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.
Unit 2 Time Advisory
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.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
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.
- Web Media: YouTube: Dr. Phil Clark’s “Big Business, Constant Rate
of Change”
Link: YouTube: Dr. Phil Clark’s “Big Business, Constant Rate of
Change” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Web Media: YouTube: Mathispower4u: “Rate and Slope” Link: YouTube: Mathispower4u: “Rate and Slope” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Web Media: YouTube: Mathispower4u: “Linear Equation Model for Submarine Depth” Link: YouTube: Mathispower4u: “Linear Equation Model for Submarine Depth” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Web Media: YouTube: Sensible Math: “Method of Substitution” Link: YouTube: Sensible Math: “Method of Substitution” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Web Media: YouTube: Khan Academy’s “Word Problem Solving 4” Link: YouTube: Khan Academy’s “Word Problem Solving 4” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Web Media: YouTube: TenMarksInstructor’s “Linear Models” Link: YouTube: TenMarksInstructor’s “Linear Models” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
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.
- Web Media: YouTube: TenMarksInstructor’s “Equations of a Best Fit
Line”
Link: YouTube: TenMarksInstructor’s “Equations of a Best Fit
Line” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Web Media: YouTube: TheMathClips: “The Correlation Coefficient” Link: YouTube: TheMathClips: “The Correlation Coefficient” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
- Web Media: YouTube: Mathispower4u: “Absolute Value Equations”
Link: YouTube: Mathispower4u: “Absolute Value
Equations” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Web Media: YouTube: Mathispower4u: “Absolute Value Inequalities” Link: YouTube: Mathispower4u: “Absolute Value Inequalities” (YouTube)
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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
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)
Instructions: Once you have completed the readings and viewed the
material for all of unit 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 assignment.
Your answers will be scored upon submission. This will provide a
review for your unit assessment.
Completing this review should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use
displayed on the webpage above.
Unit 2 Exam
- Assessment: Washington State Board for Community & Technical
Colleges: “Chapter 2: Exam A”
Link: Washington State Board for Community & Technical Colleges:
“Chapter 2: Exam
A” (HTML)
Instructions: Without the use of a calculator, complete this
assessment to gauge your understanding of the topics covered in this
unit. The correct answers will be displayed when you click the
“Submit” button.
Completing this exam should take approximately 1 hour.