Unit 1: Functions
Computing postage for a package, the gasoline consumption when driving
at 65 mph, or how much money will be in a savings account after five
years are all examples of where functions are utilized. Functions are
used to model everyday relationships between two quantities and this
unit will be your introduction to this concept using words, tables,
equations, and graphs.
Unit 1 will begin with defining the concept of function, domain,
range, function notation, and how to evaluate and graph a basic group of
functions called “toolkit functions.” You also will explore the
behaviors of functions including rate of change, local extrema, and
concavity. The remainder of the unit builds from combining functions
using composition and transformations and ends with finding a function’s
inverse. All of these skills will be used throughout the remaining
units.
Unit 1 Time Advisory
Completing this unit should take you approximately 28.25 hours.
☐ Subunit 1.1: 4.75 hours
☐ Subunit 1.2: 4 hours
☐ Subunit 1.3: 4.75 hours
☐ Subunit 1.4: 3.25 hours
☐ Subunit 1.5: 6 hours
☐ Subunit 1.6: 3 hours
☐ Review and Exam: 2.5 hours
Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- create functions from tables, graphs, formulas, and words;
- evaluate a function and an average rate of change;
- graph the toolkit functions;
- evaluate the domain and range of toolkit functions in interval and
set builder notation;
- identify where a function is increasing, decreasing, concave up or
down, and has local extrema and inflections points;
- compose two functions from equations, graphs, or tables;
- perform transformations (shifts, compressions/stretches, and
reflections) of toolkit functions;
- identify the symmetry of a function;
- compute and interpret inverse functions from a chart, graph, or
function; and
- utilize a graphing calculator to graph a function in an appropriate
window.
1.1 Functions and Function Notation
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions”
(PDF)
Instructions: Read pages 1–12 of Chapter 1 to learn about
functions, one-to-one functions, and function notation. Note that
this reading covers the material you need to know for subunits 1.1.1
through 1.1.5.
Reading this material section should take approximately 45
minutes.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-ShareAlike 3.0 United States
License](http://creativecommons.org/licenses/by-sa/3.0/us/). It is
attributed to David Lippman and Melonie Rasmussen.
1.1.1 Defining Functions Note: This subunit is covered in the reading assigned beneath subunit 1.1. To learn about functions, please read the information on pages 1 and 2.
- Web Media: YouTube: Khan Academy’s “Introduction to Functions”
Link: YouTube: Khan Academy’s “Introduction to
Functions”
(YouTube)
Instructions: Watch this video on function concepts and function notation. 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.
1.1.2 Function Notation and Evaluating Functions Note: This subunit is covered in the reading assigned beneath subunit 1.1. To learn about function notation and evaluation of functions, read the information on pages 3 through 5.
- Web Media: YouTube: Mathispower4u: “Evaluate a Function and Solve
for a Function Value Given a Table”
Link: YouTube: Mathispower4u: “Evaluate a Function and Solve for a
Function Value Given a
Table” (YouTube)
Instructions: Watch this video to learn how to evaluate a function from a table of values. 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.
1.1.3 Graphs as Functions: Vertical and Horizontal Line Tests Note: This subunit is covered in the reading assigned beneath subunit 1.1. To learn about how to use a graph to determine if a graph is a function and whether it is one-to-one, read the information on pages 5 through 7.
Web Media: YouTube: Mathispower4u: “Use the Vertical Line Test to Determine if a Graph Represents a Function” Link: YouTube: Mathispower4u: “Use the Vertical Line Test to Determine if a Graph Represents a Function” (YouTube)
Instructions: Watch this video to learn when a graph represents a 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.Web Media: YouTube: Dr. Drew Butcher’s “Horizontal Line Test to Determine if a Function is One-To-One” Link: YouTube: Dr. Drew Butcher’s “Horizontal Line Test to Determine if a Function is One-To-One” (YouTube)
Instructions: Watch this video to learn when a graph represents a one-to-one 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.Web Media: YouTube: Mathispower4u: “Evaluate a Function and Solve for a Function Value Given a Graph” Link: YouTube: Mathispower4u: “Evaluate a Function and Solve for a Function Value Given a Graph” (YouTube)
Instructions: Watch this video to learn how to use a graph to evaluate a 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.
1.1.4 Formulas as Functions Note: This subunit is covered in the reading assigned beneath subunit 1.1. To learn how to use formulas and variables to express functions, read the information on pages 7 through 10.
Web Media: Derek Owens’s “Algebra 2 2.01d – Function Notation” Link: Derek Owens’s “Algebra 2 2.01d – Function Notation” (YouTube)
Instructions: Watch this video to learn about the proper notation for function formulas and how to evaluate functions. 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: mathlablady1’s “Determining if Equations are Functions Part 1 3.1” Link: YouTube: mathlablady1’s “Determining if Equations are Functions Part 1 3.1” (YouTube)
Instructions: Watch this video to learn how determine if an equation is a function. 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: MathWOE’s “Is an Equation a Function?” Link: YouTube: MathWOE’s “Is an Equation a Function?” (YouTube)
Instructions: Watch this video to learn how to solve an equation for a particular variable to determine if it is a function. Take detailed notes and watch the video as many times as you need.
Watching this should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
1.1.5 Toolkit Functions Note: This subunit is covered in the reading assigned beneath subunit 1.1. To learn about the names, formulas, and graphs of the Toolkit Functions, read the information on pages 10 through 12.
Web Media: YouTube: waszel’s “Six Basic Functions” Link: YouTube: waszel’s “Six Basic Functions” (YouTube)
Instructions: Watch this video to see several of the Toolkit functions graphed. 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.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.1: Function and Function Notations – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.1: Function and Function Notations – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.1, complete this assessment. You will need to create a free account in order to access the material. When creating your account, enroll in course 141 Precalculus 1 Self-Study. Go to that course page, click Chapter 1, and then Graded Homework Sections 1.1-1.6. Click Section 1.1, and work your way through the questions. 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 1a: Section 1.1”
Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 1a: Section 1.1” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.1, complete this quiz. The correct answers will be displayed when you click the “Submit” button.
Completing this assessment should take approximately 30 minutes.
1.2 Domain and Range
- Reading: Reading: Lippman and Rasmussen’s Precalculus: An
Investigation of Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions”
(PDF)
Instructions: Read pages 21–30 of Chapter 1 to learn about the
domain and range of a function and how to graph a piecewise defined
function. Note that this reading covers the material you need to
know for subunits 1.2.1 through 1.2.5.
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.
1.2.1 Defining Domain and Range Note: This subunit is covered in the reading assigned beneath subunit 1.2. To learn about the domain and range of functions, read the information on pages 21 and 22.
- Web Media: YouTube: Khan Academy’s “Domain and Range of a
Function”
Link: YouTube: Khan Academy’s “Domain and Range of a
Function” (YouTube)
Instructions: Watch this video to learn how to define the domain and range of a function. 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.
1.2.2 Domain and Range Notation Note: This subunit is covered in the reading assigned beneath subunit 1.2. To learn about how to use the proper notation to write the domain and range of functions, read the information on pages 22 through 24.
- Web Media: YouTube: Mathispower4u: “Determining Domain and
Range”
Link: YouTube: Mathispower4u: “Determining Domain and
Range” (YouTube)
Instructions: Watch this video to learn the different notations to write the domain and range of a 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.
1.2.3 Domain and Range from Graphs Note: This subunit is covered in the reading assigned beneath subunit 1.2. To learn about how to determine the domain and range of functions from their graphs, read the information on pages 24 through 26.
- Web Media: YouTube: Mathispower4u: “Determine the Domain and Range
of the Graph of a Function”
Link: YouTube: Mathispower4u: “Determine the Domain and Range of
the Graph of a
Function” (YouTube)
Instructions: Watch this video to learn how to determine and write the domain and range of a function in inequality and interval notation from a graph. 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.
1.2.4 Domain and Range of Toolkit Functions Note: This subunit is covered in the reading assigned beneath subunit 1.2. To learn about the domain and range of the toolkit functions, read the information on pages 26 and 27.
Web Media: YouTube: Mathispower4u: “Determine the Domain of a Square Root Function” Link: YouTube: Mathispower4u: “Determine the Domain of a Square Root Function” (YouTube)
Instructions: Watch this video to learn how to compute and write the domain and range of square root functions from a graph and equation. 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: “The Domain of Rational Functions” Link: YouTube: Mathispower4u: “The Domain of Rational Functions” (YouTube)
Instructions: Watch this video to learn how to determine and write the domain of a rational function from an equation. 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.
1.2.5 Piecewise Functions Note: This subunit is covered in the reading assigned beneath subunit 1.2. To learn about how to write, evaluate, and graph piecewise functions, read the information on pages 27 through 30.
Web Media: YouTube: Mathispower4u: “Interpreting and Graphing Piecewise Functions” Link: YouTube: Mathispower4u: “Interpreting and Graphing Piecewise Functions” (YouTube)
Instructions: Watch this video to learn an application of a piecewise function, how to graph the functions by hand, and how to graph on a graphing calculator. 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: “Graph a Piecewise Defined Function” Link: YouTube: Mathispower4u: “Graph a Piecewise Defined Function” (YouTube)
Instructions: Watch this video for an additional example of graphing a piecewise defined 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.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.2: Domain and Range – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.2: Domain and Range – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.2, 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 2 Toolkit: Section 1.2” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 2 Toolkit: Section 1.2” (PDF)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.2, complete this quiz.
Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 2 Toolkit: Section 1.2 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.
1.3 Rates of Change and Behavior of Graphs
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions” (PDF)
Instructions: Read pages 34–44 of Chapter 1 to learn about the rate
of change of a function, when a function is increasing and
decreasing, local extrema of a function, and concavity. Note that
this reading covers the material you need to know for subunits 1.3.1
through 1.3.5.
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.
1.3.1 Defining Rate of Change Note: This subunit is covered in the reading assigned beneath subunit 1.3. To learn about the rate of change of a function, read the information on page 34.
- Web Media: YouTube: Dr. Phil Clark’s “Interpreting Average Rate of
Change”
Link: YouTube: Dr. Phil Clark’s “Interpreting Average Rate of
Change” (YouTube)
Instructions: Watch this video to see an application of an average rate of change. 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.
1.3.2 Average Rate of Change Note: This subunit is covered in the reading assigned beneath subunit 1.3. To learn about the average rate of change of a function, read the information on pages 34 through 37.
- Web Media: YouTube: Mathispower4u: “Average Rate of Change”
Link: YouTube: Mathispower4u: “Average Rate of
Change” (YouTube)
Instructions: Watch this video for a detailed explanation of the average rate of change with real-life and functional examples. 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.
1.3.3 Increasing, Decreasing, and Local Extrema Note: This subunit is covered in the reading assigned beneath subunit 1.3. To learn about how to determine when a function is increasing and decreasing, and the local extrema of a function, read the information on pages 38 and 39.
Web Media: YouTube: waszel’s “Increasing, Decreasing, Constant” Link: YouTube: waszel’s “Increasing, Decreasing, Constant” (YouTube)
Instructions: Watch this video to learn how to determine from a graph the intervals where it is increasing, decreasing, or constant. 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: “Determine Where a Function is Increasing and Decreasing” Link: YouTube: Mathispower4u: “Determine Where a Function is Increasing and Decreasing” (YouTube)
Instructions: Watch this video for a more detailed explanation of increasing and decreasing as well as how to find relative extrema with and without a 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.
1.3.4 Concavity Note: This subunit is covered in the reading assigned beneath subunit 1.3. To learn how to find the intervals where a function is concave up or concave down, read the information on pages 40 through 43.
Web Media: YouTube: M4thG33x’s “Concavity of Graphs” Link: YouTube: M4thG33x’s “Concavity of Graphs” (YouTube)
Instructions: Watch this video to learn about concavity and inflection points. 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: Dr. Phil Clark’s “Characteristics of a Function” Link: YouTube: Dr. Phil Clark’s “Characteristics of a Function” (YouTube)
Instructions: Watch this video to learn how to determine the characteristics of a function such as zeros, intervals of increase and decrease, and intervals of concavity. 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.
1.3.5 Behaviors of Toolkit Functions Note: This subunit is covered in the reading assigned beneath subunit 1.3. To learn the intervals of increase, decrease, and concavity of toolkit functions, read the information on pages 40 through 43.
Web Media: YouTube: jessiemason09’s “4.2 (3 of 4) Rate of Change, Graph Direction and Concavity.mp4” Link: YouTube: jessiemason09’s “4.2 (3 of 4) Rate of Change, Graph Direction and Concavity.mp4” (YouTube)
Instructions: Watch this video for a review of determining extrema, intervals of increase and decrease, and concavity. 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 1.3: Rates of Change and Behavior of Graphs – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.3: Rates of Change and Behaviors of Graphs – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.3, 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.
1.4 Composition of Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions” (PDF)
Instructions: Read pages 49–55 of Chapter 1 to learn about the
composition of functions using tables, graphs, and formulas. Note
that this reading covers the material you need to know for subunits
1.4.1 through 1.4.3.
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.
1.4.1 Definition of Composition of Functions Note: This subunit is covered in the reading assigned beneath subunit 1.4. To learn the definition and notation to compose functions, read the information on pages 49 and 50.
Web Media: YouTube: Dr. Phil Clark’s “Function Composition” Link: YouTube: Dr. Phil Clark’s “Function Composition” (YouTube)
Instructions: Watch this video for a brief introduction to the concept of the composition of functions. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 5 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Web Media: YouTube: Mathispower4u: “Intro Composite Function Notation Application Problem” Link: YouTube: Mathispower4u: “Intro Composite Function Notation Application Problem” (YouTube)
Instructions: Watch this video for two application examples of function composition. 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.
1.4.2 Composition of Functions using Tables and Graphs Note: This subunit is covered in the reading assigned beneath subunit 1.4. To learn how to compose functions using a table or a graph, read the information on pages 50 and 51.
Web Media: YouTube: Mathispower4u: “Evaluate Composite Functions Using Tables of Values” Link: YouTube: Mathispower4u: “Evaluate Composite Functions Using Tables of Values” (YouTube)
Instructions: Watch this video to learn how to compose functions using a table of values. 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: “Evaluate Composite Functions from Graphs” Link: YouTube: Mathispower4u: “Evaluate Composite Functions from Graphs” (YouTube)
Instructions: Watch this video to learn how to compose functions using two graphs. 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.
1.4.3 Composition using Formulas Note: This subunit is covered in the reading assigned beneath subunit 1.4. To learn how to compose functions and evaluate at a numerical value and at another function, read the information on pages 51 through 55.
Web Media: YouTube: Mathispower4u: “Composite Functions” Link: YouTube: Mathispower4u: “Composite Functions” (YouTube)
Instructions: Watch this video to learn how to compose functions using formulas. 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.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.4: Composition of Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.4: Composition of Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.4, 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.
1.5 Transformation of Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions” (PDF)
Instructions: Read pages 61–80 of Chapter 1 to learn about the
translations, transformations, and reflections of functions and
defining even and odd functions. Note that this reading covers the
material you need to know for subunits 1.5.1 through 1.5.5.
Reading this section should take approximately 1 hour and 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.
1.5.1 Shifts: Vertical and Horizontal Note: This subunit is covered in the reading assigned beneath subunit 1.5. To learn how to transform a function by shifting it horizontally or vertically, read the information on pages 61 through 67.
Web Media: YouTube: Dr. Phil Clark’s “Vertical Shifts of a Function” Link: YouTube: Dr. Phil Clark’s “Vertical Shifts of a Function” (YouTube)
Instructions: Watch this video to learn how to translate the graph of a function vertically. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 5 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Web Media: YouTube: Dr. Phil Clark’s “Horizontal Shifts” Link: YouTube: Dr. Phil Clark’s “Horizontal Shifts” (YouTube)
Instructions: Watch this video to learn how to translate the graph of a function horizontally. 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: “Function Transformations: Horizontal and Vertical Translations” Link: YouTube: Mathispower4u: “Function Transformations: Horizontal and Vertical Translations” (YouTube)
Instructions: Watch this video for a review of translations with several examples of the translations of the toolkit functions. 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.
1.5.2 Reflections Note: This subunit is covered in the reading assigned beneath subunit 1.5. To learn how to reflect or create mirror images of a function across the x or y axis, read the information on pages 67 through 70.
- Web Media: YouTube: Mathispower4u: “Function Transformations:
Reflections Across the x-axis and y-axis”
Link: YouTube: Mathispower4u: “Function Transformations:
Reflections Across the x-axis and
y-axis” (YouTube)
Instructions: Watch this video to learn how to reflect the graph of the function across the axes. 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.
1.5.3 Even and Odd Functions Note: This subunit is covered in the reading assigned beneath subunit 1.5. To learn what conditions make some functions “odd” or “even,” read the information on pages 70 and 71.
- Web Media: YouTube: Khan Academy’s “Recognizing Odd and Even
Functions”
Link: YouTube: Khan Academy’s “Recognizing Odd and Even
Functions” (YouTube)
Instructions: Watch this video to learn the definition of even and odd functions and how to recognize their graphs. 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.
1.5.4 Vertical and Horizontal Stretch/Compression Note: This subunit is covered in the reading assigned beneath subunit 1.5. To learn how to transform a function multiplying the independent or dependent variable by a constant, read the information on pages 72 through 76.
Web Media: YouTube: utexascnsquest’s “Graphing Functions: Stretching and Compressing with Constants” Link: YouTube: utexascnsquest’s “Graphing Functions: Stretching and Compressing with Constants” (YouTube)
Instructions: Watch this video to learn how to graph a function when the x- or y-variable is multiplied by a constant. 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: “Function Transformations: Horizontal and Vertical Stretches and Compressions” Link: YouTube: Mathispower4u: “Function Transformations: Horizontal and Vertical Stretches and Compressions” (YouTube)
Instructions: Watch this video to review how to stretch and compress a general function and a few of the Toolkit functions. 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.
1.5.5 Combining Transformations Note: This subunit is covered in the reading assigned beneath subunit 1.5. To practice combining the transformations in previous sections, read the information on pages 76 through 80.
Web Media: YouTube: Mathispower4u: “Graphing Multiple Function Transformations – Part 1 of 2” Link: YouTube: Mathispower4u: “Graphing Multiple Function Transformations – Part 1 of 2” (YouTube)
Instructions: Watch this video to learn how to combine translations and transformations to graph a function. 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 Transformed Quadratic Function from a Graph” Link: YouTube: Mathispower4u: “Find the Equation of a Transformed Quadratic Function from a Graph” (YouTube)
Instructions: Watch this video to learn how to write an equation of a transformed function from its graph. 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.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.5: Transformations – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.5: Transformations – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.5, 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 3a: Sections 1.3–1.5” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 3a: Sections 1.3–1.5” (PDF)
Instructions: Once you have completed the readings and viewed the material for all of subunits 1.3–1.5, complete this quiz.
Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 3a: Sections 1.3-1.5 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.
1.6 Inverse Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1: Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 1:
Functions” (PDF)
Instructions: Read pages 90–94 of Chapter 1 to learn about the
inverse of a function and deriving the inverse and properties of
inverse functions. Note that this reading covers the material you
need to know for subunits 1.6.1 through 1.6.2.
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.
1.6.1 Defining and Interpreting Inverse Functions Note: This subunit is covered in the reading assigned beneath subunit 1.6. To learn the definition of an inverse, evaluate an inverse function, and compute the inverse as a function, read the information on pages 90 through 93.
Web Media: YouTube: Khan Academy’s “Introduction to Function Inverses” Link: YouTube: Khan Academy’s “Introduction to Function Inverses” (YouTube)
Instructions: Watch this video to learn the definition, notation, and how to graph the inverse of a function.
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: Mathispower4u: “Find the Inverse of a Function” Link: Mathispower4u: “Find the Inverse of a Function” (YouTube)
Instructions: Watch this video to learn how to find the inverse function formula from a given function. 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.
1.6.2 Properties of Inverse Functions Note: This subunit is covered in the reading assigned beneath subunit 1.6. To learn how one-to-one functions and inverses are related, read the information on page 94.
Web Media: YouTube: Mathispower4u: “Inverse Functions” Link: YouTube: YouTube: Mathispower4u: “Inverse Functions” (YouTube)
Instructions: Watch this video to learn the relationship between one-to-one function and inverses, how to find the equation of an inverse, and how to graph equations on a 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.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.6: Inverse Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 1.6: Inverse Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.6, 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.
Unit 1 Review - Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 1 Review” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 1 Review” (HTML)
Instructions: Once you have completed the readings and viewed the
material for all of unit 1, 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 and 30
minutes.
Terms of use: Please respect the copyright and terms of use
displayed on the webpage above.
Unit 1 Exam
- Assessment: Washington State Board for Community & Technical
Colleges: “Chapter 1: Exam A”
Link: Washington State Board for Community & Technical Colleges:
“Chapter 1: 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.