**Unit 1: Functions**
*Functions create a basic structure in mathematics. Some mathematicians
describe them as well-functioning machines in which you can choose an
input and count on the machine always producing the same output for that
particular input. If a soda machine functions properly, you can input
money and press cola, and the outcome is a cola. It won’t be root beer -
you can count on it! A mathematical function is predictable in the same
way. For example, maybe a function gives an output of 3.47 when you
input the number -6. If it is truly a function, it will always give an
output of 3.47 when you input the number -6. This first unit focuses on
defining functions, identifying specific notation that we will be using,
and helping you develop an understanding of certain characteristics of
functions and their graphs. *

**Unit 1 Time Advisory**

Completing this unit should take approximately 19 hours and 45
minutes.

☐ Subunit 1.1: 6 hours and 10 minutes

☐ Subunit 1.1.1: 15 minutes

☐ Subunit 1.1.2: 1 hour and 25 minutes

☐ Subunit 1.1.3: 25 minutes

☐ Subunit 1.1.4: 40 minutes

☐ Subunit 1.1.5: 25 minutes

☐ Subunit 1.1.6: 3 hours and 5 minutes

☐ Subunit 1.2: 2 hours and 20 minutes

☐ Subunit 1.2.1: 40 minutes

☐ Subunit 1.2.2: 30 minutes

☐ Subunit 1.2.3: 1 hour and 30 minutes

☐ Subunit 1.3: 2 hours and 10 minutes

☐ Subunit 1.3.1: 35 minutes

☐ Subunit 1.3.2: 1 hour and 35 minutes

☐ Subunit 1.4: 2 hours and 30 minutes

☐ Subunit 1.4.1: 35 minutes

☐ Subunit 1.4.2: 1 hour and 55 minutes

☐ Subunit 1.5: 4 hours and 45 minutes

☐ Subunit 1.5.1: 1 hour

☐ Subunit 1.5.2: 35 minutes

☐ Subunit 1.5.3: 1 hour and 15 minutes

☐ Subunit 1.5.4: 1 hour and 55 minutes

☐ Subunit 1.6: 1 hour and 50 minutes

☐ Subunit 1.6.1: 1 hour and 15 minutes

**Unit1 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- Create examples of functions and nonfunctions.

- Describe attributes of relations that are not functions.

- Determine the attributes of domain and range of a given function.

- Determine whether a function is one-to-one.

- Demonstrate facility with function notation.

- Demonstrate facility with functions in words, tables, graphs, and
formulas.

- Execute the horizontal and vertical line tests.

- Evaluate a function at a specific input value.

- Solve a function given a specific output value.

- Evaluate piecewise functions.

- Calculate rates of change.

- Describe basic behaviors of graphs of functions.

- Compose functions and determine the resultant domain and range of
the new function.

- Transform functions.

Standards Addressed (*Common Core*):

- CCSS.Math.Content.HSA-APR.B.3
- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSA-REI.D.10
- 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.5
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.Math.Content.HSF-BF.B.4
- 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.11-12.5
- CCSS.ELA-Literacy.RST.11-12.7
- CCSS.ELA-Literacy.RST.11-12.10

**1.1 Functions and Function Notation**
*At a picnic, the number of sandwiches depends upon the number of
picnickers. In a theater, the number of dollars collected depends upon
the number of tickets sold. The cost of a taxi depends upon the number
of miles driven. These are examples of functions, which are relations
that create a unique output for any particular input. This subunit
focuses on the definition of function both formally and informally.*

*The notation may be intimidating at first, but it makes later topics
easier. Remember that you already think in notation. The problem,
“Fifty-four minus thirty-nine,” is easier to solve when written in the
notation: 54 - 39. Mathematics uses notation to simplify things and make
it easier to think; it just takes time for your brain to learn new
notational language.*

*Subunit 1.1 begins by defining functions and explaining a variety of
ways to describe them, such as notation, graphs, and charts. It
continues by introducing a set of important toolkit functions that will
be extremely useful in subsequent parts of the course.*

**1.1.1 What Is a Function?**
*This subunit launches the course by introducing an important basic
concept of mathematics - the function. Functions are dependable - if you
know the output for a particular input, you will always get that same
output for that particular input. It’s an easy idea but it can be
difficult to explain.*

*Suppose you have a company building a particular toy car that has
three wheels. You could write a function for wheels based on how many
cars you produced. If you make one car, you need three wheels. If you
make two cars, you need six wheels. If you make 300 cars, you need 900
wheels. The output depends upon the input and any one input will ALWAYS
yield the same output. *

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

Instructions: Click on the link above to access the text. In order to download the text, scroll down to the chart and click on “PDF,” which is located across from “Whole Book.” Using the text later in the course will be easier if you save it after downloading. Read pages 1 - 2 of this text, focusing on “What is a Function.” As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on page 2. Answers are on page 12.

Completing this activity should take approximately 15 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.5
- 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.

**1.1.2 Function Notation**
*At first, function notation often seems a pointless complication, but
the notation is useful and efficient for later mathematical activity. In
earlier courses, you focused on an input of x and an output of y.
Function notation uses the same input (x) but calls the output f(x)
instead of y. This allows us to identify the name of the function f and
also to identify the actual input used. For example: if f(x) = x + 2,
then f(3) = 3 + 2 = 5, and f(-1) = 1. In time, it gets easier to read
f(4) = 6 instead of the wordy “When x = 4, then y = 6.”*

**Explanation: YouTube: Tyler Wallace’s “Functions: Function Notation”**Link: YouTube: Tyler Wallace’s “Functions: Function Notation” (YouTube)

Instructions: Click on the link above and watch the video, which explains function notation.

Watching this video should take approximately 5 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.5
- CCSS.ELA-Literacy.RST.11-12.4

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

**Activity: Khan Academy’s “Understanding Function Notation”**Link: Khan Academy’s “Understanding Function Notation” (YouTube)

Instructions: Click on the link above to practice function notation. On the graph presented, you will be asked to find the value of a function. Remember that when you find*f*(5), that means you need to find the y-value for when x = 5. Place your answer to each problem in the block to the right, and click on the link to check you answer. Work until you can get at least five correct answers in a row.

Completing this activity should take approximately 20 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.5
- CCSS.ELA-Literacy.RST.11-12.4

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 2 - 4 of the text, focusing on function notation andtables as functions. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem 3 on page 5. Answers are on page 12.

Completing this activity should take approximately 15 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.5
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4

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

**Activity: Changing Names: y = f(x).**Instructions: Instead of using “y” to name some relations involving “x”, we are now using f(x) to show a relation is a function. Imaging explaining function notation to an algebra student as you write a paragraph identifying the difference between the following two forms ofnotation: y = 3x + 5 and f(x) = 3x + 5. Continue the activity with a second paragraph clarifying the difficulties in learning function notation. When you think you are finished, have someone read it to help you edit your explanations to make sense.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):

**1.1.3 Solving and Evaluating Functions**
*This subunit gives you a chance to practice finding outputs for given
functions. It is material you have covered before but probably in a
little different format. It gives you more opportunity to become
comfortable with function notation.*

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

Instructions: Read page 5 of the text, focusing on “Solving and Evaluating Functions.” As you complete the reading, use pencil and paper to work through the examples and do the four problems on page- Answers are on page 12.

Completing this activity should take approximately 15 minutes.

Standards Addressed (*Common Core*): - CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4

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

- Answers are on page 12.
**Explanation: Khan Academy’s “Evaluating a Function”**Link: Khan Academy’s “Evaluating a Function” (YouTube)

Instructions: Click on the link above and watch the video, which demonstrates how to evaluate a function.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4

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

**1.1.4 Graphs as Functions**
*Functions can be described not only algebraically but also graphically.
This subsection connects these two types of function descriptions. When
working with functions, it is often helpful to use both formats, because
each gives a different perspective of the relation you are exploring.*

**Activity: Khan Academy’s “Evaluating Expressions with Function Notation”**Link: Khan Academy’s “Evaluating Expressions with Function Notation” (YouTube)

Instructions: Click on the link above and practice using function notation. Continue the task until you have the concept and get at least five correct answers in a row.

Completing this activity should take approximately 15 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: YouTube: Rhodes0502: “Day 9: Vertical Line Test”**Link: YouTube: Rhodes0502: “Day 9: Vertical Line Test” (YouTube)

Instructions: Watch this video for an explanation of the vertical line test for functionality.

Watching this video should take approximately 5 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.5
- CCSS.ELA-Literacy.RST.11-12.4

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

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

Instructions: Read pages 5 - 7 of the text. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 7. Answers are on page 12.

Completing this activity should take approximately 20 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4

**1.1.5 Formulas as Functions**
*Think of it: If you know the diameter of a circle, you can find the
circumference. It is a function because one particular diameter will
ALWAYS yield one particular circumference - it won’t change. This
subunit looks at relations and how to rewrite them to clarify the
functional relation. For example: if x + y = 5, you can rearrange the
algebra to write y = 5 - x. This would identify a function we might call
g, and the algebraic description would be g(x) = 5 - x. *

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

Instructions: Using the text later in the course will be easier if you save it after downloading. Read pages 7 - 10 of the text, focusing on formulas as functions. As you complete the reading, use pencil and paper to work through theexamples and do the “Try It Now” problem on page 10. Answers are on page 12.

Completing this activity should take approximately 25 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4

**1.1.6 Basic Toolkit Functions**
*Knowing the basic characteristics of the “toolkit functions” will be
useful throughout the rest of your mathematical experiences. This
subunit identifies a number of functions and helps give an overview of
their behavior. Pay attention, because it will make many things much
easier later on.*

**Explanation: YouTube: Integralmath: “Precalculus 1: Intro; Toolkit Functions”**Link: YouTube: Integralmath: “Precalculus 1: Intro; Toolkit Functions” (YouTube)

Instructions: Click on the link above and take notes as you watch the video. The presentation will review the basics of functions and then focus on important basic functions and the shapes of their graphs. This information will be essential throughout your mathematical experiences in this course and beyond, because the toolkit functions are building blocks that will be used to make more complex functions. If you know the shape of each toolkit graph, you can use it to build a graph of a more complex function by shifting and/or stretching rather than tediously plotting a lot more points.

Watching the video and taking notes should take approximately 55 minutes.Standards Addressed (

*Common Core*):- CCSS.Math.Content.HSA-APR.B.3
- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4

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

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

Instructions: Read pages 10 - 12 of your text. As you complete the reading, use pencil and paper to work through the examples. Pay close attention to the shapes of these important functions, as this will make graphing more complex functions much easier.

Completing this activity should take approximately 30 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.5
- CCSS.Math.Content.HSF-IF.C.7

**Activity: Comparing Toolkit Functions**Instructions: Make a chart of the toolkit functions to help you remember them. At the bottom of the chart, write a few paragraphs (or a bulleted list) comparing and contrasting the different types of functions and their graphs. Keep this chart and the writing in a handy place so that you can refer to it in later units of the course. Alter the chart and writing as your perceptions and understandings change over time.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):**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 - 47, beginning on page 13 of the text. The answers begin on page 519.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-APR.B.3
- CCSS.Math.Content.HSA-CED.4
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.5
- CCSS.ELA-Literacy.RST.11-12.10

**1.2 Domain and Range**
*A function takes an input, acts upon it, and produces an output. The
set of inputs is called the domain, and the set of outputs is called the
range.* *Notice that the video in the previous subunit offered a
beginning look at these important characteristics of functions. Here you
will reinforce those ideas, develop notation to identify the domain and
range, and address how to determine domains and ranges from graphs.The
final part of this subunit introduces piecewise functions, which are
functions defined in pieces. *

**1.2.1 Introduction and Notation**
*This subunit defines the terms domain and range and also develops
notation for identifying sets of numbers. Remember that although
notation can be confusing at first, in time it makes mathematical
communication and thinking easier because it presents information in an
efficient manner. *

**Web Media: YouTube: Grand Valley State University, Math Department: “Set-Builder Notation”**Link: YouTube: Grand Valley State University, Math Department: “Set-Builder Notation” (YouTube)

Instructions: Click on the link above and watch the video to learn to read set-builder notation.

Watching this video should take approximately 10 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.4
- CCSS.Math.Content.HSF-IF.B.5

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

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

Instructions: Read pages 21 - 24. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on page 22 and page 24. Answers are on page 30.

Completing this activity should take approximately 30 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.4
- CCSS.Math.Content.HSF-IF.B.5

**1.2.2 Domains and Ranges from Graphs**
*It’s difficult to determine the domain and range of a function by
looking at an algebraic representation of a function. It’s often easier
to look at the graph. This subunit will help you build skills in using
graphs to characterize behavior of functions. While working through the
material, try to connect the algebraic and geometric representations in
your mind. It will make understanding functions easier if you try to see
functions in this broad perspective.*

**Explanation: YouTube: patrickJMT: “Finding Domain and Range of a Function Using a Graph”**Link: YouTube: patrickJMT: “Finding Domain and Range of a Function Using a Graph” (YouTube)

Instructions: Click on the link above and watch the video. Focus carefully on how the speaker looks at graphs in order to find domain and range of the functions he demonstrates.

Watching this video should take approximately 10 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.4
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.7

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 24 - 27. As you complete the reading, use pencil and paper towork through the examples and do the “Try It Now” problem on page 26. Answers are on page 30.

Completing this activity should take approximately 20 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.4
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.7

**1.2.3 Piecewise Functions**
*Suppose bananas are $2 a pound, but if you purchase 10 pounds or more,
the price is reduced to $1.50 a pound. The cost of bananas is a function
of the pounds purchased. Notice, however, that there is a significant
change in the function when you reach 10 pounds. This is a piecewise
function, because it has to be described in pieces.*

**Explanation: Free Video Lectures: Professor Derek Owens’s “Real World Piecewise Functions”**Link: Free Video Lectures: Professor Derek Owens’s “Real World Piecewise Functions” (YouTube)

Instructions: Click on the link above and watch the video. Focus carefully on the format of a piecewise function and how it appears in real-world situations.

Watching this video should take approximately 10 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.4
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4

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 28 - 30. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 30. Answers are on page 30.

Completing this activity should take approximately 20 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.4
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.4

**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 - 35 on pages 31 - 33 of the text. The answers begin on page 520.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- 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.5
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.7

**1.3 Rates of Change and Behavior of Graphs**
*How quickly a function changes shows a lot about a function, and the
concept is a basic notion in the development of calculus. This subunit
begins with average rate of change and then focuses on graphs,
identifying characteristics like intervals on increase and decrease,
maximums and minimums, concavity, and the trend of a function as x gets
very large.*

**1.3.1 Rates of Change**
*Rate of change looks at how fast function output values change. If you
travel 60 miles and it takes two hours, then your average rate of change
is 30 miles per hour. This section develops skills for determining rate
of change algebraically. This concept is a major concept used in
calculus.*

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

Instructions: Click on the link above and watch the video. Notice that the formula for average rate of change is the same as a slope formula but in different notation. The fraction’s numerator is the difference in two output (or*y*) values, and the denominator is the difference in two input (or*x*) values.

Watching this video should take approximately 5 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.4
- CCSS.Math.Content.HSF-IF.B.6

Terms of Use: This resource is licensed under Creative Commons Attribution 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 34 - 37. As you complete the reading, use pencil and paper towork through the examples and do the “Try It Now” problems on pages 35 - 37. Answers are on page 44.

Completing this activity should take approximately 30 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.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7

**1.3.2 Behaviors of Graphs**
*A graph of a function shows the overall behavior, while the algebraic
form indicates the specifics. In this subunit, you will identify
specific behaviors of functions by investigating graphs.*

*In discussing graphs, students sometimes are confused when to identify
x-values and when it is appropriate to use y-values in an answer. The
x-values identify location. If a problem asks “where,” your answers will
be x-values. If a problem asks the value of a function or size, then the
answer will be y-values. The x coordinates indicate location, and the
y-coordinates indicates size or value of the function. This subsection
focuses on characteristics of graphs and discusses the characteristic
behavior of the toolkit functions.*

**Activity: mathcentre: “Introduction to Functions: Diagnostics”**Link: mathcentre: “Introduction to Functions: Diagnostics” (Flash)

Instructions: Click on the link above and check your skill identifying functions and their characteristics. Note that when you submit a problem, a check indicates that you are correct and an x indicates that you are incorrect. Go back and alter you answers until each one has a check mark.

Completing this activity should take approximately5 minutes.

Standards Addressed (*Common Core*):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 43 - 45. As you complete the reading, use pencil andpaper to work through the examples and do the “Try It Now” problems on pages 40, 42, and 43. Answers are on page 30.

Completing this activity should take approximately 30 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.4
- CCSS.Math.Content.HSF-IF.B.5
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7

**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 - 41 on pages 46 - 48 of the text. The answers begin on page 521.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- 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.5
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.5
- CCSS.ELA-Literacy.RST.11-12.10

**1.4 Composition of Functions**
*New functions are often built by putting functions together, and this
subunit builds the ideas for creating those new functions. Imagine a
company creating a paycheck for an employee who makes $12 per hour. A
wage function can be defined as w(h) = 12h, where h is the number of
hours that employee worked. Suppose further that the company has to take
20% of that amount for taxes. That would mean taking the wage and
reducing it by 20%; this could be a tax function, t(w) = 0.20 w. A
composition of functions makes this process easier by creating a new
function by taking the output of the first function and using it as the
input of the second function and then simplifying. What we want is 0.20w
= 0.20(12h) = 2.4h, which shows the company taking out $2.40 for every
hour the employee worked. This subunit will help you build skills to
develop composed functions and restrict domains of those functions when
needed.*

**1.4.1 Composition of Functions**
*The key in composition is to remember that you take the output of one
function and use it as the input of another. This subunit introduces the
topic and gives a significant number of examples of how to apply this
important concept.*

**Explanation: YouTube: wallaceopenmath: “Composition of Functions at a Point”**Link: YouTube: wallaceopenmath: “Composition of Functions at a Point” (YouTube)

Instructions: Click on the link above and take notes as you watch the video. Focus carefully on how the presenter puts one function into another and simplifies.

Watching the video and taking notes should take approximately 5 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.4
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.4

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

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

Instructions: Read pages 49 - 51. As you completethe reading, use pencil and paper to work through the examples and do the “Try It Now” problem on pages 50 - 51. Answers are on page 43.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.4
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**1.4.2 Composition Using Formulas**
*As you learned in Subunit 1.4.1, composition entails taking the output
of one function and placing it into another. This subunit simplifies the
process by completing the process algebraically. It’s a little confusing
at first, but keep with it. It helps to write down all the pieces rather
than making your head juggle too much information. *

**Explanation: YouTube: Melanie Kennelly’s “Composition of Functions (Math)”**Link: YouTube: Melanie Kennelly’s “Composition of Functions (Math)” (YouTube)

Instructions: Click on the link above and take notes as you watch the video. Focus carefully on how the presenter puts one function into another and simplifies.

Watching the video and taking notes should take approximately 10 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.4

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 51 - 55. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on pages 52 - 54. Answers are on page 57.

Completing this activity should take approximately 45 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.4
- 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- 43 on pages 57 - 60 of the text. The answers begin on page 522.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-BF.A.1
- CCSS.Math.Content.HSF-BF.B.4
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.5
- CCSS.ELA-Literacy.RST.11-12.10

**1.5 Transformations of Functions**
*You will be using your toolkit functions to develop graphs of other
functions. You know that the graph of f(x) = x ^{2} is a parabola
with the vertex at the origin. That function is similar to g(x) =
x^{2} + 5. Each of the output values is 5 larger than the output
values of the first function. By simply shifting the original graph up 5
units vertically, you will have the graph of the second function. This
subunit helps you build an internal vision of how to see toolkit
functions and use that knowledge to build graphs of more complex
functions. *

**1.5.1 Shifts**
*If you know the graph of x ^{3}, then the graph of
(x-9)^{3} can be determined easily using a horizontal shift. The
first transformation techniques you will explore are vertical and
horizontal shifts in a graph. *

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

Instructions: Read pages 61 - 67. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 63 and 66. Answers are on page 81.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.10

**Activity: Connexions: Debbie Trahan’s “Transformation of Functions (Graphically)”**Link: Connexions: Debbie Trahan’s “Transformation of Functions (Graphically)” (HTML)

Instructions: Click on the link above and practice as you complete the exercises. Be sure to check your work with the given solutions.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3

Terms of Use: This resource is licensed under a Creative Commons Attribution License. It is attributed to Debbie Trahan.

**1.5.2 Reflections**
*Remember putting paint on a paper, folding it in half, and seeing the
wonderful symmetric shapes you could make? This subunit looks at how
reflections and symmetry of graphs can be shown in the algebraic form of
a function. It also gives terms for particular types of symmetric
functions. The odd functions are reflective about the y-axis [just like
the graph of f(x) = x ^{2]}, and the even functions are a
rotation about the origin [just like the graph of g(x) =
x^{3]}.*

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

Instructions: Read pages 67 - 71. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 70. Answers are on page 81.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.10

**Explanation: YouTube: TheIntegralCALC: “Determining if a Function Is Even, Odd or Neither”**Link: YouTube: TheIntegralCALC: “Determining if a Function Is Even, Odd or Neither” (YouTube)

Instructions: Click on the link above to watch the video on how to determine algebraically whether a function is even or odd. The introduction to reflections was in the “Transformation of Functions” video in Subunit 1.5.1.

Watching this video should take approximately 5 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3

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

**1.5.3 Stretches and Compressions**
*Think of having a graph of a function on a sheet of rubber. If you pull
on opposite sides, you could stretch the function either horizontally or
vertically. This subunit explores how an algebraic chance in a function
can create a stretch or a compression of a graph.*

**Activity: Connexions: Debbie Trahan’s “Transformation of Functions (Verbally)”**Link: Connexions: Debbie Trahan’s “Transformation of Functions (Verbally)” (HTML)

Instructions: Click on the link above and work through the graphs and functions that are explained in a concrete way. Note that the answers are shown.

Completing this activity should take approximately 15 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3

Terms of Use: This resource is licensed under a Creative Commons Attribution License. It is attributed to Debbie Trahan.

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

Instructions: Read pages 71 - 76. As you complete the reading, use pencil and paperto work through the examples and do the “Try It Now” problem on page 76. Answers are on page 81.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.10

**1.5.4 Combining Transformations**
*Finally, you will put it all together. This subunit highlights the
power of the toolkit functions and how understanding transformations can
give you insight to the behavior of more complex functions. This portion
of the course offers a way to visualize graphs of many functions that
you will encounter throughout all further mathematics courses.*

**Web Media: YouTube: MrNichollTV’s “Transformations of Graphs”**Link: YouTube: MrNichollTV’s “Transformations of Graphs” (YouTube)

Instructions: Click on the link above and take notes as you watch the video, which explains how changes in a function influence changes in its graph.

Watching the video and taking notes 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.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.3

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 76 - 81. As you complete the reading, use pencil and paper to work through the examples and do the two “Try It Now” problems on page 80. Answers are on page 81.

Completing this activity should take approximately 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.3

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

Instructions: Complete every other odd problem, 1 - 99, on pages 82 - 89 of the text. If you are stumbling after that, do the odd problems that you skipped. The answers begin on page 522.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.10

**1.6 Inverse Functions**
*You get dressed putting on socks and then your shoes. You invert the
process later by taking off the shoes first and then the socks. In this
case, you start with bare feet and complete the first task. You come
back later and do the task in reverse and end up where you started -
with bare feet! Inverse functions are similar. If two functions are
inverses, then if you take the output of one and put it in the other,
you end up where you started. For example: The two functions f(x) = x +
10 and g(x) = x -10 are inverses because one will “undo” the other and
give back your original input (x). Inverse functions are a major focus
in precalculus, and this subunit crafts a vision of how to develop
inverses and what the graphs of inverses look like.*

**Web Media: Khan Academy’s “Introduction to Function Inverses”**Link: Khan Academy’s “Introduction to Function Inverses” (YouTube)

Instructions: Click on the link above and watch the video. Focus on what a function inverse is, how a function and its inverse are related, and how the graphs of the two functions are associated.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.4
- CCSS.ELA-Literacy.RST.11-12.4

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

Instructions: Read pages 90 - 95. As you complete the reading, use pencil and paper to work through the examples and do the two “Try It Now” problems on pages 91 - 93. Answers are on page 95.

Completing this activity should take approximately 40 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.4
- 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 - 23 on pages 96 - 97 of the text. The answers begin on page 526.

Completing this activity should take approximately 1 hour.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.A.2
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSF-BF.B.3
- CCSS.Math.Content.HSF-BF.B.4
- CCSS.ELA-Literacy.RST.11-12.3
- CCSS.ELA-Literacy.RST.11-12.4
- CCSS.ELA-Literacy.RST.11-12.5
- CCSS.ELA-Literacy.RST.11-12.10