Skip navigation.

1st Quarter, Math Level 2 (10th grade math)

00.00 Start Here (Math Level 2)

 

Course Description

The fundamental purpose of Mathematics II is to formalize and extend the mathematics that students learned in 9th grade.  Students will focus on quadratic expressions, equations, and functions but also be introduced to radical, piecewise, absolute value, inverse and exponential functions. They will extend the set of rational numbers to the set of complex numbers and link probability and data through conditional probability and counting methods. Finally, they will study similarity and right triangle trigonometry and circles with their quadratic algebraic representations. The course ties together the algebraic and geometric ideas studied. Students should experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Class Overview

This integrated Secondary Mathematics II course is based on the New Utah State Standards Initiative.

Credit

This course is worth .25 credits, or nine weeks of Mathematics I. There are four Mathematics II quarter classes available. Taking all four will add up to one credit or one year of Mathematics I. In order to earn credit for each quarter, you must commit to following the EHS Honor Code: "As a student of the Electronic High School, I agree to turn in my assignments in a timely manner, do my own work, not share my work with others, and treat all students, teachers, and staff with respect." This course is for tenth grade students. After completing the work for the class, students must pass a proctored final exam to earn credit. There is not a paper-based textbook assigned for this course. If you find that having a textbook is useful, you can check out a textbook from most local libraries. You can also search for topics on the Internet to find many useful resources.

Prerequisites

You should have successfully completed 9th grade math.

Supplies needed

  • Graph paper (this can also be downloaded)
  • Scientific or Graphing Calculator (You can download scientific and graphing calculator simulators or find online versions, but as you will need a graphing calculator for the rest of your high school career, you may consider buying one now.)
  • Access to a printer to print the daily assignments is vital. Most assignments are NOT interactive and must be printed out to complete.
  • Ability to scan or photograph a completed assignment to submit electronically.

Organization of Secondary Math Level 2

Units: There are 11 units for the full credit of Secondary Math 2.

Quarter 1 has four units.
Quarter 2 has one unit.
Quarter 3 has three units.
​Quarter 4 has three units.

Schedule: When you enroll in a quarter class, you are given 10 weeks to finish all the requirements needed to earn the credit for a quarter. There are no "required" due dates for the assignments in this course. However, there is a pacing guide provided for you that will help you stay on track to being successful and finishing the course within the 10 week time frame. The pacing guide is located in the Syllabus in Module 1. Before you begin, go over the pacing guide to help you set up your own due dates for the assignments. Give your parents permission to nag you about it. You don't want to be one of those students who does a whole lot of work, but never finishes the course.

This Quarter Class

The units in this class have lessons, assignments, quizzes and a unit test.

Lessons: Each lesson provides instruction on a given topic. Many include instructional videos (hosted on YouTube) and one or more assignments for independent practice.

Assignments: Print, then complete the practice worksheet, showing how you arrived at your answer. In order to earn full points on each problem, you are required to show step by step the process needed to arrive at your answer. Then circle your answer so it is easily seen. I do give partial credit for showing your work--so ALWAYS show your work even if you think it is obvious. Assignments are submitted by uploading the digital assignment through the course website by following the instructions within each assignment. Under some circumstances, you may snail-mail a hard copy of the assignment to the instructor. If you choose this option, be sure to make a copy for yourself, as the instructor will NOT return your assignment. Also, please send the instructor an e-mail if you must mail an assignment.

Quizzes: Some quizzes are taken online. Others are more like the assignments. Print, then complete the quiz, showing all of your work as to how you arrived at your answer. In order to earn full points on each problem, you are required to show step by step the process needed to arrive at your answer. Then circle your answer so it is easily seen. I do give partial credit for showing your work--so ALWAYS show your work even if you think it is obvious. Quizzes are submitted by uploading the digital assignment through the course website by following the instructions within each quiz.

Proctored Final

Each quarter class has a proctored final exam and is worth 25% of the final grade.

Information about the Final Exam

  1. You must get approval from your teacher before you are allowed to take the final exam.
  2. You must complete every assignment and have an overall grade of C in the course to be approved to take the final exam.
  3. The final exam is a comprehensive exam that must be taken with an approved proctor.
  4. You are allowed to have a page of notes. You also will need a calculator and scratch paper.
  5. You must pass the final exam with a 60% in order to pass the class.
  6. The final exam is worth 25% of your final grade.
  7. The exam is timed. You will have 2.0 hours to complete the exam. You must finish it in one attempt.

Final Grade

Assignments and quizzes are worth 75% of the final grade. The proctored final test is worth 25% of the final grade.

Grading Scale

You earn a grade based on a modified total points percentage method. This means that the total number of points you earned is divided by the total number of points possible, times 100%. That will make up 75% of your final grade. The final exam is the remaining 25%. These scores are combined for a total percentage of the class. This percentage is translated into a grade based on this standard scale:

94-100% A
90-93% A-
87-89% B+
83-86% B
80-82% B-
77-79% C+
73-76% C
70-72% C-
67-69% D+
63-66% D
60-62% D-
0-59% no credit

00.00 *Student supplies for Secondary Math 2 (Math2)

There is no textbook assigned for this course. If you find that having a textbook is useful, you will want to select a textbook that aligns with the Common Core State Standards Initiative. You can purchase textbooks online, at some bookstores (selection is limited), or you can check out a textbook from most local libraries. Supplies needed:

  • Graph paper (this can also be downloaded)
  • Tracing paper or transparency film (if you use transparencies you will need transparency markers).
  • Straight edge (A ruler is an example of a straight edge. I have used a scrap of mat board. A piece of paper will not work! My favorite is a drafting ruler with the cork on the bottom; this gives a very clean line, and doesn't slide around.)
  • Compass (There is a very wide range of compass prices. You don't need a drafting quality compass, but you will be happier if you don't buy the cheapest compass you can find. You want something that doesn't change size as you draw with it.)
  • Graphing Calculator (You can download graphing calculator simulators, but as you will need a graphing calculator for the rest of your high school career, you may as well buy one now.)
  • Geometric software.

 

00.01 Curriculum Standards (Math Level 2)

Overview information on the Utah Mathematics Level II Core is here.

00.01.01 Student Software Needs

 

Students need access to a robust internet connection and a modern web browser.

This class may also require the Apple QuickTime plug-in to view media.

For students using a school-issued Chromebook, ask your technical support folks to download the QuickTime plug-in and enable the plug-in for your Chromebook.

$0.00

00.02 About Me (Math Level 2)

teacher-scored 10 points possible 10 minutes

About Me Assignment: This assignment gives me, as your teacher, a chance to get to know you better! To complete and submit this assignment copy the material between the asterisks into a blank word-processing document. Answer the questions using complete sentences, appropriate punctuation and sentence structure. Please write your answers in either BOLD or a {\color{Blue}DIFFERENT } {\color{Blue}COLOR }. Save the document. Finally, select all, copy, then paste the entire document into the box that opens when you click to submit this assignment.

**********************************************************************************

1. What is your full name, what name do you prefer to go by, your parent's/guardian's names, and contact information for both you and your parents? (email addresses and phone numbers.)

2. What high school do you attend and what grade are you in? What is the name of the last math class you completed?

3. Why have you chosen to take this math class with EHS?

4. What is your counselor's full name and contact information?

5. Have you read the EHS Honor Code and do you commit to following it? EHS Honor Code "As a student of the Electronic High School, I agree to turn in my assignments in a timely manner, do my own work, not share my work with others, and treat all students, teachers, and staff with respect."

6. Are you committed to finishing the class within the 10 week time frame, completing your final exam in week 9?

7. Now tell me about you! What are your likes/dislikes etc. Please be sure to include anything you think I need to know as your teacher.

*************************************************************************

I am excited to learn more about you!

Grading criteria:

1. All requested information is included.

2. Complete sentences, correct punctuation and correct grammar are used.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


01.00 Radicals and Rational Exponents (Math Level 2)

The goal of all math classes is for students to become mathematically proficient. In particular, students need to be able to:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

As you work through this unit, keep these proficiencies in mind!

This unit will review the counting, natural, whole, integer, rational, irrational, and real number sets. In particular, the focus will be on rational and irrational numbers including identifying and comparing them and performing arithmetic operations on them.

At the conclusion of this unit you must be able to state without hesitation that:

  • I can identify and define counting, natural, whole, integer, rational, irrational, and real numbers.
  • I can identify the subset(s) of the real numbers that a given number belongs to.
  • I can locate points on a number line.
  • I can compare rational numbers.
  • I can identify rational and irrational numbers.
  • I can identify the subset(s) of the real numbers that a given number belongs to.
  • I can locate points on a number line.
  • I can compare rational numbers.
  • I can identify rational and irrational numbers.
  • I can find the sums and products of rational and irrational numbers.
  • I can determine if the sum and product of rational and irrational numbers are rational or irrational.

01.01 Review Number Sets (Math Level 2)

Identify and define counting, natural, whole, integer, rational, irrational, and real numbers. Locate points on a number line, compare rational numbers, and identify rational and irrational numbers.

Mathematicians recognize several sets of numbers that share certain characteristics. These categories are helpful to know when only certain kinds of numbers are valid for values and variables. Our understanding and classification of the different sets of numbers has developed over thousands of years.

The numbers that can be represented on a number line are called real numbers. These numbers can be separated into two sets that have no numbers in common: the irrational numbers and the rational numbers. Irrational numbers have decimal forms that are nonterminating and nonrepeating. Rational numbers have decimal forms that are terminating or repeating. Inside the set of rational numbers are several smaller, nested sets: integers, whole numbers, and natural numbers.

NROC Image

NROC Image

You’ve worked with fractions and decimals, like 3.8 and 21 \small \frac{2}{3} . These numbers can be found between the integer numbers on a number line. There are other numbers that can be found on a number line, too. When you include all the numbers that can be put on a number line, you have the real number line. Let's dig deeper into the number line and see what those numbers look like.

NROC Image

The numbers that can be represented on a number line are called real numbers. These numbers can be separated into two sets that have no numbers in common: the irrational numbers and the rational numbers. Irrational numbers have decimal forms that are nonterminating and nonrepeating. Rational numbers have decimal forms that are terminating or repeating. Inside the set of rational numbers are several smaller, nested sets: integers, whole numbers, and natural numbers.

In this lesson, you will take closer look to see where these numbers fall on the number line.

If after completing this topic you can state without hesitation that...

  • I can identify and define counting, natural, whole, integer, rational, irrational, and real numbers.

Identify the subset(s) of the real numbers that a given number belongs to.

  • I can locate points on a number line.
  • I can compare rational numbers.
  • I can identify rational and irrational numbers.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the links below to begin viewing the lesson videos and to try the practice problems. Be sure to go into each link to view BOTH videos needed to learn this material.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

 

01.01 Review Number Sets – Quiz (Math Level 2)

computer-scored 33 points possible 30 minutes
Pacing: complete this by the end of Week 1 of your enrollment date for this class.


01.02 Simplifying Radicals (Math Level 2)

Identify the subset(s) of the real numbers that a given number belongs to, locate points on a number line, compare rational numbers and identify rational and irrational numbers.

Radical expressions are expressions that include a radical, which is the symbol for taking a root. They come in many forms, from simple and familiar to quite complicated:

In any case, we can use what we know about exponents to make sense of such expressions.

A radical expression is a mathematical way of representing the nth root of a number. To simplify radical expressions, look for exponential terms within the radical, and then use the property below to pull out quantities.

But keep in mind that while this property is always true if x ≥ 0, and it is always true if n is odd, different things start to happen when  x < 0 or n is even. All rules of exponents apply when simplifying radical expressions.

If after completing this topic you can state without hesitation that...

  • I can identify the subset(s) of the real numbers that a given number belongs to.
  • I can locate points on a number line.
  • I can compare rational numbers.
  • I can identify rational and irrational numbers.

…you are ready for the next topic! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

01.02 Simplifying Radicals - Assignment (Math Level 2)

teacher-scored 60 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


01.03 Exploring Sums and Products of Rational and Irrational Numbers (Math Level 2)

Find the sums and products of rational and irrational numbers and determine if the sum and product of rational and irrational numbers are rational or irrational.

The following content is taken from CPALMS, the State of Florida’s official source for standards information and course descriptions. It is Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Copy and complete each table.

CPALMS Image

Based on the above information, conjecture which of the statements is ALWAYS true, which is SOMETIMES true, and which is NEVER true?

  ALWAY SOMETIMES NEVER
i. The sum of a rational number and a rational number is rational.      
ii. The sum of a rational number and an irrational number is irrational.      
iii. The sum of an irrational number and an irrational number is irrational.      
iv. The product of a rational number and a rational number is rational.      
v. The product of a rational number and an irrational number is irrational.      
vi. The product of an irrational number and an irrational number is irrational.      

 

Click on the link below for answers.

If after completing this topic you can state without hesitation that...

  • I can find the sums and products of rational and irrational numbers.
  • I can determine if the sum and product of rational and irrational numbers are rational or irrational.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

01.03 Exploring Sums and Products of Rational and Irrational Numbers - Assignment (Math Level 2)

teacher-scored 71 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


01.03 Unit 1 Review Quiz (Math Level 2)

computer-scored 62 points possible 45 minutes

Unit 1 Review Quiz

This quiz is worth 62 points. You will need to earn at least 43 points to pass. You are allowed multiple attempts. Be sure to answer each question before you submit the quiz.

Pacing: complete this by the end of Week 1 of your enrollment date for this class.


02.00 Exponents and Exponential Functions (Math Level 2)

The goal of all math classes is for students to become mathematically proficient. In particular, students need to be able to:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

As you work through this unit, keep these proficiencies in mind!

This unit will review exponential notation and teach the related rules. Students will then be able to express radicals as an expression with fractional exponents and understand the needed properties involved.

At the conclusion of this unit you must be able to state without hesitation that:

  • I can simplify and solve expressions in exponential notation.
  • I can convert radicals to expressions with rational exponents.
  • I can convert expressions with rational exponents to their radical equivalent.
  • I can use the laws of exponents to simplify expressions with rational exponents.
  • I can use rational exponents to simplify radical expressions.

 

02.01 Rules of Exponents (Math Level 2)

Understand the rules of exponents and simplify and solve expressions in exponential notation.

We need a common language in order to communicate mathematical ideas clearly and efficiently. Exponential notation is one example. It was developed to express repeated multiplication and to make it easier to write very long numbers. For example, growth models for populations often use exponents to manage and manipulate large numbers that change quickly over time.  

In order to work with exponents, we need to “speak the language” and learn a few rules first.

Exponential notation is composed of a base and an exponent. It is a “shorthand” way of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication.

If after completing this topic you can state without hesitation that...

  • I understand the rules of exponents.
  • I can simplify and solve expressions in exponential notation.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

02.01 Rules of Exponents – Assignment (Math Level 2)

teacher-scored 60 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


02.02 Rational Exponents (Math Level 2)

Convert radicals to expressions with rational exponents to their radical equivalent, use the laws of exponents to simplify expressions with rational exponents, and use rational exponents to simplify radical expressions.

Using your calculator, complete the following table.

  1. What did you notice about your answers located in the same row?
  2. Can you see a pattern that relates the two expressions that are in the same row? Describe the pattern.
  3. Given the expression (5²)¼, what expression could you write using a radical that would give you the same value?
  4. Given the expression cube root of 48, what expression could you write using a fractional exponent that would give you the same value?

In this lesson we are going to discuss what you discovered from the above activity about fractional exponents. Let’s start with \small 9^{\frac{1}{2}}. Based on our rules of exponents, we can rewrite this as: \small (9^{\frac{1}{2}})^{2} = 9^{1} since 1/2\small \cdotx2 = 1. So, what does that tell us about \small 9^{\frac{1}{2}}? Well, it is some number that when you square it, you get 9. 3 is a number that when you square t you get 9! So, \small 9^{\frac{1}{2}} is 3. Turns out that fractional exponents have to do with radicals. Isn’t that exciting!

Check out where the 1 and the 2 of the fraction 1/2 end up.

Let’s try another one: Let’s try another one: \small 8^{\frac{2}{3}}. Based on our rules of exponents, we can rewrite this as:\small (8^{\frac{2}{3}})^{3} = 8^{2} since 2/3 x 3 = 2. So, what does that tell us about \small 8^{\frac{2}{3}}? It is some number that when we cube it we get 82 or 64. 4 is a number that when you cube it you get 64. So, 8^{\frac{2}{3}}= \sqrt[3]{8^{2}}=\sqrt[3]{64}=4. Again, check out where the 2 and 3 of the fraction end up. Starting to see a pattern?

In general, 8^{\frac{2}{3}}= \sqrt[3]{8^{2}}=\sqrt[3]{64}=4. for any real number q ≠ 0.

Look at the following and make certain you understand the concept.

\sqrt{2}=2^{\frac{1}{2}}

\sqrt{3}=3^{\frac{1}{2}}

\sqrt[3]{4^{2}}=4^{\frac{2}{3}}

Square roots are most often written using a radical sign, like this, \sqrt{4}. But now we know another way to represent the taking of a root. You can use rational exponents, exponents that are fractions, instead of a radical. For example, \sqrt{4} can be written as 4^{\frac{1}{2}}.

Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems.

In this lesson we will explore the relationship between rational (fractional) exponents and radicals.

A radical can be expressed as an expression with a fractional exponent by following the convention:

Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.

Properties of Rational Exponents

Let a and b be real numbers and let m and n be rational numbers.

If after completing this topic you can state without hesitation that...

  • I can convert radicals to expressions with rational exponents.
  • I can convert expressions with rational exponents to their radical equivalent.
  • I can use the laws of exponents to simplify expressions with rational exponents.
  • I can use rational exponents to simplify radical expressions.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos before continuing.

02.02 Rational Exponents – Assignment (Math Level 2)

teacher-scored 84 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


02.02 Unit 2 Review Quiz (Math Level 2)

computer-scored 45 points possible 30 minutes

Complete the computerized quiz.

This quiz is worth 45 points. You are required to earn a minimum of 32 points to pass this quiz. You are allowed as many attempts as you need to earn the score you want. Be sure to answer all questions as completly as possible before you submit the quiz.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


03.00 Polynomials (Math Level 2)

The goal of all math classes is for students to become mathematically proficient. In particular, students need to be able to:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

As you work through this unit, keep these proficiencies in mind!

This unit will review the terms related to polynomials. Students will then be able to evaluate, simplify and identify the types of polynomials. In addition, you will be able to add, subtract, multiply, divide and factor polynomials. Finally, you will be introduced to a new number set – complex numbers – and learn to perform arithmetic operations on them.

At the conclusion of this unit you must be able to state without hesitation that:

  • I can identify the terms, the coefficients and the exponents of a polynomial.
  • I can evaluate a polynomial for given values of the variable.
  • I can simplify polynomials by collecting like terms.
  • I can identify monomials, binomials and polynomials.
  • I can write polynomials to describe real world situations.
  • I can add and subtract polynomials.
  • I can multiply and divide monomials.
  • I can multiply polynomials and collect the like terms of the resulting sum of monomials.
  • I can identify and multiply binomial products.
  • I can find the greatest common factor (GCF) of monomials.
  • I can factor polynomials by factoring out the greatest common factor (GCF).
  • I can factor expressions with four terms by grouping.
  • I can factor trinomials with a leading coefficient of 1.
  • I can factor trinomials with a common factor.
  • I can factor trinomials with a leading coefficient other than 1.
  • I can factor trinomials that are perfect squares.
  • I can factor binomials in the form of the difference of squares.
  • I can express roots of negative numbers in terms of i.
  • I can express imaginary numbers as bi and complex numbers as a + bi.
  • I can add complex numbers.
  • I can subtract complex numbers.
  • I can multiply complex numbers.
  • I can find conjugates of complex numbers.
  • I can divide complex numbers.

03.01 Introduction to Polynomials (Math Level 2)

Identify the terms, the coefficients and the exponents of a polynomial, evaluate a polynomial for given values of the variable, and simplify polynomials by collecting like terms.

The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.

The coefficient can be any real number, including 0. The exponent of the variable must be a whole number—0, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.

The value of the exponent is the degree of the monomial. Remember that a variable that appears to have no exponent really has an exponent of 1. And a monomial with no variable has a degree of 0. (Since x0 has the value of 1 if x ≠ 0, a number such as 3 could also be written 3x0, if x ≠ 0. as 3x0 = 3 • 1 = 3.)

Polynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into each instance of the variable, then using order of operations to complete the calculations.

If after completing this topic you can state without hesitation that...

  • I can identify the terms, the coefficients and the exponents of a polynomial.
  • I can evaluate a polynomial for given values of the variable.
  • I can simplify polynomials by collecting like terms.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.01 Introduction to Polynomials - Assignment (Math Level 2)

teacher-scored 72 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 2 of your enrollment date for this class.


03.02 Polynomials (Math Level 2)

Identify monomials, binomials and polynomials and write polynomials to describe real world situations.

Algebraic expressions containing one or more terms are called polynomials. There are several kinds of polynomials, based on how many terms they have.  For example, monomials are polynomials with one term (“mono-” is a prefix meaning one).

Polynomials are useful because they can be written to represent real world situations and to find solutions to actual problems.

One of the powers of algebra is in representing aspects of the world with algebraic expressions in order to learn more about them.  An expression that combines one or more terms to describe a situation is called a polynomial. Binomials, which are polynomials with two terms, and monomials, which are polynomials with one term, are two types of polynomials. By definition, polynomials do not have variables in the denominator or negative exponents.

If after completing this topic you can state without hesitation that...

  • I can identify monomials, binomials and polynomials.
  • I can write polynomials to describe real world situations.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.02 Polynomials - Assignment (Math Level 2)

teacher-scored 74 points possible 35 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


03.03 Adding and Subtracting Polynomials (Math Level 2)

Add and subtract polynomials.

Adding and subtracting polynomials may sound complicated, but it’s really not much different from adding and subtracting numbers. Any terms that have the same variables with the same exponents can be combined.

Adding polynomials involves combining like terms. Like terms are monomials that contain the same variable or variables raised to the same powers. The following are examples of like and unlike terms:

When adding or subtracting polynomials, look for like terms, which are terms that have the same variables raised to the same power.  Use the commutative property of addition to regroup the terms in an expression into sets of like terms. Like terms are combined by adding or subtracting the coefficients as appropriate while keeping the variables and exponents the same.

Polynomials are not considered simplified until all of the like terms have been combined.

If after completing this topic you can state without hesitation that...

  • I can add and subtract polynomials.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.03 Adding and Subtracting Polynomials - Assignment (Math Level 2)

teacher-scored 73 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


03.04 Multiplying and Dividing Monomials (Math Level 2)

Multiply and divide monomials.

A monomial is an expression that consists of a number, a variable, or the product of numbers and variables. Expressions such as 2, z, and 42p3y are monomials, while those with more than one term, like 2 + z, are not.

When monomials include both a number and a variable, the number is called the coefficient.  For example, in the monomial 8x2, 8 is the coefficient.

The variables in a monomial can have whole number exponents, but no negative exponents. Just as numbers can be multiplied and divided, monomials with variables can also be multiplied and divided following the same rules.

Multiplying and dividing monomials are done by following the rules of exponents.  To multiply monomials, multiply coefficients and add the exponents of like bases. To raise a monomial to a power, when there is a coefficient or more than one variable raised to a power of a power, each variable or number is taken to the power by multiplying the exponent of the base by the exponent of the power it is being raised to. To divide monomials, divide the coefficients and subtract the exponents of like bases.

If after completing this topic you can state without hesitation that...

  •  I can multiply and divide monomials.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.04 Multiplying and Dividing Monomials - Assignment (Math Level 2)

teacher-scored 93 points possible 35 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

 

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


03.05 Multiplying Polynomials (Math Level 2)

Multiply polynomials and collect the like terms of the resulting sum of monomials. Identify and multiply binomial products.

Multiplying polynomials involves applying the rules of exponents and the distributive property to simplify the product. This multiplication can also be illustrated with an area model and can be useful in modeling real world situations. Understanding polynomial products is an important step in factoring and solving algebraic equations.

Multiplication of binomials and polynomials requires use of the distributive property and integer operations. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients.  A product is written in simplified form if all of its like terms have been combined.

Certain types of binomial multiplication sometimes produce results that are called special products. Special products have predictable terms. Although the distributive property can always be used to multiply any binomials, recognition of those that produce special products provides a problem-solving shortcut.

Some products of multiplying binomials follow a predictable pattern that makes it easy to simplify them. These are known as special products.  There are three special products of binomials that each follow a specific formula:

Special Binomial Products

Product of a Sum: (a + b)2 = a2 + 2ab + b2

Product of a Difference: (a – b)2 = a2 – 2ab + b2

Product of a Sum & a Difference: (a + b)(a – b) = a2 – b2

Polynomials can be identified as special products by examining the characteristics of their terms.

If after completing this topic you can state without hesitation that...

  • I can multiply polynomials and collect the like terms of the resulting sum of monomials.
  • I can identify and multiply binomial products.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the links below to begin viewing the lesson videos and practice problems. Be sure to go into each link to view BOTH videos needed to learn this material.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.05 Multiplying Polynomials - Assignment (Math Level 2)

teacher-scored 94 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 3 of your enrollment date for this class.


03.06 Review Greatest Common Factor and Factor Polynomials with Four Terms (Math Level 2)

Find the greatest common factor (GCF) of monomials, factor polynomials by factoring out the greatest common factor (GCF) and factor expressions with four terms by grouping.

Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4 and 5 and 1 and 20. To factor a number is to rewrite it as a product. 20 = 4 • 5.

Likewise to factor a polynomial, you rewrite it as a product. Just as any integer can be written as the product of factors, so too can any monomial or polynomial be expressed as a product of factors. Factoring is very helpful in simplifying and solving equations using polynomials.

A prime factor is similar to a prime number—it has only itself and 1 as factors. The process of breaking a number down into its prime factors is called prime factorization.

A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in ab + ac form has been rewritten as a(b + c), where a is the GCF, the polynomial is in factored form.

When factoring a four-term polynomial using grouping, find the common factor of pairs of terms rather than the whole polynomial. Use the distributive property to rewrite the grouped terms as the common factor times a binomial. Finally, pull any common binomials out of the factored groups. The fully factored polynomial will be the product of two binomials.

If after completing this topic you can state without hesitation that...

  • I can find the greatest common factor (GCF) of monomials.
  • I can factor polynomials by factoring out the greatest common factor (GCF).
  • I can factor expressions with four terms by grouping.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend viewing the following video examples before moving on to the assignment.

 

03.06 Review Greatest Common Factor and Factor Polynomials with Four Terms - video (Math Level 2)

03.06 Review Greatest Common Factor and Factor Polynomials with Four Terms – Assignment (Math Level 2)

teacher-scored 72 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 4 of your enrollment date for this class.


03.07 Factoring Trinomials (Math Level 2)

Factor trinomials with a leading coefficient of 1, with a common factor, and with a leading coefficient other than 1. Factor trinomials that are perfect squares and binomials in the form of the difference of squares.

A polynomial with three terms is called a trinomial. Trinomials often (but not always!) have the form x2 + bx + c. At first glance, it may seem difficult to factor trinomials, but you can take advantage of some interesting mathematical patterns to factor even the most difficult-looking trinomials.

So, how do you get from 6x2 + 2x – 20 to (2x + 4)(3x −5)?  

Trinomials in the form x2 + bx + c can be factored by finding two integers, r and s, whose sum is b and whose product is c. Rewrite the trinomial as x2 + rx + sx + c and then use grouping and the distributive property to factor the polynomial.

When a trinomial is in the form of ax2 + bx + c, where a is a coefficient other than 1, look first for common factors for all three terms. Factor out the common factor first, then factor the remaining simpler trinomial. If the remaining trinomial is still of the form ax2 + bx + c, find two integers, r and s, whose sum is b and whose product is ac. Then rewrite the trinomial as ax2 + rx + sx + c and use grouping and the distributive property to factor the polynomial.

One of the keys to factoring is finding patterns between the trinomial and the factors of the trinomial. Learning to recognize a few common polynomial types will lessen the amount of time it takes to factor them. Knowing the characteristic patterns of special products—trinomials that come from squaring binomials, for example—provides a shortcut to finding their factors.

Learning to identify certain patterns in polynomials helps you factor some “special cases” of polynomials quickly. The special cases are:

trinomials that are perfect squares, a2 + 2ab + b2 and a2 – 2ab + b2, which factor as (a+ b)2 and (a – b)2, respectively;
binomials that are the difference of two squares, a2 – b2, which factors as (a + b)(a – b).

For some polynomials, you may need to combine techniques (looking for common factors, grouping, and using special products) to factor the polynomial completely.

Some equations are considered special cases. These are equations that have no solution and equations whose solution is the set of all real numbers. When you use the steps for solving an equation, and you get a false statement rather than a value for the variable, there is no solution. When you use the steps for solving an equation, have avoided multiplying both sides of the equation by zero, and you get a true statement rather than a value for the variable, the solution is all real numbers. Algebra is a powerful tool for modeling and solving real-world problems.

If after completing this topic you can state without hesitation that...

  • I can factor trinomials with a leading coefficient of 1.
  • I can factor trinomials with a common factor.
  • I can factor trinomials with a leading coefficient other than 1.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the links below to begin viewing the lesson videos and to try the practice problems. Be sure to go into each link to view BOTH videos needed to learn this material.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

03.07 Factoring Trinomials - Assignment (Math Level 2)

teacher-scored 75 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 4 of your enrollment date for this class.


03.08 Extend Polynomial Identities to Complex Numbers (Math Level 2)

Express roots of negative numbers in terms of i and imaginary numbers as bi and complex numbers as a + bi.

The material for this lesson is from NROC Developmental Math: Complex Numbers under a Creative Commons Attribution 3.0 Unported License.

Several times in your learning of mathematics, you have been introduced to new kinds of numbers. Each time, these numbers made possible something that seemed impossible! Before you learned about negative numbers, you couldn’t subtract a greater number from a lesser one, but negative numbers give us a way to do it. When you were learning to divide, you initially weren't able to do a problem like 13 divided by 5 because 13 isn't a multiple of 5. You then learned how to do this problem writing the answer as 2 remainder 3. Eventually, you were able to express this answer as 2\frac{3}{5}. Using fractions allowed you to make sense of this division.

Up to now, you’ve known it was impossible to take a square root of a negative number. This is true, using only the real numbers. But here you will learn about a new kind of number that lets you work with square roots of negative numbers! Like fractions and negative numbers, this new kind of number will let you do what was previously impossible.

Complex numbers have the form a + bi, where a and b are real numbers and i is the square root of −1. All real numbers can be written as complex numbers by setting b = 0. Imaginary numbers have the form bi and can also be written as complex numbers by setting a = 0. Square roots of negative numbers can be simplified using:

\sqrt{-1} = i and \sqrt{ab} = \sqrt{a} \sqrt{b}.

If after completing this topic you can state without hesitation that...

  • I can express roots of negative numbers in terms of i.
  • I can express imaginary numbers as bi and complex numbers as a + bi.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos before continuing.

03.08 Extend Polynomial Identities to Complex Numbers – Assignment (Math Level 2)

teacher-scored 58 points possible 30 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.


03.09 Operations with Complex Numbers (Math Level 2)

Add, subtract, multiply and divide complex numbers and find their conjugates.

Any time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, “How do you add them?” In this topic, you’ll learn how to add complex numbers and also how to subtract, multiply, and divide them.

Complex numbers can be added, subtracted, multiplied, and divided using the same ideas you used for radicals and variables. With multiplication and division, you may need to replace i2 with −1 and simplify further.

If after completing this topic you can state without hesitation that...

  • I can add complex numbers.
  • I can subtract complex numbers.
  • I can multiply complex numbers.
  • I can find conjugates of complex numbers.
  • I can divide complex numbers.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos before continuing.

03.09 Operations with Complex Numbers - Assignment (Math Level 2)

teacher-scored 90 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.


03.09 Unit 3 Review Quiz (Math Level 2)

computer-scored 86 points possible 45 minutes

Complete the computerized quiz.

This quiz is worth 86 points. You are required to earn a minimum of 60 points to pass this quiz. You are allowed as many attempts as you need to earn the score you want. When entering answers, do not put ANY SPACES between letters, numbers or symbols. Be sure to answer all questions as completly as possible before you submit the quiz.

Pacing: complete this by the end of Week 5 of your enrollment date for this class.


04.00 Functions and their Graphs (Math Level 2)

The goal of all math classes is for students to become mathematically proficient. In particular, students need to be able to:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

As you work through this unit, keep these proficiencies in mind!

This unit is about a variety of functions – quadratic, radical, piecewise, absolute value, inverse and exponential. Students will learn the necessary terms, the different forms of the equations, the domain and range, how to graph each function and their transformations.

You will need to keep extensive notes on each function type!

At the conclusion of this unit you must be able to state without hesitation that:

  • I can graph the equation y = x2 by plotting points.
  • I can define the terms "parabola", "vertex", and "axis of symmetry."
  • I can use the symmetry of parabolas to answer question about points on the parabolas.
  • I recognize "y = ax2 +bx + c" as the equation of a parabola.
  • I understand how a, b, c effects the parabola.
  • I can find factors and x-intercepts of parabolas.
  • I can change from a parabolic equation in factored form to standard form and vice-versa.
  • I know the vertex form of a parabolic equation.
  • I can change a parabolic equation from vertex form to standard form and factored form.
  • I can identify the domain and range of the quadratic functions.
  • I can graph transformations of the parent quadratic function.
  • I can graph radical functions.
  • I can identify the domain and range of radical functions.
  • I can graph radical functions using transformations of the parent radical functions.
  • I can graph piecewise functions.
  • I can identify the domain and range of absolute value functions.
  • I can graph absolute value functions and the transformation of their parent functions.
  • I can determine whether or not a function has an inverse, and find the inverse when it exists.
  • I can graph exponential equations and functions.
  • I can solve applied problems using exponential problems and their graphs.
  • I can graph the parent function of exponential functions and their transformations.
  • I can model and solve problems of growth and decay using exponential functions.
  • I can find the average rate of change of a function over a given interval.

04.01 Quadratic Functions (Math Level 2)

Graph the equation y = x^2 by plotting points, define the terms "parabola", "vertex", and "axis of symmetry," use the symmetry of parabolas to answer question about points on the parabolas, recognize "y = ax^2 +bx + c" as the equation of a parabola, understand how a, b, c effects the parabola, find factors and x-intercepts of parabolas, change from a parabolic equation in factored form to standard form and vice-versa, know the vertex form of a parabolic equation, and change a parabolic equation from vertex form to standard form and factored form. Identify the domain and range of the quadratic functions.

 

Next to linear functions, one of the most common types of functions we work with in algebra is the quadratic function.

A quadratic function is a function that can be described by an equation of the form y = ax2 + bx + c, where a ≠ 0. No term is of a degree higher than 2. Quadratic functions are useful when working with area, and they often appear in motion problems that involve gravity or acceleration.

The graphs of quadratic functions have some defining characteristics which are closely related to their symbolic forms.  As we explore these graphs, we'll learn to identify their important characteristics, and see some of the many ways quadratic equations can be structured.

The graph of a quadratic function is a U-shaped curve called a parabola. It can be drawn by plotting solutions to the equation, by finding the vertex and using the axis of symmetry to plot selected points, or by finding the roots and plotting them along with the vertex.

There are three useful forms of a quadratic function. The first form, which we introduced above, is y = ax2 + bx + c, where x represents a variable and a, b, and c are constants where a ≠ 0.  This form is called standard form. The value of a will determine if the parabola will open up or down. If a < 0 (negative), the parabola will open down. If a > 0 (positive), the parabola will open up.

The second form is called the vertex form of a quadratic function. Vertex form is y = a(x – h)2 + k where the vertex is (h, k). Note: the h in the vertex is the opposite of the h in the function!

The third form is factored form. Factored form looks like this: y = (x - a)(x - b).

The vertex is the highest or lowest point of the parabola. If the parabola opens up, the vertex will be a "minimum". If the parabola opens down, the vertex will be a "maximum". If the quadratic equation is in standard form, y = ax2 + bx + c, the vertex can be found by first finding the x value of the ordered pair using this formula: x = -\frac{b}{2a}   To find the y value of the ordered pair, substitute the x value you find into the equation and work through the math. If the quadratic equation is in vertex form, y = a(x – h)2 + k, the vertex is (h,k).

The axis of symmetry is a vertical line running through the vertex of the parabola. Every parabola is symmetric about the axis of symmetry. This means that if you were to fold the parabola on the axis of symmetry, both sides of the parabola would match up perfectly. For an equation of the form y = ax2 + bx + c, the equation of the axis of symmetry is  x = -\frac{b}{2a}. If the quadratic equation is in vertex form, y = a(x – h)2 + k, , then the axis of symmetry is x = h.

Domain and Range of Quadratic Functions:

The Domain of any function is all the x values of the function, or when looking at a graph, it is what the function is doing from left to right.

The Range of any function is all the y values of the function, or looking at the graph it is what the function is doing up and down.

You now know the vertex form of a quadratic function is f(x) = a(x – h)2 + k. The vertex is (h, k).  The domain of this quadratic function, regardless of the values of the parameters a, h, and k, is the set of all real numbers.  The range, on the other hand, depends on the values of a and k. If a This also holds true when the quadratic equation is in any of the forms.

You also know that the graph of a parabola opens upward if a > 0 and downward if a < 0.  The value of k (part of the vertex) gives you the maximum or minimum value of x depending on the value of a.

Take time to study the two graphs below and their ranges.  Make certain you understand this concept!

Now give these a try on your own. The answers will be shown below.

  1.                              2. 

               y = -\frac{1}{2}x^{2}-3                                                      y = \frac{1}{2}x^{2}-2                                                                          

 

ANSWERS:   

1. Range equals all real numbers \leq -3

2. Range equals all real numbers \geqslant -2        

               

If after completing this topic you can state without hesitation that...

  • I can graph the equation y = x2 by plotting points.
  • I can define the terms "parabola", "vertex", and "axis of symmetry."
  • I can use the symmetry of parabolas to answer question about points on the parabolas.
  • I recognize "y = ax2 +bx + c" as the equation of a parabola.
  • I understand how a, b, c effects the parabola.
  • I can find factors and x-intercepts of parabolas.
  • I can change from a parabolic equation in factored form: y = (x - a)(x - b) to standard form: y = ax2 + bx + c and vice-versa.
  • I know the vertex form: y = a(x - h)2 + k of a parabolic equation.
  • I can change a parabolic equation from vertex form: y = a(x - h)2 + k to standard form: y = ax2 + bx + c and factored form: y = (x - a)(x - b).
  • I can find the domain and range of a quadratic function.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Click the link below to begin viewing the lesson videos and to try the practice problems.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos before continuing.

04.01 Quadratic Functions - Assignment (Math Level 2)

teacher-scored 112 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed.You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


04.02 Transformations of Parent Quadratic Functions (Math Level 2)

Graph transformations of the parent quadratic function.

In this topic you will learn to transform quadratic functions given the parent function f(x) = x2

The best form of a quadratic function to use for this body of work is the vertex form. Recall that this is written f(x) = (xh)2 + k. In the parent function, both h and k equal zero. Let’s explore transformations of this parent function.

 

 

Now, just for fun, let’s make things more interesting!

Steps in transforming graphs from the parent function f(x) = x2:

  1. Make certain the quadratic function is in vertex form.
  2. Identify the value of a.
  • If a is positive, the parabola opens up.
  • If a is negative, the parabola opens down.
  • If |a| >1, the parabola is compressed.
  • If 0 < |a| < 1, the parabola stretches.

   3. Identify the value of h.

  • If h is positive, you will translate the graph to the right.
  • If h is negative, you will translate the graph to the left.

   4. Identify the value of k.

  • If the value of k is positive, you translate EVERY point on the graph UP k units.
  • If the value of k is negative, you translate EVERY point on the graph DOWN k units.

I highly recommend that you click on the links below and watch the videos before continuing.
 

If after completing this topic you can state without hesitation that...

  • I can graph transformations of the parent quadratic function.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

04.02 Transformations of Parent Quadratic Functions - Assignment (Math Level 2)

teacher-scored 66 points possible 45 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


04.03 Exploring Radical Functions (Math Level 2)

Graph radical functions. Identify the domain and range of radical functions.

You can graph radical functions (such as square root functions and cube root functions) by choosing values for x and finding points that will be on the graph. Again, it’s helpful to have some idea about what the graph will look like.

Think about the basic square root function, \small \sqrt{x}. Let’s take a look at a table of values for x and y and then graph the function. (Notice that all the values for x in the table are perfect squares. Since you are taking the square root of x, using perfect squares makes more sense than just finding the square roots of 0, 1, 2, 3, 4, etc.)

Recall that x can never be negative here because the square root of a negative number would be imaginary, and imaginary numbers cannot be graphed. There are also no values for x that will result in y being a negative number.

Look at the graph of the points from the table.

The graph of the square root function is the top (or bottom) half of a parabola laying on its side.

Now lets think about the cube root function, \small \sqrt[3]{x}. Just with the square root function, in order to graph the cube root function, we will need to make a table of values. (Notice that all the values for x in the table are perfect cubes, including negative values. Since you are taking the cube root of x, using perfect cubes makes more sense than just finding the cube roots of 0, 1, 2, 3, 4, etc.)

Now, look at the graph of the cube root function.

Domain and Range of Radical Functions.

Before we get started, let’s define a few terms that are important to this topic. Given

x is called the radicand, the number under the radical sign.
n is called the index, the degree of the root.

Remember that the domain is the set of all the x values of the function. Or if looking at a graph of the function, the domain will describe what the graph is doing from left to right.

The Range is the set of all the y values of the function. Or if looking at the graph of the function, the range will describe what the graph is doing from bottom to top.

To find the domain of a radical function, the first thing we need to do is identify the index. If the index is an even number, the domain will have to be numbers ≥ 0 because we are working with the principal (positive) roots and square roots and fourth roots, etc., must always be positive in order to yield a positive root. If the index is an odd number (≥ 3), the domain can be all real numbers.

After determining the limitations of the domain due to the index, the next step is to look at the radicand. For an even index function, the radicand must be ≥ 0 as discussed above. For an odd index, the domain is all real numbers. 

Examples:

Now, let’s look at the range. For an odd index function, the range is all real numbers. The graphs make this clear. For an even index function, the range is all the points of the y-axis that relates to the given values of the domain.

These last two examples should remind you of the vertex form of the quadratic equation. With these examples, you should be equipped to find the domain and range of any radical function.

Summary

Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for x, finding the corresponding y values, and plotting them. However, it helps to understand the basic shape of the function. Knowing the effect of changes to the basic function equation is also helpful. You can also determine the domain and range of a function by looking at its graph.

If after completing this topic you can state without hesitation that...

  • I can graph radical functions.
  • I can determine the domain and range of radical functions.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos after completing the topic.

 

04.03 Exploring Radical Functions – Quiz (Math Level 2)

computer-scored 50 points possible 30 minutes

Complete the computerized quiz.

This quiz is worth 50 points. You are required to earn a minimum of 35 points to pass this quiz. You are allowed as many attempts as you need to earn the score you want. When entering answers, do not put ANY SPACES between letters, numbers or symbols. Be sure to answer all questions as completly as possible before you submit the quiz.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


04.04 Graph Radical Functions and Transformations of Parent Functions (Math Level 2)

Graph radical functions using transformations of the parent radical functions.

Graphing radical functions and transformations of the parent functions is very similar to graphing parabolas. We can even use similar letters. Let’s begin by looking at the square root function. We’ve already looked at the graph of two in the last topic.

This following content is taken from NROC Developmental Math Unit 17 Lesson 2:

As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same. Here are some examples.

Multiplying \sqrt{x} by a positive value changes the width of the half-parabola. Multiplying \sqrt{x} by a negative number gives you the other half of a horizontal parabola.

 

Adding a value outside the radical moves the graph up or down. Think about it as adding the value to the basic y value of \sqrt{x}, so a positive value added moves the graph up.

Adding a value inside the radical moves the graph left or right. Think about it as adding a value to x before you take the square root—so the y value gets moved to a different x value. For example, for f(x)=\sqrt{x}, the square root is 3 if x = 9. For f(x)=\sqrt{x+1}, the square root is 3 when x + 1 is 9, so x is 8. Changing x to x +1 shifts the graph to the left by 1 unit (from 9 to 8). Changing x to x − 2 shifts the graph to the right by 2 units.

 

Notice that as x gets greater, adding or subtracting a number inside the square root has less of an effect on the value of y!

Example Problem:

Graph \dpi{100} \fn_phv {\color{Magenta} f(x)=-2+\sqrt{x-1}}

Before making a table of values, look at the function equation to get a general idea what the graph should look like.

 

Inside the square root, you’re subtracting 1, so the graph will move to the right 1 from the basic f(x)=\sqrt{x} graph.

You’re also adding −2 outside the square root, so the graph will move down two from the basic f(x)=\sqrt{x} graph.

Create a table of values. Choose values that will make your calculations easy. You want x – 1 to be a perfect square (0, 1, 4, 9, and so on) so you can take the square root.

Since values of x less than 1 makes the value inside the square root negative, there will be no points on the coordinate graph to the left of x = 1. There is no need to choose x values less than 1 for your table!

x f(x)
1 –2
2 –1
5 0
10 1

 

Use the table pairs to plot points.

 

Connect the points as best you can, using a smooth curve.

Answer

 

Which of the following is a graph of f(x)=-2\sqrt{x}?

A)                                                                     B)

                           

 

C)                                                                    D)

                           

Answer:

A) Incorrect.

Recall that the graph of f(x)=-\sqrt{x} is entirely under the x-axis, in Quadrant IV. The graph of f(x)=-2\sqrt{x} will be similar but wider. The correct answer is Graph C.

B) Incorrect.

In the function f(x)=-2\sqrt{x}, f(0) = 0. This point is not represented on this graph. Find a graph that looks similar to Graph B, but which goes through the point (0,0). The correct answer is Graph C.

C) Correct.

The graph of f(x)=-2\sqrt{x} will be similar to the graph of f(x)=-\sqrt{x}, although it will be a bit wider.

D) Incorrect.

This is the graph of f(x)=2\sqrt{x}. The graph of f(x)=-2\sqrt{x} will be below the x-axis. The correct answer is Graph C.

End of NROC Lesson

Now, let's look at the cube root function:

I highly recommend that you click on the links below and watch the videos before continuing:
 

IF after completing this topic you can state without hesitation that...

I can graph radical functions using transformations of the parent radical functions.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

04.04 Graph Radical Functions and Transformations of Parent Functions - Assignment (Math Level 2)

teacher-scored 66 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 6 of your enrollment date for this class.


04.05 Introduction to Piecewise Functions (Math Level 2)

Graph piecewise functions.

This content of this lesson is from “Graphing the Basic Functions”, section 2.4 from the book Advanced Algebra (v. 1.0). It is licensed under a Creative Commons by-nc-sa 3.0 license.

A piecewise function, or split function, is a function whose definition changes depending on the value in the domain. For example, we can write the absolute value function f(x)=|x| as a piecewise function:

In this case, the definition used depends on the sign of the x-value. If the x-value is positive, x ≥ 0, then the function is defined by f(x) = x. And if the x-value is negative, x < 0, then the function is defined by f(x) = −x.

Following is the graph of the two pieces on the same rectangular coordinate plane:

 

Example 1

 

Graph the following:

 

Solution:

In this case, we graph the squaring function over negative x-values and the square root function over positive x-values.

 

Notice the open dot used at the origin for the squaring function and the closed dot used for the square root function. This was determined by the inequality that defines the domain of each piece of the function. The entire function consists of each piece graphed on the same coordinate plane.

Answer:

When evaluating, the value in the domain determines the appropriate definition to use.

Example 2

 

Given the function h, find h(−5), h(0), and h(3).

Solution:

Use h(t) = 7t + 3 where t is negative, as indicated by t < 0.

h(-5)= 7(−5) + 3 = −35 + 3 = −32

Where t is greater than or equal to zero, use h(t) = −16t² + 32t.

h(0)= −16(0)² + 32(0) = 0 + 0 = 0

h(3)= −16(3)² + 32(3) = −16(9) + 96 = −144 + 96 = −48

Answer:

h(−5) = −32, h(0) = 0, and h(3) = −48

Try this! Graph:

Answer:

 

The definition of a function may be different over multiple intervals in the domain.

Example 3

 

Graph:

Solution:

In this case, graph the cubing function over the interval (−∞,0). Graph the identity function over the interval [0,4]. Finally, graph the constant function f(x)=6 over the interval (4,∞). And because f(x) = 6 where x > 4, we use an open dot at the point (4,6). Where x = 4, we use f(x) = x and thus (4,4) is a point on the graph as indicated by a closed dot.

Answer:

 

 

 

 

I highly recommend that you click on the links below and watch the videos before continuing.

Dennin, Michael. Physics 3A: Basic Physics. (UCI OpenCourseWare: University of California, Irvine): ocw.uci.edu

IF after completing this topic you can state without hesitation that...

I can graph piecewise functions.

…you are ready for the assignment! Otherwise, go back and review the material and view the additional example video links below before moving on.

04.05 Introduction to Piecewise Functions – Assignment (Math Level 2)

teacher-scored 50 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


04.06 Domain, Range, Graphs and Transformations of Absolute Value Functions (Math Level 2)

Identify the domain and range of absolute value functions and graph absolute value functions and the transformation of their parent functions.

Recall that the absolute value is defined as the distance a number is from zero. Since it is a distance, the absolute value will always be postive, or zero. Absolute value is represented by two parallel vertical lines on either side of a term. For example, |3|.

An absolute value function is a function whose rule contains an absolute value expression. The parent graph for absolute values is f(x)=|x| and it looks like this:

 

The graph is a V shape. It has a vertex and is symmetrical about the vertical line running through the vertex.

The general form of an absolute value function is y=a\left | x-h \right |+k. The absolute value function has many similarities with the quadratic function.

  • The vertex of the absolute value function is (h,k). Note that the value of h is the opposite of what is in the equation.
  • It is symmetric about the line x = h.
  • If a > 0, (a is positive), the graph opens up. If a < 0, (a is negative), the graph opens down.
  • If \left | a \right | < 1, the graph will be wider. If |a| > 1, the graph will be narrower.

You will recall that the domain is the set of all possible inputs (all the x values) of a function which allow the function to work. The range is the set of all possible outputs (all the y values) of a function.

Let’s apply this to absolute value functions.

Look at the graph of f(x) = |x|:

Absolute value function graphs always look like the letter V. All such functions will have a domain that includes all real numbers. The range will vary depending on the value of the vertex. Can you see that the range of this function is all real numbers ≥ 0?

Now, let’s graph the function f(x) = |x + 2|:

  • How does (x+2) change the domain?
  • How does (x+2) change the range?

The domain is still all real numbers and the range is still all real numbers ≥ 0. The (x+2) moved the vertex (0,0) to (-2,0).

What about the graph of f(x) = |x| + 2?

  • How does +2 change the domain?
  • How does +2 change the range?

The domain is still all real numbers and the range is all real numbers ≥ 2. The vertex is (0,2).

Here is the graph of f(x) = |x – 2|.

  • How does (x – 2 change the domain?
  • How does (x – 2) change the range?

The domain is still all real numbers and the range is all real numbers ≥ 0.

Finally, look at the graph of f(x) = |x| – 2?

  • How does –2 change the domain?
  • How does –2 change the range?

In what must be obvious to you by now, the domain is still all real numbers but the range is all real numbers ≥ –2. The vertex is (0, –2).

Have you figured out a pattern?

  • f(x) = |x + a| translates the graph to the left and does not alter the domain nor the range.
  • f(x) = |xa| translates the graph to the right and does not alter the domain nor the range.
  • f(x) = |x| + a translates the graph up and does not alter the domain. However, the range is moved to ≥ k.
  • f(x) = |x| – a translates the graph down and does not alter the domain. However, the range is moved to ≥ – k.

In this “family” of graphs, the original f(x) = |x| is called the parent function. The other graphs in the “family” are referred to as transformations of the parent function. Once you have graphed the parent function it is easy to graph the transformations.

Let's make things a bit more interesting. Look at the graph of f(x) = 2|x|:

The domain and range don’t change but the “V” is narrower.

What about the graph of f(x) = –2|x|?

Hmmm. The “V” is upside down! It is a reflection of f(x) = 2|x|.

The domain doesn’t change but the range is now all real numbers ≤ 0.

Of course, we can make things even more interesting.

How about the graph of f(x) = –3|x + 2| + 1?

Can you figure out the domain and range?

The domain is still all real numbers but the range is all real numbers ≤ 1. The vertex is (–2,1).

We can now add to our list from above:

  • f(x) = a|x| compresses the graph but does not alter the domain nor the range.
  • f(x) = –a|x| compresses the graph, then reflects it across the x-axis and does not alter the domain. However, the range is changed to all real numbers ≤ 0.
  • f(x) = a|x + h| + k translates the graph left or right depending on the value of h. It also translates the graph up or down depending on the value of k.
  • Finally, it compresses the graph depending on the value of a.

Take time to practice with these types of graphs at the website listed below.

 

I highly recommend that you click on the links below and watch the videos before continuing:

If after completing this topic can you state without hesitation that...

  • I can identify the domain and range of absolute value functions.
  • I can graph absolute value functions and the transformation of their parent functions.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

04.06 Domain, Range, Graphs and Transformations of Absolute Value Functions – Assignment (Math Level 2)

teacher-scored 66 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


04.07 Inverse Functions (Math Level 2)

Determine whether or not a function has an inverse and find the inverse when it exists.

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ(x)=y, and g(y)=x. More directly, g(ƒ(x))=x, meaning g(x) composed with ƒ(x) leaves x unchanged. Basically speaking, an inverse is found by switching the x and y coordinates.

Let's look at an example.

Let f(x) = 2x + 10. The easiest way to find the inverse function is to rewrite it y = 2x +10, and solve for x

y = 2x +10  Subtract 10 from both sides.

y - 10 = 2x Divdie both sides by 2.

y/2- 10/2 = x

1/2y - 5 = x

You solve the equation for x and then swap the y and the x. So, f–1(x) = 12x - 5. Okay, that wasn’t too bad!

Now if we were to look at the graphs of the function and its inverse on the same graph, you would see an interesting thing occuring.

The red line, indicated by g, is the original function f(x) = 2x + 10 and the blue line indicated by f is the inverse f–1(x) = 1⁄2x - 5. Notice how the graph of the inverse is a reflection of the original line over the line y = x! Every inverse will be a mirror image of the original function over the line y = x!

The lesson material for this topic is found at the Khan Academy site. Click on the links below to begin.
 

 

I highly recommend that you click on the links below and watch the videos before continuing.

IF after completing this topic you can state without hesitation that...

I can determine whether or not a function has an inverse, and find the inverse when it exists.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

04.07 Inverse Functions - Assignment (Math Level 2)

teacher-scored 84 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 7 of your enrollment date for this class.


04.08 Exponential Functions (Math Level 2)

Graph the parent function of exponential functions and their transformations.

In addition to linear, quadratic, rational, and radical functions, there are exponential functions. Exponential functions have the form f(x) = bx, where b > 0 and b ≠ 1. Just as in any exponential expression, b is called the base and x is called the exponent.

Just for a quick review, lets talk about 0 and negative numbers as the exponents.

Using the quotient rule for exponents, we can define what it means to have zero as an exponent. Consider the following calculation:

{\color{Blue} 1}=\frac{25}{25}=\frac{5^{2}}{5^{2}}=5^{2-2}={\color{Blue} 5^{0}}

Twenty-five divided by twenty-five is clearly equal to one, and when the quotient rule for exponents is applied, we see that a zero exponent results. In general, given any nonzero real number x and integer n,

1=\frac{x^{n}}{x^{n}}=x^{n-n}=x^{0}

This leads us to the definition of zero as an exponent,

x^{0}=1  x\neq 0

It is important to note that 00 is indeterminate. If the base is negative, then the result is still positive one. In other words, any nonzero base raised to the zero power is defined to be equal to one.

Noting that 2^{0}=1 we can write,

{\color{Blue} \frac{1}{2^{3}}}=\frac{2^{0}}{2^{3}}=2^{0-3}={\color{Blue} 2^{-3}}

In general, given any nonzero real number x and integer n,

\frac{1}{x^{n}}=\frac{x^{0}}{x^{n}}=x^{0-n}=x^{-n},x\neq 0

This leads us to the definition of negative exponents:

x^{-n}=\frac{1}{x^{n}}, x\neq 0

An expression is completely simplified if it does not contain any negative exponents.

“Rules of Exponents and Scientific Notation”, section 1.5 from the book Advanced Algebra (v. 1.0). It is licensed under a Creative Commons by-nc-sa 3.0 license.

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2x bacteria after x hours. This can be written as f(x) = 2x.

Before you start,    f(0) = 20 = 1

After 1 hour           f(1) = 21 = 2

In 2 hours              f(2) = 22 = 4

In 3 hours              f(3) = 23 = 8

and so on.

With the definition f(x) = bx and the restrictions that b > 0 and that b ≠ 1, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows f(x) = 2x.

It is important to point out that as x approaches negative infinity, the results become very small but never actually attain zero. For example,

f(-5)=2^{-5}=\frac{1}{2^{5}}\approx 0.03125

f(-10)=2^{-10}=\frac{1}{2^{10}}\approx 0.0009766

f(-15)=2^{-15}=\frac{1}{2^{15}}\approx 0.00003052

This describes a horizontal asymptote at y=0, the x-axis, and defines a lower bound for the range of the function: (0 ,\infty ).

The base b of an exponential function affects the rate at which it grows. Below we have graphed y=2^{x}, y=3^{x}, and y=10^{x} on the same set of axes.

Note that all of these exponential functions have the same y-intercept, namely (0,1). This is because f(0)=b^{0}=1 for any function defined using the form f(x)=b^{x}. As the functions are read from left to right, they are interpreted as increasing or growing exponentially. Furthermore, any exponential function of this form will have a domain that consists of all real numbers (-\infty ,\infty ) and a range that consists of positive values (0 ,\infty ) bounded by a horizontal asymptote at y=0.

Next consider exponential functions with fractional bases 0< b< 1. For example, f(x)=(\frac{1}{2})^{x} is an exponential function with base b=\frac{1}{2}.

x    y   f(x)=(\frac{1}{2})^{x}                       {\color{Blue}solution }

-2    {\color{Blue} 4}   f(\frac{1}{2})=(\frac{1}{2})^{-2}=\frac{1^{-2}}{2^{-2}}=\frac{2^{2}}{1^{2}}=4(-2,4)

-1    {\color{Blue} 2}   f(\frac{1}{2})=(\frac{1}{2})^{-1}=\frac{1^{-1}}{2^{-1}}=\frac{2^{1}}{1^{1}}=2 (-1,2)

0    {\color{Blue} 1}   f(\frac{1}{2})=(\frac{1}{2})^{0}=1                  (0,1) 

1    {\color{Blue} \frac{1}{2}}   f(\frac{1}{2})=(\frac{1}{2})^{1}=\frac{1}{2}                 (1,\frac{1}{2})

2    {\color{Blue} \frac{1}{4}}   f(\frac{1}{2})=(\frac{1}{2})^{2}=\frac{1}{4}                 (2,\frac{1}{4})

Plotting points we have,

Reading the graph from left to right, it is interpreted as decreasing exponentially. The base affects the rate at which the exponential function decreases or decays. Below we have graphed y=(\frac{1}{2})^{x}, y=(\frac{1}{3})^{x}, and y=(\frac{1}{10})^{x} on the same set of axes.

"Exponential Functions and Their Graphs”, section 7.2 from the book Advanced Algebra (v. 1.0).  It is licensed under a Creative Commons by-nc-sa 3.0 license.

Exponential functions of the form f(x) = bx appear in different contexts, including finance and radioactive decay. The base b must be a positive number and cannot be 1. The graphs of these functions are curves that increase (from left to right) if b > 1, showing exponential growth, and decrease if 0 < b < 1, showing exponential decay.

The basic (or parent) form of the general exponential function is f(x)=bx. However, when considering transformations, the parent form is: f(x)=a(b^{x}-h)+k. Does this remind you of the quadratic parent form?

Initially, we will keep a = 1 and b = 2. We will vary the value of first k and then h.

The graph of f(x) of our initial parent function will then be f(x) = 2x. Both k and h = 0.

Now, let’s look at f(x) = 2x + 2. The value of k = 2 but h still equals 0. This function is graphed in blue. The parent function will continue to be in red.

As you can see the new graph is a vertical translation up 2 units. The y-intercept changes from (0, 1) to (0,3).

Now, let’s look at f(x) = 2x – 2. This time k = – 2 and h = 0.

As you can see the new graph is a vertical translation down 2 units. The y-intercept changed to (0,-2)

We can now summarize what we have discovered: For f(x) = bx ± k, the graph will be a vertical translation up or down k units. The y-intercept will be (0, ± k).

Now, let’s vary the exponent part of our parent function. Keeping with the pattern, we will look at the graph of f(x) = 2(x-2). This means that h = – 2 and k = 0.

The new graph is a horizontal translation right two units. Given the shift, the y-intercept was changed from (0,1) to (0, ¼). Think about that while we look at another example.

Here’s the graph of f(x) = 2(x+ 2). This means that h = 2 and k = 0.

This transformation is a horizontal translation left two units. Also, given the shift, the y-intercept also changed from (0,1) to (0,4). Hmmm.

Let’s let x = 0 in both of the transformations.

f(x) = 2x+2 f(x) = 2x-2  

 

f(0) = 22 = 4

 

f(0) = 2-2 = 1/4 or 0.25

 

 

Now you can see why the y-intercepts changed the way they did!

We can now summarize what we have discovered: For f(x) = bx±h, the graph will be a horizontal translation left or right h units. The y-intercept is (0, bh).

Here’s another transformation we must consider. Let’s consider f(x) = 6x. Before it is graphed, we could think of this as f(x) = 3∙2x to keep with the parent function we’ve been working with in our examples. In this case, a = 3, b = 2, and both k and h = 0.

Here’s the graph:

This transformation is a vertical compression of the graph of the parent function. The y-intercept (0,1) changed to (0, 3). Think of it as the red graph being pushed towards the y-axis.

Now, let’s look at (1⁄2)x. We could think of this as 1⁄4∙2x in keeping with the parent function. DO NOT SIMPLIFY THE FUNCTION!

Here’s the graph of 0.25 ∙ 2x:

This transformation is a vertical stretch of the graph of the parent function. The y-intercept is (0, 0.25). Think of the red function being pulled away from the y-axis.

We can now summarize what we have discovered: For f(x) = a∙bx, the graph will be:

  • Vertically compressed if a > 1
  • Vertically stretched if 0 > a > 1

What we have done in this case is multiply each y-coordinate of the function by b.

I’m certain you are hoping we have exhausted this topic. But you would be so wrong! Sorry!

Let’s look at f(x) = – 2x:

This transformation is a reflection about the x-axis. Nice! What if we put the negative sign in the exponent?

Let’s look at f(x) = 2–x:

This transformation is a reflection about the y-axis.

Okay, there are more that we could consider but I figure you are sick of this. The important thing is that you can graph the transformations of the parent function without having to plot a bunch of points.

Summary of Transformations of Exponential Functions

IF after completing this topic you can state without hesitation that...

  • I can graph exponential equations and functions and their transformations.
  • I can solve applied problems using exponential problems and their graphs.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

Each topic is divided into sections that include the following:

Warm Up - questions to answer to see if you are ready for the lesson.
Presentation - high quality video with excellent illustrations that teaches the topic.
Worked Examples - examples that are worked out step-by-step with narration.
Practice - quiz problems on the topic covered.
Review - practice test to check your knowledge before moving on.

You are not required to complete every section. However, REMEMBER the goal is to MASTER the material!!

I highly recommend that you click on the links below and watch the videos before continuing.

04.08 Exponential Functions – Assignment (Math Level 2)

teacher-scored 68 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


04.09 Growth and Decay Problems (Math Level 2)

Model problems of growth and decay using exponential functions.

There are many applications involving exponential functions. However, right this minute, admittedly, not very many. We will be concentrating on exponential growth and decay in this lesson.

Exponential growth examples include population growth, spread of contagious diseases, and compound interest to name a few. See, all useful, real-world experiences!

 

 

 

 

 

 

Exponential decay examples include radioactive decay, decline of HIV in a person after antiviral treatment, and population decline. Once again, all useful, real-world examples!

Exponential Growth

An equation for exponential growth is: y(t) = aekt, a > 0, where a is the initial amount, k is the growth rate, t is the time and y is the final amount.

First, we will apply this equation to population growth.

Populations tend to grow exponentially but at very different rates. In poorer countries the rates tend to be higher. In China there is population decline rather than growth.

Example 1: A small island has a population of 345 people. If the annual population growth was estimated to be 3%, what will its population be in 5 years?

y = aekt = 345e(.03)(5) = 401 rounded to the nearest person.

Example 2: The population in Iron County, Utah in 1990 was 20,789. The growth rate over the previous twenty years was 2.67%. If that rate continued, what would the predicted population be for 2010?

y(t) = aekt = 20,789e(0.026)(20) = 34,968 rounded to the nearest person.

The actual population in Iron County in 2010 was 46,163. Looks like there was a population boom!

Example 3: The population in Salt Lake County in 1900 was 77,725. By 1950, the population had increased to 274,895. The growth rate over that period was 2.53%. If this growth rate continued, what would the predicted population of Salt Lake County be in 2000?

y(t) = aekt = 274,895e(0.0253)(50) = 973,978 rounded to the nearest person. The actual population in 2000 was 898,387. Looks like the population growth slowed down during the second half of the century.

Example 4: China's one-child policy was implemented in 1978 with a goal of reducing China's population to 700 million by 2050. China's 2000 population was about 1.2 billion. Suppose that China's population declines at a rate of 0.5% per year (-0.005). Will this rate be sufficient to meet the original goal? We will cut off the millions and add them back at the end.

y = 1200000000e(-0.005)(72) = 837 million rounded to the nearest person.

The goal will not be met at that rate!

We will now move on to compound interest on money. This is a plan that pays interest on interest already earned. The value of an investment depends on the interest rate AND how frequently interest is compounded. Interest can be compounded daily, monthly, four times a year, or annually. Here’s the formula:

A = P(1 + )nt, where A is the account balance, P is the initial deposit, r is the annual interest rate, t is the number of years, and n is the number of compoundings per year.

Let’s say you put $1,000 into a regular savings account with a 5% interest rate. The formula for regular savings accounts is I = prt, where I is the interest you will earn, p is the principal or amount invested, r is the rate of interest, and t is the time period of the investment. After one year, you would earn 1000 x 0.05x1 = $50 in interest. Your account would have $1,050.

Now, let’s look at the same amounts but compound the interest annually. Therefore, n would be 12.

A = 1000(1 +  )12*1 = $1,051.15 to the nearest cent.

Again, using the same data let’s compound the interest four times a year:

A = 1000(1 +  )4*1 = $1,050.95 to the nearest cent.

Finally, there is such a thing as compounded continuously! This formula is:

A = Pert In one year, this would mean: A = 1000 e.05 = $1,051.27 to the nearest cent.

All this looks like a lot of fuss for such small differences in amounts. However, if you leave the money to grow over several years the difference is much greater.

Now, you try one on your own!

Example 4: Which investment earns more interest: $5,000 at 6% interest compounded quarterly (4 times a year) for 20 years, or $5,000 at 6% interest compounded continuously for 20 years?

Answers:

Part 1: $5,000 at 6% interest compounded quarterly (4 times a year) for 20 years. A = P(1 +  )nt = 5000(1 +  )4*20 = 5000(1.015)80 = $16,453.31 rounded to the nearest cent.

Part 2: A = Pert = 5000e(0.06)(20) = 5000e1.2 = $16,600.58 rounded to the nearest cent.

Compounded continuously earns more interest.

Exponential Decay

The formula for exponential decay is the really the same as the one for growth. The only difference is all of the rates will be negative numbers.

y = ae-kt

Example 1: Let’s suppose your friend’s dad bought a new car for $30,000. The car depreciates (decreases in value) approximately 15% of its value each year (decay rate). What will the car be worth in 8 years when he plans on turning it in for a new car?

y = ae-kt = $30,000e(-.15)(8) = $9035.83 rounded to the nearest cent.

Example 2: If a person takes 25 milligrams of a drug, whose concentration decreases by 25% each hour, what is the approximate concentration of the drug in his bloodstream after 3 hours?

y = ae-kt = 25e(-0.25)(3) = 10 rounded to the nearest milligram.

Okay, I’m done – I hope you are as well! This means, of course, you understand it, not that you don’t want to do any more. ☺

There are two online calculators that can help you with this section. Use them to CHECK your answers only – since you won’t be able to use them on the final!!!

Take time to view the two Khan Academy lesson videos that follow to get a feel for exponential growth and decay functions.

Finally, click on the link that follows to practice solving exponential growth and decay problems.

04.09 Growth and Decay Problems Supplemental Links (Math Level 2)

04.09.01 Growth and Decay Problems Videos (Math Level 2)

I highly recommend that you click on the links below and watch the videos after completing the topic.
 

04.09.02 Growth and Decay Problems, Videos (Math Level 2)

04.09.03 Growth and Decay Problems – Assignment (Math Level 2)

teacher-scored 50 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


04.10 Average Rate of Change (Math Level 2)

Find the average rate of change of a function over a given interval.

Let’s begin by reviewing the differences between a linear and exponential function:

Slope is also a term you should already know. Here is the formula:

m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}

This formula works well for the linear function above.

Let’s pick two points on our linear function example: (0, 0) and (2, 4). Now, we will find the slope:

m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-0}{2-0}=2

The slope is the same no matter which two points you choose to use. How will we find the slope of the second function? There is another formula that will work for it:

Average rate of change:

\overline{m}=\frac{f(b)-f(a)}{b-a}

where [a, b] marks the interval selected.

Look at the graph below:

In this case, a = 0, b = 2, f(a) = 1, and f(b) = 4.

\overline{m}=\frac{f(b)-f(a)}{b-a}=\frac{4-1}{2-0}=\frac{3}{2}

This formula allows you could find the “slope” or average rate of change over a given interval for any function, linear or not.

The next part of the lesson is taken from Rates of Change and Behavior of Graphs from the UC Davis MathWiki a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

Since functions represent how an output quantity varies with an input quantity, it is natural to ask about the rate at which the values of the function are changing. 

For example, the function C(t) below gives the average cost, in dollars, of a gallon of gasoline t years after 2000.

If we were interested in how the gas prices had changed between 2002 and 2009, we could compute that the cost per gallon had increased from $1.47 to $2.14, an increase of  $0.67.  While this is interesting, it might be more useful to look at how much the price changed per year.  You are probably noticing that the price didn’t change the same amount each year, so we would be finding the average rate of change over a specified amount of time.

The gas price increased by $0.67 from 2002 to 2009, over 7 years, for an average of 0.0957 dollars per year. On average, the price of gas increased by about 9.6 cents each year.

Rate of Change

A rate of change describes how the output quantity changes in relation to the input quantity.  The units on a rate of change are “output units per input units.

Some other examples of rates of change would be quantities like:

  • A population of rats increases by 40 rats per week
  • A barista earns $9 per hour (dollars per hour)
  • A farmer plants 60,000 onions per acre
  • A car can drive 27 miles per gallon
  • A population of grey whales decreases by 8 whales per year
  • The amount of money in your college account decreases by $4,000 per quarter

Example 1

Using the cost-of-gas function from earlier, find the average rate of change between 2007 and 2009.

From the table, in 2007 the cost of gas was $2.64.  In 2009 the cost was $2.14.

The average rate of change is:

IF after completing this topic you can state without hesitation that...

  • I can find the average rate of change of a function over a given interval.

…you are ready for the assignment! Otherwise, go back and review the material before moving on.

I highly recommend that you click on the links below and watch the videos before continuing.

04.10 Average Rate of Change – Assignment (Math Level 2)

teacher-scored 65 points possible 40 minutes

Activity for this lesson

Complete the attached worksheet.

  1. Print the worksheet and complete the assignment in the space provided. You may use additional paper if needed. Work all the problems showing ALL your steps.
  2. Once you have completed the assignment, digitize (scan or take digital photo, up close and clear) and save it to the computer and convert it to an image file such as .pdf or .jpg.
  3. Finally, upload the image using the assignment submission window under the assignment link on your math home page for this assignment.

Pacing: complete this by the end of Week 8 of your enrollment date for this class.


04.10 Unit 4 Review Quiz (Math Level 2)

computer-scored 77 points possible 40 minutes
Pacing: complete this by the end of Week 8 of your enrollment date for this class.