the graph by plotting and labeling the y-intercept and using what you know about the direction the graph opens. Label the axes with words. B. Determine the maximum height of the ball and when it occurred. 3.5 – Modeling with Quadratic Functions 91 Part 2: Quadratic Modeling Reference Materials: Open Stax: College Algebra 2e by Jay Abramson Chapter 5 Section 5.1: Quadratic Functions Objectives: • Be able to create quadratic function models to represent applied situations • Be able to use the formula / procedure from Part 1 to answer maximum / minimum value questions • Be able to determine an output value corresponding to a given input value for a quadratic model • Be able to use the quadratic formula to determine input values corresponding to a given output value for a quadratic model • Be able to determine whether a modeling question is asking about the vertex, input value(s) or an output value for a quadratic model Optimization Word Problems – Strategy Recommendations 1 First read through the entire problem to get an “overview” perspective. 2. Find the question or task sentence and use it to: • Determine what is to be maximized or minimized. That is what you need to create a function / model for if one is not already provided. • Determine the input variable for the output quantity that is to be maximized or minimized. 3. Strategies to help you create a function: • Draw and label a picture; • Calculate output values for some simple numerical input values in order to help you discover and understand mathematical relationships; • Write down mathematical relationships in words BEFORE trying to express them algebraically; • Test your function / model with simple cases to see if it makes sense. 4. Draw a rough sketch of the graph of your function / model. 5. Use 2x b a= − to determine the input-coordinate of the vertex of the graph. 6. Use ( 2 )y f b a= − to determine the output-coordinate of the vertex of the graph. Business Terms Price: What consumers pay for a product. Cost: Business expenses to make and sell a product. Revenue: Money coming in Profit: Amount kept after subtracting business expenses Revenue = Price x Quantity Sold Profit = Revenue – Cost https://openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions 3.5 – Modeling with Quadratic Functions 92 Understanding Business Terms & Their Relationships The graphs of monthly revenue, ( )R x , and cost, ( )C x , in thousands of dollars from producing and selling x items are shown below. A. What production levels result in the company making a profit? B. How many items must the company produce and sell to break even? C. What production levels result in the company taking a monthly loss? Examples Example 1: During the summer months, Bozeman Bike & Ski found that they could rent an average of 8 bicycles per day at a rate of $30 per day. But, if they decreased their price to $25, they would rent 18 bicycles each day. Furthermore, BB&S noticed that there is a linear relationship between the price they charge to rent a bicycle and the number of bicycles they could expect to rent out each day. BB&S wants to know what to charge for a bicycle rental in order to have the largest daily revenue. A. Determine an algebraic expression for Bozeman Bike & Ski’s daily revenue from bicycle rentals when they charge x dollars per day to rent a bicycle. ( )R x = 3.5 – Modeling with Quadratic Functions 93 From Part A: ( )R x = where x = B. Determine Bozeman Bike & Ski’s largest possible daily revenue from bicycle rentals and the amount they should charge in order to obtain the maximum revenue. Round your answers to the nearest cent. C. Bozeman Bike and Ski estimates that its average daily cost to provide bike rentals is $200. To show appreciation to the local community, a couple of days a year they offer bike rentals to people who live in Bozeman at a very low rate. What is the lowest price that BB&S can charge and still break even for the day, i.e. have enough revenue to meet their expenses? Round your answer to the nearest cent. 3.5 – Modeling with Quadratic Functions 94 Example 2: Betty has 72 linear feet of chicken-wire fencing and wants to use it to make a portable rectangular frame with a divider down the middle (as shown in the diagram) in order to provide separate outdoor exercise spaces for her laying hens and pet rabbits. What dimensions should Betty use to make the rectangular frame if she wants to give her animals as much space to roam as possible? You Try! Example 3: A landscaping business wants to build a series of four open-front bins for storing various types of mulch as shown in the diagram below. They have enough materials to build 130 linear feet of walls. A. Determine a function for the total area of the set of bins in terms of their depth, x. 3.5 – Modeling with Quadratic Functions 95 From Part A: ( )A x = where x = B. What is the largest total area that can be enclosed in the bin set and what overall dimensions should be used to build the bin set in order to achieve it? C. If the landscaping business wants each bin to enclose 150 ft2, what dimensions should be used to build the set of bins? 3.6 – X-intercepts of a Quadratic Function 96 Section 3.6: X-intercepts of a Quadratic Function Part 1: Factored Form of a Quadratic Function Reference Materials: See reference material provided in D2L. Objectives: • Know connections between the factored form of a quadratic function and its graph • Be able to sketch the graph of a quadratic function that is presented in factored form • Be able to determine the factored form of a quadratic function based on the x-intercepts of its graph and another point on its graph Important Vocabulary Words: Factored Form, X-intercepts Factored Form of a Quadratic Function: ))(()( nxmxaxf −−= ; where a, m, n are real numbers. 0a  : opens up 0a  : opens down x-intercepts: ( ) ( ),0 , ,0m n ; x-coordinate of vertex 2 m n+ = ; y-coordinate of vertex 2 m n f +  =     Example 1: Determine the x-intercepts and the vertex of graph of ( ) 2( 1)( 5)f x x x= − − and use them to sketch the graph. 3.6 – X-intercepts of a Quadratic Function 97 Example 2: Determine the algebraic expression in factored form of the quadratic function whose graph is shown. 3.6 – X-intercepts of a Quadratic Function 98 You Try! Example 3: The graph of a quadratic function 𝑓 has x-intercepts at ( 9,0)− and (5,0) and passes through the point (1,8) . A. Determine an algebraic expression for ( )f x . ( )f x = B. Determine the vertex of the graph of ( )y f x= . Vertex: REVIEW - You Try! Example 4: Let 2( ) 4 7g x x x= − + − . A. Find the x- and y-intercepts for the graph of ( )y g x= . Write your answers using coordinate pair notation. x-intercept: y-intercept: B. Determine the vertex for the graph of ( )y g x= . Vertex: 3.6 – X-intercepts of a Quadratic Function 99 C. Sketch the graph of ( )y g x= . Part 2: Using the Quadratic Formula to Factor Reference Materials: See reference material provided in D2L. Objectives: • Be able to convert a quadratic function from general form to factored form; especially by using the quadratic formula as “bridge” in the process Important Vocabulary Words: Factored Form, Quadratic Formula Example 1: Find the x-intercepts for the graph of 2( ) 2 18 36h x x x= − + . Write your answers using coordinate pair notation. 3.6 – X-intercepts of a Quadratic Function 100 Example 2: Use the quadratic formula to find the x-intercepts for 2( ) 3 144 5184f x x x= − − . Then rewrite ( )f x in factored form: ( ) ( )( )f x a x m x n= − − . Factored form: ( )f x = You Try! Example 3: Use the quadratic formula to rewrite 2( ) 5 37 24g x x x= − − in factored form: ( ) ( )( )f x a x m x n= − − . Factored form: ( )g x = 3.6 – X-intercepts of a Quadratic Function 101 Part 3: The Discriminant Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 9 Section 9.6: Graph Quadratic Functions Using Properties Objectives: • Be able to use the discriminant to determine the number of real solutions to a quadratic equation and hence use it to determine the number of x-intercepts for the graph of a quadratic function Important Vocabulary Words: Discriminant The Discriminant: The Discriminant of the quadratic equation 2 0ax bx c+ + = is D = b2 – 4ac ; i.e. the part under the square root in the Quadratic Formula. • If 0D  (negative), then the equation has no real solutions. • If 0D = , then the equation has exactly one real solution. • If 0D  (positive), then the equation has two real solutions. Example Graphs: Example 1: Use the discriminant to determine the number of x-intercepts for the graphs of the following quadratic functions: A. 2( ) 2 5f x x x= + + B. 2( ) 4 4 1g x x x= − + C. 2( ) 2 8 3h x x x= − − https://openstax.org/books/intermediate-algebra-2e/pages/9-6-graph-quadratic-functions-using-properties 3.7 – Polynomial Functions 102 Section 3.7: Polynomial Functions Reference Materials: Khan Academy Resources. All Khan Academy content is available for free at www.khanacademy.org . • YouTube Video: Polynomials intro • YouTube Video: Zeros of Polynomials: plotting zeros • YouTube Video: Zeros of Polynomials (with factoring): common factor • Exposition / Lesson: Positive & negative intervals of polynomials • YouTube Video: Positive and negative intervals of polynomials Objectives: • Know what polynomial functions are, polynomial vocabulary terms and what the graphs of polynomial functions look like • Be able to determine the zeros / x-intercepts of a simple polynomial function by using factoring and the Zero Product Property • Be able to determine intervals where a polynomial function is positive and negative and make graphical connections Important Vocabulary Words: Polynomial, Degree, Leading Term, Leading Coefficient, Zeros, X-intercepts, Zero Product Property Definition of a Polynomial Function: A polynomial function is a function of the form: 1 1 1 0( ) . . .n n n np x a x a x a x a− −= + + + + ; where 0 1, , . . . , na a a are real numbers and n is a positive whole number. n (the largest exponent) is called the degree of the polynomial. n na x is called the leading term of the polynomial. na is called the leading coefficient of the polynomial. Examples: Non-Examples http://www.khanacademy.org/ https://youtu.be/Vm7H0VTlIco https://youtu.be/5qQUN1fugXQ https://youtu.be/SONskIMkYdU https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/a/positive-and-negative-intervals-of-polynomials https://youtu.be/z3faK7OV3Ew 3.7 – Polynomial Functions 103 Graphs of polynomials look like smooth curves with no jumps or breaks. Zeros and X-intercepts The zeros of a polynomial ( )p x are the solutions to ( ) 0p x = If q is a zero of ( )p x , then ( ) 0p q = and ( ,0)q is an x-intercept for the graph of ( )y p x= . Fact: A polynomial of degree n can have up to n zeros / x-intercepts. Finding Zeros / X-intercepts of a Polynomial of Degree > 3 Solve ( ) 0p x = by factoring ( )p x and using the Zero Product Property. Zero Product Property 0a b = if and only if 0a = or 0b = Example 1: Given polynomial: 3 2( ) 2 16 30p x x x x= − + − A. Determine the zeros of ( )p x and the x-intercepts of the graph of ( )y p x= . Use coordinate pair notation to write the x-intercepts. 3.7 – Polynomial Functions 104 A function ( )f x is positive over an interval if ( ) 0f x  for all values of x in the interval. A function ( )f x is negative over an interval if ( ) 0f x  for all values of x in the interval. B. Determine the intervals over which ( )p x is positive and negative. C. Draw a rough sketch of the graph of ( )y p x= . You Try! Example 2: Given polynomial: 3 2( ) 3 75q x x x= − A. Determine the x-intercepts of the graph of ( )y q x= . Write your answers using coordinate pair notation. 3.7 – Polynomial Functions 105 B. Determine the intervals over which ( )q x is positive and negative. C. Draw a rough sketch of the graph of ( )y q x= . You Try! Example 3: Find the zeros of 5 4 3( ) 7 21 7r x x x x= − + + . 3.7 – Polynomial Functions 106 CHAPTER 4 COVER PAGE 107 Chapter 4 Rational Functions, Combining Functions & Function Notation - Beyond the Basics Sections 4.1 – 4.6 CHAPTER 4 COVER PAGE 108 4.1 – Rational Expressions 109 Section 4.1: Rational Expressions Part 1: Introduction to Rational Expressions Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 7 Section 7.1: Multiply and Divide Rational Expressions Objectives: • Know what is meant by a rational expression • Be able to state the restrictions of a rational expression • Be able to simplify rational expressions Important Vocabulary Words: Rational Expression, Simplify Definition of a Rational Expression: A rational expression is an expression of the form ( ) ( ) p x q x where ( )p x and ( )q x are polynomials and ( ) 0q x  . To avoid ( ) 0q x = , the value of x may need to be restricted. Example 1: What are restrictions on x for 2 3 7 16 x x + − ? Like fractions, rational expressions can be simplified. Example 2: A. Simplify: 20 15 B. Simplify: 2 2 5 3 10 x x x x − − − Restrictions: https://openstax.org/books/intermediate-algebra-2e/pages/7-1-multiply-and-divide-rational-expressions 4.1 – Rational Expressions 110 Part 2: Multiply & Divide Rational Expressions Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 7 Section 7.1: Multiply and Divide Rational Expressions Objectives: • Be able to multiply and divide rational expressions and write the answer in simplest form Because they are essentially algebraic fractions, multiplication and division of rational expressions mirror multiplication and division of fractions. Example 1: A. Multiply and simplify: 6 7 5 9  B. Multiply and simplify: 2 2 4 9 3 2 10 x x x x x −  + − Example 2: A. Divide and simplify: 10 20 3 21  B. Divide and simplify: 2 2 2 2 6 5 5 x x x x x x − −  + + + https://openstax.org/books/intermediate-algebra-2e/pages/7-1-multiply-and-divide-rational-expressions 4.1 – Rational Expressions 111 You Try! Example 3: Divide and simplify: 2 2 2 5 6 3 x x x x x x +  − − − Part 3: Add and Subtract Rational Expressions Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 7 Section 7.2: Add and Subtract Rational Expressions Objectives: • Be able to add and subtract rational expressions and write the answer in simplest form Addition and subtraction of rational expressions mirror addition and subtraction of fractions ↔ need to utilize a common denominator! Example 1: Perform the operations and simplify your answers. A. 2 4 3 5 + B. 2 3 x x x + + Example 2: Perform the operations and simplify your answers. A. 7 1 15 6 − B. 2 2 5 3 2 4x x x − − − https://openstax.org/books/intermediate-algebra-2e/pages/7-2-add-and-subtract-rational-expressions 4.1 – Rational Expressions 112 You Try! Example 3: Perform the operations and simplify your answers. A. 5 4 x x − + HINT: 5 5 1 = B. 2 1 5 2 15 x x x x + − − − 4.2 – Rational Equations 113 Section 4.2: Rational Equations Reference Materials: Open Stax: Intermediate Algebra 2e by Maracek & Honeycutt Mathis Chapter 7 Section 7.4: Solve Rational Equations Objectives: • Be able to solve equations and formulas that involve rational expressions Important Vocabulary Words: Clear / eliminate the fractions Eliminate the Fractions Strategy: Multiply both sides of the equation by the LCM of the denominators. Then undo the division on both sides of the equation by using the LCM to cancel out the denominators. Example 1: Let 1 ( ) 5 2 x f x x − = + . Solve for x: 4 ( ) 3 f x = Are there any restrictions on x to consider? You Try! Example 2: Let 2 6 ( ) ( 7) g x x = + . Solve for x: ( ) 2g x = Are there any restrictions on x to consider? https://openstax.org/books/intermediate-algebra-2e/pages/7-4-solve-rational-equations 4.2 – Rational Equations 114 Example 3: Solve for x: 3 1 1 2 2x x + = + − . Are there any restrictions on x to consider? You Try! Example 4: Solve for x: 5 1 0 3x x − = − . Are there any restrictions on x to consider? 4.2 – Rational Equations 115 You Try! Example 5: Two resistors with resistances A and B are connected in parallel leading to a combined resistance C. The relationship between A, B and C is given by AB C A B = + . Solve the formula for B. 4.3 – Rational Functions 116 Section 4.3: Rational Functions Part 1: Introduction to Rational Functions Reference Materials: Scottsdale Community College Mathematics Division OER Resources Website College Algebra, An Investigation of Functions by David Lippman, Melonie Rasmussen & Jay Abramson Chapter 3 Section 3.4: Rational Functions pages 201 – 208 of the pdf-file (printed pages 193 – 200) NOTE: We are not using rational functions content on pages beyond those stated above. Objectives: • Know what rational functions, vertical asymptotes and horizontal asymptotes are • Be able to identify the domain, vertical asymptote(s) and horizontal asymptote from the graph of a rational function • Be able to determine the domain, vertical asymptote(s) and x- and y-intercepts of a rational function when given its algebraic expression Important Vocabulary Words: Rational Function, Vertical Asymptote, Horizontal Asymptote, X-intercept, Y-intercept Features of Rational Functions Rational Function: A function of the form ( ) ( ) ( ) p x f x q x = where ( )p x and ( )q x are polynomial functions. Example 1: The graph of 3 6 ( ) 4 x f x x − = + is shown below. What do you notice? Domain: x ( )y f x= 4.1− 183 4.01− 1803 4.001− 18003 x ( )y f x= 3.9− 177− 3.99− 1797− 3.999− 17997− x ( )y f x= 100− 3.1875 1,000− 3.0181 10,000− 3.0018 x ( )y f x= 100 2.8269 1,000 2.9821 10,000 2.9982 https://sccmath.wordpress.com/ https://sccmath.wordpress.com/wp-content/uploads/2017/07/college-algebra-proof-4-with-links-ed4-fall-2017.pdf 4.3 – Rational Functions 117 Vertical Asymptote: A vertical line x a= where the output values tend towards positive or negative infinity as the input values get closer and closer to a . (Function graphs never cross a vertical asymptote.) A function 𝑓(𝑥) has a vertical asymptote at x a= if _________________ as ___________ . Horizontal Asymptote: A horizontal line y b= where the graph gets closer and closer to the line as the input values get larger and larger in either the positive or negative direction; i.e. on the far right or far left sides of the graph. (Function graphs can cross a horizontal asymptote.) A function 𝑓(𝑥) has a horizontal asymptote at y b= if ___________ as _______________ . You Try! Example 2: Determine the domain, vertical asymptote(s) and horizontal asymptote, y- intercept and x-intercept(s) for the rational function whose graph is shown below. Domain: Vertical Asymptote(s): Horizontal Asymptote: y-intercept: Approximate the x-intercept(s): Example 3: Find the following for 2 4 5 ( ) 2 x x f x x − − = − . A. Domain B. Vertical Asymptote(s) C. Y-intercept D. X-intercept(s) 4.3 – Rational Functions 118 To find the vertical asymptote(s) of a rational function: To find the y-intercept of a rational function: To find the x-intercept(s) of a rational function: Unusual Cases Example: Find the vertical asymptote(s) for 2 3 ( ) 2 3 x f x x x + = + − . Work: Look for when denominator = 0 ↔ Solve for x: 2 2 3 0x x+ − = ( 3)( 1) 0 3 0 or 1 0 3 or 1 x x x x x x → + − = → + = − = → = − = So, one expects the graph of ( )y f x= to have two vertical asymptotes. BUT, the graph of ( )y f x= is: What is happening?! Simplify: 2 3 3 1 ( ) 2 3 ( 3)( 1) 1 x x f x x x x x x + + = = = + − + − − for 3x  − . So, the graph of 2 3 ( ) 2 3 x f x x x + = + − is the same as the graph of 1 ( ) 1 g x x = − except that the graph of ( )y f x= has a hole at ( 3, 1 4)− − . ** It is necessary to simplify rational functions BEFORE finding vertical asymptotes. GOOD NEWS: In this course we will only ask you to work with rational functions that are already simplified. We will not expect you to recognize when graphs have holes. 4.3 – Rational Functions 119 Practice You Try! Example 4 A & B: Find the domain, vertical asymptote(s), y-intercept and x-intercept(s) for each of the following rational functions. [You will find it easier if you first factor whenever possible.] A. 2 3 ( ) 4 1 x f x x + = + B. 2 1 ( ) 5 6 x g x x x − + = − + Domain: Domain: Vertical Asymptote(s): Vertical Asymptote(s): Y-intercept: Y-intercept: X-intercept(s): X-intercept(s): 4.3 – Rational Functions 120 Part 2: