Multiple Pathways To Success

Quarter 1 Learning Module

Algebra 2

Unit 1: Quadratic Functions

and Radical Equations

Copyright July 31, 2014 – Drafted November 5, 2015

Prince George’s County Public Schools

Board of Education of Prince George’s County, Maryland

Dear Scholars, As you move through the Algebra II curriculum, the level of academic rigor will increase. This could potentially lead to gaps in your understanding. Therefore, this learning module was designed to assist you in acquiring and strengthening the essential skills needed for successful completion of Algebra II Common Core. Your experiences with this module will also help to remediate misconceptions, confusion, and rebuild areas of weakness. Sincerely, Writers of the Multiple Pathways to Success Modules

Part I: Solving Systems of Equations

Student Learning Objectives · Solve a simple system containing a linear equation and a quadratic equation in two

variables (conic sections possible) graphically and symbolically. · Write and solve 3-variable systems of linear equations to solve real-world scenarios. · Solve systems of equations comprised of a linear and a quadratic equation algebraically. · Solve systems of equations comprised of a linear and a quadratic equation graphically. · Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any

combination of linear, quadratic, rational, absolute value, exponential, and logarithmic functions. · Find solution(s) by using technology to graph the equations and determine their point of

intersection. · Find solution(s) by using tables of values or successive approximations that become

closer and closer to the actual value. Mathematical Practices · MP1: Make Sense of a Problem and Persevere · MP2: Reason Abstractly and Quantitatively · MP3: Construct a viable argument and critique the reasoning of others · MP4: Model with Mathematics · MP5: Use tools strategically · MP6: Attend to Precision · MP7: Make Use of Structure Resources/Websites · Before you engage in the Algebra II experience, it is extremely important that you

re-familiarize yourself with the concepts of Solving Systems of Linear Equations and Inequalities with graphing, substitution, and eliminations methods. Please refer to textbook pages 134 – 153 for assistance. · The YouTube Video “System of 3 Equations, 3 Unknowns Using Elimination – Ex 2” will

demonstrate how to apply the Elimination Method to solve 3 variable systems of linear equations and how to express the solutions as ordered triples. (https://www.youtube.com/watch?v=JZDQRJVxw30&spfreload=10) · The next two pages contain examples of how to solve non-linear systems of equations

that involve linear equations and circles and linear equations and parabolas. (http://www.purplemath.com/modules/syseqgen4.htm)

Solving Non-Linear Systems of Equations with LInear Equations and Circles

In general, the method of solution for general systems of equations is to solve one of the equations (you choose which) for one of the variables (again, you choose which). Then you plug the resulting expression into the other equation for the chosen variable, and solve for the values of the other variable. Then you plug those solutions back into the first equation, and solve for the values of the first variable. Here are some examples. Copyright © 2002-2011

Elizabeth Stapel All Rights Reserved Example #1: Solve the system: y = –x – 3

x2 + y2 = 17

Graphically, this system is a straight line crossing a circle centered at the origin:

○ There appear to be two solutions. We will proceed algebraically to confirm this impression, and to get the exact values. Since the first equation is already solved for y, plug "–x – 3" in for "y" in the second equation, and solve for the values of x:

x2 + y2 = 17 x2 + (–x – 3)2 = 17

x2 + (–x – 3)(–x – 3) = 17 x2 + (x2 + 6x + 9) = 17 2x2 + 6x + 9 = 17 2x2 + 6x – 8 = 0 x2 + 3x – 4 = 0 (x + 4)(x – 1) = 0 x = –4, x = 1

So, when x = –4, y = –x – 3 = –(–4) – 3 = 4 – 3 = 1. When x = 1, y = –x – 3 = –(1) – 3 = –4.

Therefore, the solution consists of the points (–4, 1) and (1, –4).

*Note the procedure: I solved one of the equations (the first equation looked easier) for one of the variables (solving for "y=" looked easier), and then plugged the resulting expression back into the other equation. This gave me one equation in one variable (the variable happened to be x), and a one-variable equation is something I know how to solve. Once I had the solution values for x, I back-solved for the corresponding y-values. I emphasize "corresponding" because you have to keep track of which y-value goes with which x-value. In the example above, the points (–4, –4) and (1, 1) are not solutions. Even though I came up with x = –4 and 1 and y = –4 and 1, the x = –4 did not go with the y = –4, and the x = 1 did not go with the y = 1. Warning: You must match the x-values and y-values correctly!

Solving Non-Linear Systems of Equations with LInear Equations and Parabolas Example #2: Solve the system: y = (1/2 )x – 5

y = x2 + 2x – 15 Since both equations are already solved for y, I'll set them equal and solve for the values of x:

(1/2)x – 5 = x2 + 2x – 15 x – 10 = 2x2 + 4x – 30

0 = 2x2 + 3x – 20 0 = (2x – 5)(x + 4)

x = 5/2, x = –4

When x = 5/2: y = (1/2)x – 5 = (1/2)(5/2) – 5 = 5/4 – 20/4 = – 15/4 = –3.75

When x = –4: y = (1/2)x – 5 = (1/2)(–4) – 5 = –2 – 5 = –7

Therefore, the solutions are the points ( 5/2, –15/4 ) and (–4, –7).

Graphically, the above system looks like this:

1. Solve the system below in the space provided. y = 5x + 1 x2 + y2 = 9

2. Mike is taking the Accuplacer and has 2 hours to complete the test. It takes him 5 minutes to complete multiple-choice questions, 5 minutes to complete calculation questions and 10 minutes to complete word problems. He must complete 20 questions and as many as multiple-choice questions as calculation questions and word problems combined. Write and solve a 3-variable system of equations to determine how many of each question type he should complete.

3. Let f(x) = ax2 where a > 0, and let g(x) = mx + b where m > 0 and b < 0. The equation f(x) = g(x) has n district real solution(s). What are all the possible values of n? Justify your answers. Place your answers and your justification in the space provided below.

Scoring Rubric/Success Criteria

Question #1: You will receive 2 points* for appropriately graphing the given linear equation. You will receive 1 point for correctly identifying the slope and the y-intercept. You will receive 2 points* for accurately graphing the given circle. You will receive 1 point for correctly identifying the center and the radius. You will receive 1 point for precisely identifying the number of times both graphs intersected. You will receive 8 points* for accurately solving the system points are broken down below. You will receive 1 point for substituting the linear equation into the circle. You will receive 1 point for correctly simplifying and getting 0 on one side. You will receive 1 point for using the quadratic formula correctly. You will receive 1 point for simplifying the expression.

You will receive 1 point for back substituting into the linear equation. You will receive 1 point for simplifying the expression. You will receive 2 points for each solution point. Question #2: You will receive 3 points* for writing a 3-variable system of equations that accurately reflects the given scenario. You will receive 1 point for appropriately defining all unknown variables. You will receive 3 points* for properly applying the Elimination Method to determine how many of each question type should be completed. You will receive 1 point for explaining what the solution means within the context of the problem. Question #3: You will receive 1 point for every properly identified value of n. You will receive 1 point for every justification that supports why its respective solution is a possible value of n. * = Partial credit is available.

Part II: Solving Quadratic Equations With Complex Roots

Student Learning Objectives · · Understand complex numbers and how they relate to quadratic functions. · Know that every number is a complex number of the form , where and are real numbers. · Identify, graph, and perform operations with complex numbers. · Know the definition of the imaginary unit is and is used to express the square root of a negative number. · Write complex numbers in form and identify the real and imaginary parts of the number.

· Use the associative, commutative, and distributive properties to add, subtract, and multiply complex numbers. · Identify, graph, and perform operations with complex numbers. · Know the definition of the imaginary unit is and is used to express the square root of a negative number. · Find complex number solutions of quadratic equations. · Solve quadratic equations with real coefficients that have solutions of the form . · Find complex number solutions of quadratic equations. · Identify, graph, and perform operations with complex numbers. Mathematical Practices · MP1: Make Sense of a Problem and Persevere · MP2: Reason Abstractly and Quantitatively · MP3: Construct a viable argument and critique the reasoning of others · MP4: Model with Mathematics · MP5: Use tools strategically · MP6: Attend to Precision · MP7: Make Use of Structure Resources/Websites · The Virtual Nerd Videos “Complex Numbers” will demonstrate how to add, subtract, and multiply complex numbers.

(http://www.virtualnerd.com/algebra-2/quadratics/complex-numbers) · The Powerpoint will demonstrate Complex Numbers.

(https://drive.google.com/open?id=0BwV-rRDNmsGld3NQN3hiOUZmSzg)

1) Complete the proof.

a. ______________________

b. ______________________

c. ______________________

2) Comparing Methods Describe the two different methods shown for writing the complex expression in

standard form. Which method do you prefer? Explain. Method 1

𝟒𝒊(𝟐   −  𝟑𝒊)  +  𝟒𝒊(𝟏   −  𝟐𝒊)  =  𝟖𝒊   −  𝟏𝟐𝒊𝟐  +  𝟒𝒊   −  𝟖𝒊𝟐                                                                                    =  𝟖𝒊   −  𝟏𝟐(−𝟏)  +  𝟒𝒊   −  𝟖(−𝟏) =  𝟐𝟎   +  𝟏𝟐𝒊

Method 2

𝟒𝒊(𝟐   −  𝟑𝒊)  +  𝟒𝒊(𝟏   −  𝟐𝒊)  =  𝟒𝒊[(𝟐   −  𝟑𝒊)  +  (𝟏   −  𝟐𝒊)]

                                                                                   =  𝟒𝒊[𝟑   −  𝟓𝒊] =  𝟏𝟐𝒊   −  𝟐𝟎  𝒊𝟐

                                                                                                                   =  𝟏𝟐𝒊   −  𝟐𝟎(−𝟏)

                                                                                     =  𝟐𝟎   +  𝟏𝟐𝒊

3)

Explain Lily's error and correct the step, which contains her error. 4) An equation is shown below. (𝒂   +  𝒃𝒊)(𝟒   −  𝟐𝒊)  =  𝟒𝟎 What is the value of 𝒃?

5) Solve the equations and express their roots in standard form.

a) 2𝑥!  =  −3(2𝑥  +  3)

b) 3𝑥!  +  3   =  5𝑥

Scoring Rubric/Success Criteria

Question #1: You will receive a total of 3 points. You will receive 1 point for each correct property. Question #2: You will receive 2 points* for describing the two different methods shown. You will receive 2 points* for the method you prefer and your explanation. Question #3: You will receive a total of 2 points* for explaining the error and correcting the steps. Question #4: You will receive a total of 2 points*. You will receive 1 point for doing the Distributive Property or FOIL Method. You will receive 1 point for solving for b. Question #5:

a. You will receive 5 points* for solving the equation (3 points) and express their roots in standard form (2 points).

b. You will receive 5 points* for solving the equation (3 points) and express their roots in standard form (2 points).

* = Partial credit is available.

Part III: Solving Radical Equations

Student Learning Objectives ● Rewrite expressions involving radicals and rational exponents using the properties of

exponents. ● Relate fractional exponents to integer exponents. ● Perform operations and solve variables and equations with rational exponents. ● Simplify radical expressions by factoring. ● Estimate the value of square roots.

Mathematical Practices · MP1: Make Sense of a Problem and Persevere · MP2: Reason Abstractly and Quantitatively · MP3: Construct a viable argument and critique the reasoning of others · MP4: Model with Mathematics · MP5: Use tools strategically · MP6: Attend to Precision · MP7: Make Use of Structure Resources/Websites ● The following notes on purplemath show how to solve radical

equations.http://www.purplemath.com/modules/solverad.htm ● Visit Coolmath for a review of the properties of

exponents.http://www.coolmath.com/algebra/01-exponents ● Visit this Khan Academy Playlist for videos that pertain to radical equations.

https://www.khanacademy.org/math/algebra2/radical-equations-and-functions

1) A biology student is studying bacterial growth. She was surprised to find that the population of the bacteria doubled every hour.

a. Complete the following table and plot the data.

b. Write an equation for �, the population of the bacteria, as a function of time,  �, and verify that it produces correct populations for �   =  1, 2, 3,and 4.

c. The student conducting the study wants to create a table with more entries; specifically, she wants to fill in the population at each half hour. However, she forgot to make these measurements so she wants to estimate the values. Instead she notes that the population increases by the same factor each hour, and reasons that this property should hold over each half-hour interval as well. Fill in the part of the below table that you've already computed, and decide what constant factor she should multiply the population by each half hour in order to produce consistent results. Use this multiplier to complete the table:

d. What if the student wanted to estimate the population every 20minutes instead of every 30 minutes. What multiplier would be necessary to be consistent with the population doubling every hour? Use this multiplier to complete the following table:

e. Use the population context to explain why it makes sense that we define 21/2to be 2 and 21/2as 2� .

f. Another student working on the bacteria population problem makes the following claim:

Comment on this idea. How does it compare to the multipliers generated in the previous problems? For what kind of function would this reasoning work?

Question part a: You will receive a total of 2 points. You will receive 1 point for each correct property. Question part b: You will receive 2 points* , one point for the correct equation and one point for verification of all 4 points. Question part c: You will receive a total of 1 point* for completion of the table. Question part d: You will receive a total of 2 points*. You will receive 1 point for the correct multiplier. You will receive 1 point for correctly completing the chart. Question part e: You will receive 2 points* for stating your position and providing evidence that it makes sense. Question part f: You will receive 3 points* . You will receive 1 point for an insightful comment. You will receive 1 point for the comparison. You will receive 1 point for the identification of the function. * = Partial credit is available.