Engaging Mathematics: TEKS-Based Activities Algebra II ... · Solving Quadratics ... Engaging Mathematics: ... What strategies could you use to help you determine the solution to

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TEKS-BasedTEKS-BasedActivitiesActivities

Algebra IIAlgebra II

ISBN: 978-1-934950-67-8

www.theansweris4.net

Engaging Mathematics:TEKS-Based Activities

Algebra II

Teacher Edition

Product ID: 407-1603

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educational excellence®

Revolutionizing education to inspire and advance future

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Table of Contents

Introduction ......................................................................................................... i–vii Overview of Materials ............................................................................................ i Texas Essential Knowledge and Skills (TEKS) Alignment Chart .......................... iv

Foundations for Functions ............................................................................... 1–27 Functions .............................................................................................................. 2 Using the Calculator to Evaluate Expressions ...................................................... 4 Discrete and Continuous Functions ...................................................................... 6 Domain and Range, Activity 1 .............................................................................. 8 Domain and Range, Activity 2 ............................................................................ 10 Views of a Function ............................................................................................ 14 Interpreting Scatterplots ..................................................................................... 18 Transformations .................................................................................................. 20 Identifying Linear, Absolute Value, and Quadratic Functions ............................. 22 Transformations of Parent Functions .................................................................. 24

Linear Functions ............................................................................................ 28–103 Graphing Linear Functions ................................................................................. 28 Finding the Equation of a Line ............................................................................ 30 Solving Equations ............................................................................................... 34 Standard Form and Slope-Intercept Form .......................................................... 36 Standard Form of an Equation ............................................................................ 40 Standard Form and Slope-Intercept Equations Defined ..................................... 42 Graphing Functions ............................................................................................ 44 Writing Equations of Lines .................................................................................. 46 Slope and y-Intercept ......................................................................................... 48 Linear Inequalities, Activity 1 .............................................................................. 50 Linear Inequalities, Activity 2 .............................................................................. 52 Reflections .......................................................................................................... 54 Absolute Value Function ..................................................................................... 56 Graphing Absolute Value Functions ................................................................... 60 Linear and Absolute Value Graphs and Equations ............................................. 62 Problem Solving ................................................................................................. 66 Introduction to Linear Systems of Equations ...................................................... 68 Attributes of Linear Functions ............................................................................. 70 Systems of Equations, Activity 1 ......................................................................... 72 Systems of Equations, Activity 2 ......................................................................... 74 Systems of Linear Equations .............................................................................. 78 Systems of Equations Concept Map ................................................................... 80 Matrices, Activity 1 .............................................................................................. 82 Matrices, Activity 2 .............................................................................................. 84 Writing Equations ............................................................................................... 86 Matrices, Activity 3 .............................................................................................. 88 Linear Inequalities, Activity 1 .............................................................................. 92 Linear Inequalities, Activity 2 .............................................................................. 96 Systems of Linear Inequalities .......................................................................... 100

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Quadratic and Square Root Functions ....................................................... 104–189 Quadratics ........................................................................................................ 104 Modeling Quadratics ......................................................................................... 106 Identifying Quadratic Functions ........................................................................ 108 Graphs of Quadratic Functions ......................................................................... 112 Definition of a Quadratic Function ..................................................................... 114 Transformations on Quadratic Functions .......................................................... 116 Quadratic Transformations ............................................................................... 118 Graphing Quadratic Equations .......................................................................... 120 Applying Graphs to Quadratics ......................................................................... 122 Quadratic Functions .......................................................................................... 124 Quadratic Functions—Vertex Form ................................................................... 126 Quadratic Equations, Activity 1 ......................................................................... 128 Quadratic Equations, Activity 2 ......................................................................... 130 Quadratic Inequalities ....................................................................................... 132 Factoring Quadratics ......................................................................................... 134 Solving Quadratics—Inverse Operations .......................................................... 138 Solving Quadratics—Factoring ......................................................................... 142 Quadratic Formula ............................................................................................ 144 Solving Quadratics—Quadratic Formula ........................................................... 146 Linear and Quadratic Graphs ............................................................................ 148 Solving Quadratics ............................................................................................ 150 Solving Quadratic Equations ............................................................................. 152 Quadratic Functions .......................................................................................... 156 Using the Discriminant, Activity 1 ...................................................................... 160 Using the Discriminant, Activity 2 ...................................................................... 162 Real and Imaginary Roots ................................................................................ 164 Adding Complex Numbers ................................................................................ 168 Introduction to Inverse Functions ...................................................................... 170 Inverse Functions .............................................................................................. 172 Simplifying Radicals .......................................................................................... 174 Square Root Functions ..................................................................................... 176 Graphing Square Root Functions, Activity 1 ..................................................... 180 Graphing Square Root Functions, Activity 2 ..................................................... 182 Square Root Activity ......................................................................................... 184 Square Root Equations ..................................................................................... 186 Square Root Inequalities ................................................................................... 188

Exponential and Logarithmic Functions .................................................... 190–225 Exponent Properties, Activity 1 ......................................................................... 190 Exponent Properties, Activity 2 ......................................................................... 192 Rational Exponents ........................................................................................... 196 Rewriting Exponential Expressions ................................................................... 198 Tables of Exponential Functions ....................................................................... 200 Logarithmic Activity ........................................................................................... 202 Introduction to Exponential Functions ............................................................... 206 Writing Exponential Functions .......................................................................... 208

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Graphs of Exponential Functions ...................................................................... 210 Exponential Functions ...................................................................................... 214 Parent Functions .............................................................................................. 216 Graphing Transformations of the Logarithmic Parent Function ........................ 218 Exponential and Logarithmic Graphs ................................................................ 220 Applications of Exponential Functions .............................................................. 222 Exponential Graphs .......................................................................................... 224

Rational Functions ....................................................................................... 226–259 Rational Functions, Activity 1............................................................................ 226 Rational Functions, Activity 2............................................................................ 228 Rational Functions, Activity 3............................................................................ 230 Direct Variation and Inverse Variation .............................................................. 232 Variation ........................................................................................................... 234 Asymptotes of Rational Functions .................................................................... 236 Graphing Rational Functions ............................................................................ 238 Identifying Vertical Asymptotes of Rational Functions ...................................... 242 Writing Equations for Rational Functions, Activity 1 ......................................... 244 Writing Equations for Rational Functions, Activity 2 ......................................... 246 Rational Function Activity ................................................................................. 248 Solving Rational Equations ............................................................................... 250 Lines of Symmetry ............................................................................................ 252 Completing the Square ..................................................................................... 256

Conic Sections ............................................................................................. 260–283 Domain and Range of Conic Sections .............................................................. 260 Intersection of a Plane and a Pyramid .............................................................. 262 The Ellipse ........................................................................................................ 264 The Hyperbola .................................................................................................. 268 The Parabola .................................................................................................... 272 Graphing Ellipses ............................................................................................. 276 Graphing Conic Sections with a Graphing Calculator ....................................... 278 Summary of Conic Sections ............................................................................. 282 SAMPLE

Engaging Mathematics: Algebra II TEKS-Based Activities

© Region 4 Education Service Center All rights reserved.

Activity Objective The student will match systems of equations with their solutions.

Materials Systems of Equations Loop Systems of Equations Cards—one set per student Tape or glue Scissors

Facilitation Questions What strategies could you use to help you determine the solution to the puzzle?

I could solve the equations in terms of y and use the calculator to graph the system and find the point of intersection.

How could you verify that the ordered pair is a solution?

I could substitute the x and y values into the equations that represent the system and see if they create a true statement.

Answers

Systems of Equations

Solution

x y

x y

6

2 (4, 2)

y x

x y

2

3 5 (5, 10)

x y

x y

3

3 6 9 (3, 0)

y x

y x

2

2 1

3 2

(3, 1)

x y

x y

8 2 4

2 4 (–1, 2)

Systems of Equations

Solution

y x

y x

2

31

2

(–3, –5)

y x

6x 8y 12

6 4 6 (6, –3)

x y

y x

3 10

2 9 5 (–1, 7)

3 2

8

y x

y x (2.5, 5.5)

Systems of Equations, Activity 2

Allow

students to cut out the

cards.

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Student Name: ________________________________________ Date: ________________

Systems of Equations Loop 1. Cut apart the Systems of Equations Cards. 2. Determine which systems of equations match the solution. 3. Tape the bottom of the card that contains the systems of equations to the top of the card

that contains the solution. 4. Continue this process for the remaining systems of equations. 5. When complete, the taped cards should form a loop.

Communicating About Mathematics Explain the strategy you used to determine your card matches.

My Workspace:

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Systems of Equations Cards Cut along the dotted lines. Do not cut along the solid lines.

6

2

x y

x y

2

3 5

y x

x y

3

3 6 9

x y

x y

(–3, –5) (6, –3) (3, 1)

2

3 1

2 2

y x

y x

8 2 4

2 4

x y

x y

3 2

8

y x

y x

(–1, 2) (5, 10) (4, 2)

3 10

2 9 5

x y

y x

6 8 12

6 4 6

x y

y x

2

21

3

y x

y x

(2.5, 5.5) (3, 0) (–1, 7)

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Activity Objective The student will use clues to determine the size and elements of a matrix.

Materials Mystery Matrix

Facilitation Questions How can you distinguish between rows and columns?

The rows of a matrix are horizontal, and the columns of a matrix are vertical.

What is an element of a matrix? An element is a value within a matrix.

Answer

2 3 4

3 8 3

5 4 3

Matrices, Activity 2

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Mystery Matrix Use the clues to determine the size and elements of a matrix. Clues

I am a matrix containing only whole numbers less than or equal to 8. The number of rows is equal to the number of columns. I have a total of 9 elements. The values in the first row are consecutive numbers. The value in the third row, first column is 5. The value in the second row, second column is 8. The sum of the elements in the second row is one less than the sum of the elements in

the second column. The elements of the third row are in descending order. The sum of the elements in the first column is a multiple of 10. The sum of the elements in the second column is 15.

Communicating About Mathematics If a matrix has a total of 20 elements, describe the possible sizes the matrix could be. Justify your answer. _______________________________________________________________

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Activity Objective The student will be able to match a quadratic function with its graph and zeros.

Materials Where Are the Zeros? Zero Cards Tape or glue Scissors Graphing calculator

Facilitation Questions How could you use the equation to find the zeros?

I could factor the equation, set each factor equal to zero, and find the solutions.

How could you use the graph to find the zeros? I can use the graph to locate the x-intercepts.

How could you use a graphing calculator to help you? I could enter the function into the equation editor and look at the graph for the x-intercepts or use the table feature to find the x values for when y = 0.

Answers

Zeros at –5 and 5 Zeros at 1 and 3 Zero at 6

Quadratic Functions

f x x2( )= 25 f x x x2( )= + 4 3 f x x x2( )= 12 +36

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Where Are the Zeros?

Cut apart the Zero Cards. Determine where to place each of the cards in order to match the function, graph and zeros. Not all cards will be used. Attach the cards when you have found the correct solution.

Function Graph Zeros

2( ) 25f x x

6

Communicating About Mathematics Describe how you could use the factored form of 2( ) 4 3f x x x to find the zeros.

_______________________________________________________________

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Zero Cards Cut along the bold dotted lines.

2( ) 4 3f x x x

1, 3

2( ) 12 36f x x x

–3, 5

–5, 5

2( ) 10f x x

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Activity Objective The student will analyze values near the vertical asymptote of a rational function from a table of data.

Materials Zooming In Graphing calculator

Facilitation Questions Do you need to use parentheses when entering the function into the graphing

calculator? Why or why not? Since the denominator contains an expression, I will need to use parentheses around the expression so the calculator knows the entire expression is the denominator.

Answers

x y

–4.06 –33.33

–4.05 –40

–4.04 –50

–4.03 –66.67

–4.02 –100

x y

0 3

1 ERROR

2 –3

3 –1.5

4 –1

As the x-values approach –4 from the left, the y-values decrease greatly. As the x-values approach –4 from the right, the y-values increase greatly.

As x approaches 1 from the left, the y-values increase greatly. As x approaches 1 from the right, the y-values decrease greatly.

Identifying Vertical Asymptotes of Rational Functions

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Zooming In Graph each function in your calculator. Then complete the tables of data for each function.

1 2

2 3Y = Y =

+4 1x x

Describe what happens to the Describe what happens to the y-values as x approaches –4. y-values as x approaches 1. Communicating About Mathematics Explain why you had error messages for some values in the tables above.

x y

–4.06

–4.05

–4.04

–4.03

–4.02

x y

0

1

2

3

4

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Engaging Mathematics: TEKS-Based Activities Algebra II ... · Solving Quadratics ... Engaging Mathematics: ... What strategies could you use to help you determine the solution to