**Unit 2: Linear Functions**
*Math students often ask, “How can I use this? What’s it for?” The
answer is “modeling.” Mathematics is designed to help model real-world
phenomena and trends. Modeling allows us to apply mathematics to
situations such as predicting population growth, balancing costs and
profits, developing inventions, or clarifying the shape of terrain where
a road must go. As you learn mathematics, it is important to acquire
skill in creating functions to describe our world. The power of
mathematics is in its application to real life.*

*As you delve into specific functions in this and additional units, you
will not only learn mathematical processes but also about using
functions to model situations. You begin with a familiar form - linear.
Linear functions are well named because their graphs are lines. This
unit begins by describing linear functions and their graphs. It applies
those concepts to modeling as you learn to fit a line to a set of data
points and measure how close to being linear those points actually are.
The unit closes by exploring absolute value functions and their
graphs. *

**Unit 2 Time Advisory**

Completing this unit should take approximately 12 hours and 40
minutes.

☐ Subunit 2.1: 1 hour and 55 minutes

☐ Subunit 2.1.1: 25 minutes

☐ Subunit 2.1.2: 1 hour and 30 minutes

☐ Subunit 2.2: 3 hours

☐ Subunit 2.2.1: 50 minutes

☐ Subunit 2.2.2: 45 minutes

☐ Subunit 2.2.3: 1 hour and 25 minutes

☐ Subunit 2.3: 1 hour and 55 minutes

☐ Subunit 2.4: 2 hours and 40 minutes

☐ Subunit 2.4.1: 35 minutes

☐ Subunit 2.4.2: 30 minutes

☐ Subunit 2.4.3: 1 hour and 35 minutes

☐ Subunit 2.5: 3 hours and 10 minutes

☐ Subunit 2.5.1: 25 minutes

☐ Subunit 2.5.2: 30 minutes

☐ Subunit 2.5.3: 2 hours and 15 minutes

**Unit2 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- Determine the slope of a linear function.

- Calculate the rate of change of a linear function.

- Create graphs of linear functions.

- Create equations to model given graphs of linear functions.

- Fit linear models to real-world data.

- Evaluate a correlation coefficient.

- Solve absolute value equations.

- Graph absolute value functions.

Standards Addressed (*Common Core*):
- CCSS.Math.Content.6-NS.C.7.c
- CCSS.Math.Content.8-SP.A.1
- CCSS.Math.Content.HSA-SSE.A.1.a
- CCSS.Math.Content.HSA-CED.A.1
- CCSS.Math.Content.HSA-CED.A.3
- CCSS.Math.Content.HSA-CED.A.4
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7

- CCSS
Math.Content.HSF-IF.C.8
- CCSS-Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.Math.Content.HSF-LE.A.2
- CCSS.Math.Content.HSS-ID.B.6
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.Math.Content.HSS-ID.C.8
- CCSS.ELA-Literacy.RST.11-12.2
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.6
- CCSS.ELA-Literacy.RST.11-12.9
- CCSS.ELA-Literacy.RST.11-12.10
- CCSS.ELA-Literacy.WHST.11-12.2.d
- CCSS.ELA-Literacy.WHST.11-12.4

**2.1 Linear Functions**
*Some of the concepts in this subunit will be familiar and part of the
material will be review. Focus carefully; there are seeds of ideas in
this unit that will grow into bigger ideas as they are applied to more
complex functions in the later parts of the course. This subunit defines
linear functions and investigates how to find the rate of change of a
function.*

**2.1.1 Linear Functions**
*Suppose the local fitness center charges $80 to join and $35 every
month that you are a member. The relationship between cost and the
number of months you are a member is linear and can be described as C(n)
= 80 + 35 n. This function would predict your total fitness expense
until the gym raises prices. The subunit begins by defining linear
functions and their characteristics and then investigates how to find
the rate of change.*

**Did I Get This? Activity: mathcentre: “Linear Functions: Diagnostics”**Link: mathcentre: “Linear Functions: Diagnostics” (Flash)

Instructions: Click on the link above for a short quiz that will check your knowledge and help you identify areas in which you need to focus as you address linear functions in this subunit.

Completing this activity should take approximately 5 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.4
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.C.7

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 99 - 101 of the text. Focus on the characteristics of linear functions. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem at the top of page 102. Answers are on page 106.

Completing this activity should take approximately 20 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: This resource is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.

**2.1.2 Rate of Change**
*One of the important descriptors of a function is the rate of change.
Linear functions have a constant rate of change. Although there is an
easy way to “see” the rate of change of the algebraic form of a linear
function, it is important to focus on how to determine the rate of
change algebraically. This process will help you understand rate of
change and will be important when you apply it to more complex functions
in subsequent units of the text. This subunit will define rate of change
and apply it to linear functions.*

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 102 - 106 of the text, focusing on rates of change. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 103 and 105. Also complete the Flashback problems on page 105. All answers are on page 106.

Completing this activity should take approximately 25 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: This resource is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.

**Explanation: mathispower4u: Average Rate of Change Application - Hot Air Balloon Function**Link: mathispower4u: “Average Rate of Change Application - Hot Air Balloon Function” (YouTube)

Instructions: Click on the link above and watch the YouTube video, which gives an example of how to determine rate of change.

Completing this activity should take approximately 5 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: This resource is licensed under Creative Commons Attribution 3.0 Unported License.

**Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Complete the odd problems 1 - 49, beginning on page 107 of the text. The answers begin on page 526.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.4
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: This resource is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.

**2.2 Graphs of Linear Functions**
*As you explore linear functions further, you will apply ideas from Unit
1 to these familiar graphs. In the previous subunit, you learned about
transformations of toolkit functions. This subunit applies those ideas
as it delves into graphs of lines and explores relationships in parallel
and perpendicular lines. It closes by focusing on the intersections of
lines.*

**2.2.1 Graphing Linear Functions**
*This subunit begins by reviewing the topics of slopes and intercepts.
It extends the exploration by applying graph transformations as we focus
on shifts and compressions. The subunit closes by identifying the
special types of equations that describe horizontal and vertical lines.*

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 111 - 117 of your text. Focus on graphs of lines and the characteristics of their equations. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 113 and 117. Answers are on page 121.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-SSE.A.1.a
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Activity: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions”**Link: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions” (HTML)

Instructions: Click on the link above to see transformations you can control. Below the graph, find where f(x) is identified and change the function to*f*(*x*) =*x*. Change the*g*function to*g*(*x*) =*a*x*+*b*. Focus on the left-hand graph, and ignore the one on the right. Try to think in terms of transformations while moving the sliding bars to alter values of*a*and*b*. It will be tempting to think in terms of slope and intercepts, but this activity is designed to help you connect transformation concepts with linear functions. Applying these concepts to linear functions will help you with more complex functions later in the course.

Completing this activity should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 License.

**2.2.2 Parallel and Perpendicular Lines**
*The relationship among parallel lines is very specific. This subunit
helps you see the connections among algebraic descriptions of functions
that create parallel lines. Similarly it looks at connections among
descriptions of functions creating perpendicular lines. *

**Activity: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions”**Link: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions” (HTML)

Instructions: Click on the link above to see more complex transformations. Below the graph, find where f(x) is identified and change the function to*f*(*x*) = a**x+b*. Change the*g*function to*g*(*x*) =*a*x*+*c*. Focus on the left-hand graph, and ignore the one on the right. Move the sliding bars to see the transformations as you alter values of*a*,*b, c,*and*x*. Try to determine which of the bars controls whether the functions are parallel.

After working with these two functions, rename the*g*function to be*g*(*x*) = -(1/*a*)x + c. Move the sliding bars to see the transformations as you alter values of*a*,*b*,*c*, and*x*. Try to determine which of the bars controls whether the functions are perpendicular.

Completing this activity should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 License.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 117 - 119 of the text, focusing on the relationships between parallel lines and between perpendicular lines. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 119. Answers are on page 121.

Completing this activity should take approximately 15 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.4
- CCSS.Math.Content.HSF-IF.B.4
- CCSS Math.Content.HSF-IF.C.8
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Activity: Getting the Slant on Parallel and Perpendicular Lines**Instructions: Address an explanation to a math student who is about to begin this course as you write two paragraphs in answer to the following: Why do parallel lines have the same slope? With perpendicular lines, why would one slope always be positive and the other always be negative? Have someone read your paragraphs and help you polish them to clarity.

Completing this activity should take approximately 20 minutes.

Standards Addressed (*Common Core*):

**2.2.3 Intersections of Lines**
*The connections between graphs and algebraic forms come together in
this subunit. A graph gives the big picture of the behavior of
functions, but it is the algebraic form that gives specifics. It is
impossible from a small graph to tell if a point is at 2 or 2.00146, but
the algebra can pin things down. This subunit uses graphs to illustrate
the relation between two distinct linear functions and the place at
which they have points in common.*

**Explanation: YouTube: Khan Academy’s “Solving Systems by Graphing”**Link: YouTube: Khan Academy’s “Solving Systems by Graphing” (YouTube)

Instructions: Watch the video as you focus on the graphical solution of a set of two linear functions.

Watching this video should take approximately 5 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)Instructions: Read pages 119 - 121 of the text, focusing on the connections between algebraic and graphic forms of functions and their intersections. As you complete the reading, use pencil and paper to work through the examples and do the “Try it Now” problem on page 120. Answers are on page 121.

Completing this activity should take approximately 20 minutes.

Standards Addressed (

*Common Core*):- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Web Media: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Complete the odd problems 1 - 51, beginning on page 122 of the text. The answers begin on page 527.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.4
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.C.7
- CCSS Math.Content.HSF-IF.C.8
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.Math.Content.HSF-LE.A.2
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**2.3 Modeling with Linear Functions**
*Modeling is where mathematics is applied. This subunit focuses on
problem solving and writing models to describe life situations. One of
the secrets of writing an equation lies in translation. Suppose you want
an algebraic description of the following declarative sentence: Doubling
the height is the same as adding five to the width. The key to an
equation is to find the words in the sentence that represent the equal
sign. Everything in front of the words meaning “equal” will be in front
of the equal sign; everything after the words meaning “equal” will be
after the equal sign. Seldom is it necessary to change the order when
translating. The translation is as follows:*

* Doubling **the**height **is the same as**adding five **to the**width**.*

*2**h **=**5 + **w*

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 126 - 133. As you complete the reading, use pencil and paper to work through the examples.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.1
- CCSS-Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Complete the odd problems 1 - 21, beginning on page 134 of the text. The answers begin on page 528.

Completing this activity should take approximately 45 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.A.1
- CCSS-Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Activity: Translation--English to Mathematics**Instructions: Write a letter to a future student, explaining how to write an equation to describe the following situation mathematically.*The number of elephants is five times the number of camels.*Find a friend to read and help you edit your explanation.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):

**2.4 Fitting Linear Models to Data**
*If you wrote down the outside temperature at noon every day from spring
to midsummer, it would tend to get warmer every day, but the data points
would not be on an exact straight line. The points might be
approximately linear, though. This subunit begins with creating a line
on a set of data points to approximate a relation and looks at when it’s
appropriate to predict other values from that line. It explores the use
of technology to create a line of best fit and then introduces the
correlation coefficient, which is a way to measure how well a line
describes the relation among a set of data points.*

**2.4.1 Approximately Linear - Trend Lines**
*Many relations are only approximately linear, but sometimes we can use
lines to estimate them. Suppose a dieter gets weighed every Friday. The
weights over time could be approximately linear, but not fall exactly on
a line. This subunit places lines on sets of data, identifies the
equation of each line, and then uses the equation to identify other
values. If, for instance our dieter missed getting weighed one Friday,
the equation of the trend line would make it possible to estimate that
Friday’s weight later on. *

**Activity: Illustrative Mathematics: “Birds’ Eggs”**Link: Illustrative Mathematics: “Birds’ Eggs” (HTML)

Instructions: Click on the link above for an introduction to trend lines. Answer the five questions (a - e), and check your answers in the commentary section.

Completing this activity should take approximately 10 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSS-ID.B.6
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: Content on this site is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 138 - 139. As you complete the reading, use pencil and paper to work through the examples and do the “Flashback” problems on page 139. Answers are on page 143.

Completing this activity should take approximately 15 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSS-ID.B.6
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Did I Get This? Activity: Quizlet: Scatter: Algebra Linear Functions**Link: Quizlet: “Scatter: Algebra Linear Functions” (HTML)

Instructions: Click on the link above to check your knowledge about linear functions and lines of best fit.

Completing this activity should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**2.4.2 Interpolation and Extrapolation**
*Imagine a girl who is 85 centimeters tall at age 2 and 148 centimeters
at age 9. We could develop a trend line of height as a function of age:
h(a) = 9a + 67. We could use that function to estimate the girl’s height
at age 4, which would be h(4) = 9(4) + 67 = 103 centimeters. That would
probably be a close approximation. This process is called
interpolation.*

*If, however, we used the function to predict her height at age 70, we
would have h(70) = 9(70) + 67 = 697 centimeters. That would mean that
this girl would grow to nearly 23 feet tall! Extrapolation is using a
trend outside the boundaries of the original input data collected, and
it often leads to wild predictions.*

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 139 - 140. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 140. Answers are on page 143.

Completing this activity should take approximately 20 minutes.

Standards Addressed (*Common Core*):**Reading: Lock Haven University: “Mark Twain - Except from**Link: Lock Haven University: “Mark Twain - Except from*Life on the Mississippi*”*Life on the Mississippi”*(HTML)

Instructions: Click on the link above and read the excerpt by Mark Twain, who offers a humorous discussion of the length of the Mississippi River and how extrapolation can bring us to mistaken assumptions.

Reading this material should take approximately 10 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.8-SP.A.1
- CCSS.ELA-Literacy.RST.11-12.2
- CCSS.ELA-Literacy.RST.11-12.6
- CCSS.ELA-Literacy.RST.11-12.10

Terms of Use: This resource is in the public domain.

**2.4.3 Technology and Linear Models**
*Trend lines like the ones you created in the previous subunit are rough
estimates. Statisticians have developed a complex process to find the
“line of best fit,” also called the “least squares regression line.”
Luckily, technology enables us to find the equation of that line while
avoiding the* *intricate mathematical process. This subunit will develop
these regression lines and also introduce the correlation coefficient,
which is a value that indicates how close to a line a set of data points
actually are.*

**Interactive Lab: Michael Buescher: GeoGebra: Correlation Experimentation**Link: Michael Buescher: GeoGebra: “Correlation Experimentation” (HTML)

Instructions: Click on the link above and download the correlation experimentation. Move the points and see how the correlation coefficient changes. After exploring awhile, note the difference in the correlation coefficient when you move far points closer and the difference when you move close points farther.

Completing this activity should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under Creative Commons Attribution-ShareAlike 3.0 Unported License.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 140 - 143. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 143. Answers are also on page 143.

Completing this activity should take approximately 25 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSS-ID.B.6
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.Math.Content.HSS-ID.C.8
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Complete the odd problems 1 - 13, beginning on page 144 of the text. The answers begin on page 529.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSS-ID.B.6
- CCSS.Math.Content.HSS-ID.C.7
- CCSS.Math.Content.HSS-ID.C.8
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.6
- CCSS.ELA-Literacy.RST.11-12.10

**2.5 Absolute Value Functions**
*Think of absolute value as distance - that’s why it’s never negative.
As you work through the material in this subunit, it is helpful to keep
the concept of distance in your mind. You will begin with absolute value
and its important features and progress to solving absolute value
equations and inequalities.*

**2.5.1 Defining Absolute Value Functions**
*A standard absolute value function can be described with a piecewise
function of two linear functions. This subunit describes the concept of
absolute value and identifies the important features of this type of
function.*

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 146 - 148. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 147 and 148. Answers are on page 152.

Completing this activity should take approximately 25 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**2.5.2 Solving Absolute Value Equations**
*If the absolute value of (x - a) is 7, that means that the distance
between x and a is seven units. So consider the absolute value of (x -
3): if it’s 7, then the distance between x and 3 is seven, meaning x is
seven units away from three - that means that x is at 10 or -4.
Sometimes a number line is a useful way to consider absolute value
equations. This subunit focuses on an algebraic way to solve absolute
value equations and teaches a step-by-step process to address these
sometimes confusing equations.*

**Explanation: Khan Academy’s “Absolute Value Equation Example 2”**Link: Khan Academy’s “Absolute Value Equation Example 2” (YouTube)

Instructions: Watch the video. Focus on the connection between distance and absolute value as you review how to solve this type of equation.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Read pages 148 - 149. Focus on how to solve absolute value equations. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page- Answers are on page 152.

Completing this activity should take approximately 20 minutes.

Standards Addressed (*Common Core*): - CCSS.Math.Content.6-NS.C.7.c
- CCSS.Math.Content.HSA-SSE.A.1.a
- CCSS.Math.Content.HSA-CED.A.3
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

- Answers are on page 152.

**2.5.3 Solving Absolute Value Inequalities**
*Inequalities look a lot like equations, but they do not use an equal
sign (=) as the verb. Inequalities use things like <, >, and <.
Keeping the distance idea in mind with absolute value inequalities is
very helpful. In the inequality |x – 3| < 2, the distance idea can be
useful. Think of this problem as saying the distance between x and 3 is
less than 2 units. That means that x can be 2 units away from three, but
no farther; thus, x has to be somewhere between 1 and 5, inclusive. If
the verb is >, and we have |x – 3| > 2, then we know that x is at
least 2 units away from 3, meaning it must be greater than 5 or less
than 1. This subunit focuses on algebraic methods to solving
inequalities that have absolute value statements within them. If you
remember that you can think of absolute value in terms of distance, it
may be much easier to solve the problems*

**Explanation: Net Texts: Algebra II: “Absolute Value Inequalities”**Link: Net Texts: Algebra II: “Absolute Value Inequalities” (HTML)Instructions: Click on the link above and then click on the link that says “Inequalities and Absolute Values.” Scroll down to the link entitled “Absolute Value Inequalities.” (Don’t get confused and select “Inequalities and Absolute Value.” Choose the correct title further down on the list. It has a picture of a video camera to the left.) Watch the “Absolute Value Inequality” video, and focus on connecting the idea of distance and absolute value in understanding these inequalities.

Watching this video should take approximately 5 minutes.

Standards Addressed (

*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Khan Academy.

**Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)Instructions: Read pages 149 - 152, focusing on absolute value inequalities and how to solve them. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 152. Answers are also on page 152.

Completing this activity should take approximately 30 minutes.

Standards Addressed (

*Common Core*):- CCSS.Math.Content.6-NS.C.7.c
- CCSS.Math.Content.HSA-SSE.A.1.a
- CCSS.Math.Content.HSA-CED.A.3
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions*** Link: Lippman and Rasmussen’s*Precalculus: An Investigation of Functions*(PDF)

Instructions: Complete the odd problems 1 - 25, beginning on page 153 of the text. The answers begin on page 529.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.6-NS.C.7.c
- CCSS.Math.Content.HSA-SSE.A.1.a
- CCSS.Math.Content.HSA-CED.A.3
- CCSS.Math.Content.HSA-REI.B.3
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**Activity: Clarifying the Difference: Inequalities vs Equations**Instructions: An equation always has an equal sign for the verb in the mathematical sentence. An inequality has a different verb; some examples are: <, >, <, >, and ≠. Solving an inequality is similar to solving an equation, but there are some distinct differences. Write a short essay explaining how they are alike and how they are different. Be sure to clarify why those differences occur.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):