# MA002: Precalculus I

Unit 2: Linear Functions   Is there a relationship between the number of hours of TV a person watches and the number of sit-ups that person can do? Exploring data and determining if a pattern exists is an important part of mathematical modeling. If a relationship is found, it can be used to predict current and future behaviors of the variables.

In this unit, the relationship of variables with a constant rate of change is explored. Linear functions are used widely in applications of mathematics and your study will begin with the slope of a line, computing linear equations based on various forms of data and how to graph using both analytic methods and technology. Further exploration of data and technology are learned by using least-square regression on data sets to determine if a linear relationship exists and the strength of the trend. Other relationships will be explored in future units as the data sets become more complex.

This unit should take approximately 20.75 hours to complete.

☐    Subunit 2.1: 3.75 hours

☐    Subunit 2.2: 4.5 hours

☐    Subunit 2.3: 3.75 hours

☐    Subunit 2.4: 3 hours

☐    Subunit 2.5: 3.75 hours

☐    Review and Exam: 2 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - compute a slope of a line and write its equation from data or graphs; - graph constant, linear, and absolute value equations; - identify an appropriate approach to solve linear equations and then interpret the solution; - utilize technology to find the best fit linear equation for a set of data and use it to predict other data; and - solve linear and absolute value equations and inequalities.

2.1 Linear Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” (PDF)

Instructions: Read pages 99–106 of Chapter 2 to learn about lines (linear functions), slopes, graphs of lines, and how to find where lines intersect. You will also learn how to apply lines to real-life situations and find the best fit line to data. Note that this reading covers the material you need to know for subunits 2.1.1 through 2.1.3.

Reading this section should take approximately 30 minutes.

2.1.1 Defining Linear Functions   Note: This subunit is covered in the reading assigned beneath subunit 2.1. To read some examples of real-world situations that use linear functions and the formula for a linear function, read the information on pages 99 through 101.

Instructions: Watch this video to learn the process of modeling data with a linear function. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

2.1.2 Slopes and Rate of Change   Note: This subunit is covered in the reading assigned beneath subunit 2.1. To learn how to find a slope or rate of change of a linear function and how this can determine if a line is increasing or decreasing, read the information on pages 100 through 103.

Instructions: Watch this video to see examples of rates of change and how this relates to the slope of a line. Demonstrations of computing slopes of lines from data and graphs are included. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

• Web Media: YouTube: Mathispower4u: “A line in y = mx + b form” Link: YouTube: Mathispower4u: “A line in y = mx + b form” (YouTube)

Instructions: Watch this video to see through an animation how changes to the slope or the y-intercept affect the graph of the line. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

2.1.3 Finding the Equation of the Line   Note: This subunit is covered in the reading assigned beneath subunit 2.1. To learn how to use the slope and formula for a line to write equations of lines for specific situations, read the information on pages 103 through 106.

• Web Media: YouTube: NEAPortal: “Write an Equation in Slope-Intercept Form Given Two Points” Link: YouTube: NEAPortal: “Write an Equation in Slope-Intercept Form Given Two Points” (YouTube)

Instructions: Watch this video to learn how to find the equation of a line given two points on the line. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

• Web Media: YouTube: Mathispower4u: “Find the Equation of a Line in Slope Intercept Form Given the Graph of a Line” Link: YouTube: YouTube: Mathispower4u: “Find the Equation of a Line in Slope Intercept Form Given the Graph of a Line” (YouTube)

Instructions: Watch this video to learn how to write the equation of the line in y = mx + b form from the graph of a line. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

Instructions: Watch this video to learn how to use the equation of a line to model the depth of a submarine at different times. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.1: Linear Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.1: Linear Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.1, complete this activity. You will need to create a free account in order to access the assessment. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

2.2 Graphs of Linear Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” (PDF)

Instructions: Read pages 111–120 of Chapter 2 to learn about different techniques to graph lines, special equations for horizontal and vertical lines, connections between slopes and parallel or perpendicular lines, and how to find the intersection of two lines. Note that this reading covers the material you need to know for subunits 2.2.1 through 2.2.4.

Reading this section should take approximately 45 minutes.

2.2.1 Graphing Techniques   Note: This subunit is covered in the reading assigned beneath subunit 2.2. To learn how to graph a linear function in slope-intercept form and match linear equations to graphs, read the information on pages 111 through 115.

• Web Media: YouTube: Picrustable’s “Graph Linear Equations Using y = mx + b” Link: YouTube: Picrustable’s “Graph Linear Equations Using y = mx + b” (YouTube)

Instructions: Watch this video to see how to graph a linear function from an equation in slope intercept form. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

2.2.2 Horizontal and Vertical Lines   Note: This subunit is covered in the reading assigned beneath subunit 2.2. To learn the unique equations for horizontal and vertical lines, read the information on pages 116 and 117.

• Web Media: YouTube: mathwithmrbarnes’s “Equations of Horizontal and Vertical Lines” Link: mathwithmrbarnes’s “Equations of Horizontal and Vertical Lines” (YouTube)

Instructions: Watch this video to learn how to write and graph the unique equation of horizontal and vertical lines. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 15 minutes.

2.2.3 Parallel and Perpendicular Lines   Note: This subunit is covered in the reading assigned beneath subunit 2.2. To learn the relationship between parallel and perpendicular lines and their slopes, read the information on pages 117 through 119.

• Web Media: YouTube: Mathispower4u: “Parallel and Perpendicular Lines – Part I” Link: YouTube: Mathispower4u: “Parallel and Perpendicular Lines – Part I” (YouTube)

Instructions: Watch this video to learn about the relationship between slopes of two parallel lines and between two perpendicular lines. You will also learn how to determine if two lines are parallel, perpendicular, or neither from the equations of the lines. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 25 minutes.

• Web Media: YouTube: Mathispower4u: “Parallel and Perpendicular Lines – Part II” Link: YouTube: Mathispower4u: “Parallel and Perpendicular Lines – Part II” (YouTube)

Instructions: Watch this video to review the concepts of parallel and perpendicular lines and to learn how to write equations of parallel and perpendicular lines given sufficient information. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

2.2.4 Intersections of Lines   Note: This subunit is covered in the reading assigned beneath subunit 2.2. To learn how to find where two linear functions intersect, read the information on pages 119 and 120.

Instructions: Watch this video to see how to solve for the intersection of two lines when they are both in y = mx + b form. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.2: Graphs of Linear Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.2: Graphs of Linear Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.2, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 4a: Sections 2.1–2.2” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 4a: Sections 2.1–2.2” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunits 2.1 and 2.2, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 4a: Sections 2.1-2.2 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

2.3 Modeling with Linear Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” (PDF)

Instructions: Read pages 126–133 of Chapter 2 to learn how to take real-world scenarios and model them with a linear function. Note that this reading covers the material you need to know for subunits 2.3.1 through 2.3.2.

Reading this section should take approximately 30 minutes.

2.3.1 Problem-Solving Strategy   Note: This subunit is covered in the reading assigned beneath subunit 2.3. To learn a seven-step process for problem solving for any application and how it is applied to a real-life problem, read the information on pages 126 and 127.

• Web Media: YouTube: GreeneMath.com: “Algebra I Help: Solving Word Problems I” Link: YouTube: GreeneMath.com: “Algebra I Help: Solving Word Problems I” (YouTube)

Instructions: Watch this video to see how to apply a five-step process to solve applications that involve creating and solving linear equations. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 30 minutes.

Instructions: Watch this video to learn how to solve linear application problems using linear equation and graphing. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 25 minutes.

2.3.2 Linear Modeling Applications   Note: This subunit is covered in the reading assigned beneath subunit 2.3. To read through several examples where the problem-solving process is applied by using linear functions, read the information on pages 128 through 132.

Instructions: Watch this video to see how to take a table of values or data and find the best type of model or equation that fits that data. Then you will learn how to find the equation or model for the data. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 25 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.3: Modeling with Linear Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.2: Modeling with Linear Functions – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.3, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 5a: Section 2.3” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 5a: Section 2.3” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.3, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 5a: Section 2.3 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

2.4 Fitting Linear Models to Data   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” (PDF)

Instructions: Read pages 138–143 of Chapter 2 to learn how to take a table of data and use technology to find the best fit line to model the data. Note that this reading covers the material you need to know for subunits 2.4.1 through 2.4.3.

Reading this section should take approximately 15 minutes.

2.4.1 Linear Modeling by Hand   Note: This subunit is covered in the reading assigned beneath subunit 2.4. To learn how to use two points in a set of numbers to model the data and how to interpolate and extrapolate using the model, read the information on pages 138 through 140.

Instructions: Watch this video to find out how to create the best fit line when data appears to be linear but does not have an exact equation that correlates the independent variable and the dependent variable. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

2.4.2 Linear Modeling using Technology   Note: This subunit is covered in the reading assigned beneath subunit 2.4. To learn how to use least-squares regression to find a linear model to a data set, read the information on pages 140 and 141.

• Web Media: YouTube: Mathispower4u: “Linear Regression on the TI84 – Example 1” Link: YouTube: Mathispower4u: “Linear Regression on the TI84 – Example 1” (YouTube)

Instructions: Watch this video on how to use a TI-83 or TI-84 graphing calculator to find the equation of the line of best fit for a set of data. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

2.4.3 The Correlation Coefficient   Note: This subunit is covered in the reading assigned beneath subunit 2.4. To learn how to find and interpret the correlation coefficient for a model, read the information on pages 141 through 143.

Instructions: Watch this video to learn about the correlation coefficient and to practice matching correlation value to scatterplots. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.4: Fitting Linear Models to Data – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.4: Fitting Linear Models to Data – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.4, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

2.5 Absolute Value Function   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 2: Graphs of Linear Functions” (PDF)

Instructions: Read pages 146–152 of Chapter 2 to learn about the absolute value function, its graph, and how to solve absolute value equations. Note that this reading covers the material you need to know for subunits 2.5.1 through 2.5.3.

Reading this section should take approximately 30 minutes.

2.5.1 The Absolute Value Function: Graph and Features   Note: This subunit is covered in the reading assigned beneath subunit 2.5. To learn the definition of absolute value as a distance and the special graphical features of this function, read the information on pages 146 through 148.

• Web Media: YouTube: MuchoMath: “Absolute Value & the Opposite of a Number” Link: YouTube: MuchoMath: “Absolute Value & the Opposite of a Number” (YouTube)

Instructions: Watch this video to review how to take the absolute value of a number and its relationship to a distance on the number line. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

• Web Media: YouTube: Mathispower4u: “Graph an Absolute Value Function Using a Table of Values” Link: YouTube: Mathispower4u: “Graph an Absolute Value Function Using a Table of Values” (YouTube)

Instructions: Watch this video to see how to plot points to graph a linear absolute value function. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 10 minutes.

2.5.2 Solving Absolute Value Equations   Note: This subunit is covered in the reading assigned beneath subunit 2.5. To learn how to solve absolute value equations that can have zero, one, or two solutions, read the information on pages 148 and 149.

Instructions: Watch this video to see how to algebraically solve absolute value equations and also solve them graphically on the TI-84 graphing calculator. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 20 minutes.

2.5.3 Solving Absolute Value Inequalities   Note: This subunit is covered in the reading assigned beneath subunit 2.5. To learn how to solve absolute value inequalities using a graph and algebra, read the information on pages 149 through 152.

Instructions: Watch this video to see how to algebraically solve absolute value inequalities. You will also learn how to express the solution on a number line, with inequalities and interval notation. Take detailed notes and watch the video as many times as you need.

Watching this video should take approximately 25 minutes.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.5: Absolute Value Function – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 2.5: Absolute Value Function – Graded Homework” (HTML)

Instructions: Once you have completed the readings and viewed the material for all of subunit 2.5, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.

Completing this assessment should take approximately 1 hour.

• Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 6: Sections 2.3–2.5” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 6: Sections 2.3–2.5” (PDF)

Instructions: Once you have completed the readings and viewed the material for all of subunits 2.3–2.5, complete this quiz.

Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 6: Sections 2.3-2.5 Answer Key” (PDF).

Completing this assessment should take approximately 30 minutes.

Unit 2 Review   - Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 2 Review Problems” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 2 Review Problems” (HTML)

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