Common Core State Standards
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Common Core State Standards
W
ith American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. — Common Core State Standards Initiative
What is the goal of the Common Core State Standards? The mission of the Common Core State Standards is to provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that students need for success in college and careers.
Who wrote the standards? The National Governors Association Center for Best Practices and the Council of Chief State School Officers worked with representatives from participating states, a wide range of educators, content experts, researchers, national organizations, and community groups.
uadratic Functions Functions uadratic nd Relations Relations nd
At the high school level the Common Core State Standards are organized by conceptual category. To ease implementation four model course pathways were created: traditional, integrated, accelerated traditional and accelerated integrated. Glencoe Algebra 1, Glencoe Geometry, and Glencoe Algebra 2 follow the traditional pathway.
What are the major points of the standards? The standards seek to develop both students’ mathematical understanding and their procedural skill. The Standards for Mathematical Practice describe varieties of expertise that mathematics teachers at all levels should seek to develop in their students. The Standards for Mathematical Content define what students should understand and be able to do at each level in their study of mathematics.
Mathematical Content Unit 2
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Quadratic, Polynomial, and Radical FFunctions unctio and Relations
Quadratic Quadr Q uadrrat Functions Functions Quadratic and and nd Relations R Re Relations el and
A.SSE.1.a, F.IF.9 F IF 4
4-1 Graphing Quadratic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Extend: Graphing Technology Lab Modeling Real-World Data . . . . . . . . . . . . . . . . . . . 228
4-2 Solving Quadratic Equations by Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Extend: Graphing Technology Lab Solving Quadratic Equations by Graphing . . . . . . . . 237
How do I implement the standards?
A.SSE.1.a, F.IF.9 F.IF.4 A.CED.2, F.IF.4 A.REI.11
4-3 Solving Quadratic Equations by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.SSE.2, F.IF.8.a
4-4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
N.CN.1, N.CN.2
Extend: Algebra Lab The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4-5 Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Extend: Graphing Technology Lab Solving Quadratic Equations. . . . . . . . . . . . . . . . . . 263
4-6 The Quadratic Formula and the Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . 264 Explore: Graphing Technology Lab Families of Parabolas . . . . . . . . . . . . . . . . . . . . . . 273
4-7 Transformations of Quadratic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Extend: Algebra Lab Quadratics and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . 281
4-8 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Black 100/Allsport Concepts/Getty Images
The Common Core State Standards are shared goals and expectations for what knowledge and skills your students need to succeed. You as a teacher, in partnership with your colleagues, principals, superintendents, decide how the standards are to be met. Glencoe Algebra 2 is designed to help you devise lesson plans and tailor instruction to the individual needs of the students in your classroom as you meet the Common Core State Standards.
Mathematical Content
Get Ready for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
N.CN.7, F.IF.8.a N.CN.7 N.CN.7, A.SSE.1.b F.IF.4, F.BF.3 F.IF.8.a, F.BF.3 F.IF.4, F.IF.6 A.CED.1, A.CED.3
Extend: Graphing Technology Lab Modeling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Assessment Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Preparing for Standardized Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Standardized Test Practice, Chapters 1–4 . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Virtual Manipulatives
Graphing Calculator
pp. 137, 219
pp. 205, 237
Foldables pp. 134, 218
Self-Check Practice pp. 183, 295
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Domain Names
How do I decode the standards? This diagram provides clarity for decoding the standard identifiers.
A.REI.2 Conceptual Category N = Number and Quantity A = Algebra F = Functions S = Statistics and Probability
Domain Standard
Graphing Technology Lab coverage in Glencoe Algebra 2 ensures Complete standards
Modeling Data that you have allUsing the content you need to teach the standards, Polynomial Functions C Content Standards F.IF.4 For a function that models a relations Extend and tables in5-4terms of the quantities, and sk Graphing Technology Lab of the relationship. Modeling Data Using F.IF.7.c Graph polynomial functions, identify 1 Focus Polynomial Functions showing Objective end behavior. Use a graphing calculator to You can use a TI-83/84 Plus graphing calculator model data whose curve of best fit is a to model data dat points when a curve of best fit is a Mathematical Practices polynomial function. fu polynomial function. 5 Use appropriate tools strategically.
SSE
Arithmetic with Polynomials and Rational Expressions
APR
Creating Equations
CED
Reasoning with Equations and Inequalities
REI
Interpreting Functions
IF
Building Functions
BF
Linear, Quadratic, and Exponential Models
LE
Trigonometric Functions
TF
Interpreting Categorical and Quantitative Data
ID
Making Inferences and Justifying Conclusions
IC
Correlations that show at a glance where each standard is addressed in Glencoe Algebra 2.
You can also visit connectED.mcgraw-hill.com to learn more about the Common Core State Standards. There you can choose from an extensive collection of resources to use when planning instruction.
Lesson 5-7
Lesson 5-7 Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.
Example The table shows the distance a seismic wave produced by an earthquake travels from the epicenter. Draw a scatter plot and a curve of best fit to show how the distance is related to time. Then determine approximately how far away from the epicenter a seismic wave will be felt 8.5 minutes after an earthquake occurs.
Travel Time (min)
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Distance (km) 400 800 2500 3900 6250 8400 10,000
800
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[STAT PLOT] 1
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Put students in pairs, mixing abilities. Then have pairs complete all steps in the Example.
Practice Have students complete Exercises 1–3.
5
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screen. If rounded, the regression The equation is shown in the equation shown on the calculator can be written as the algebraic equation y = 0.7x 4 - 17x 3 + 161x 2 - 21x + 293.
Step 4 Use the [CALC] feature to find the value of the function for x = 8.5. KEYSTROKES:
ConceptSummary Zeros, Factors, Roots, and Intercepts Let P (x ) = a nx n + … + a 1x + a 0 be a polynomial function. Then the following statements are equivalent.
Words
• c is a zero of P (x ). • c is a root or solution of P (x ) = 0. • x - c is a factor of a nx n + … + a 1x + a 0. • If c is a real number, then (c, 0) is an x-intercept of the graph of P (x ). Consider the polynomial function P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12. The zeros of P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12 are -3, -2, 1, and 2.
P(x) 12
The roots of x 4 + 2x 3 - 7x 2 - 8x + 12 = 0 are -3, -2, 1, and 2. 4
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8
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The factors of x + 2x - 7x - 8x + 12 are (x + 3), (x + 2), (x - 1), and (x - 2).
4 −4
The x-intercepts of the graph of P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12 are (-3, 0), (-2, 0), (1, 0), and (2, 0).
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4x
When solving a polynomial equation with degree greater than zero, there may be one or more real roots or no real roots (the roots are imaginary numbers). Since real numbers and imaginary numbers both belong to the set of complex numbers, all polynomial equations with degree greater than zero will have at least one root in the set of complex numbers. This is the Fundamental Theorem of Algebra.
[CALC] 1 8.5
The table gives the distance for 7 minutes as 3900 and the distance for 10 minutes as 6250. Since 8.5 is halfway between 7 and 10 a reasonable estimate for the distance is halfway between 3900 and 6250.
KeyConcept Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. (continued on the next page)
338 | Extend 5-4 | Graphing Technology Lab: Modeling Data Using Polynomial Functions
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Extension The nature of polynomial functions makes them perfect for inventing funny stories about how things change over time. Challenge students to create stories about something that is growing or shrinking (or both) and display the growth graphically. Extend the problem by having them create scales for the graphs that match the context of their stories.
Resource Teacher Edition
Approaching Level AL
Chapter Resource Masters
Study Guide and Intervention,pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45
338 | Extend 5-4 | Graphing Technology Lab: Modeling Data Using Polynomial Functions
On Level OL358
Beyond Level BL
0358_0365_ALG2_S_C05_L07_663990.indd
Differentiated Instruction, p. 361
338
Synthetic Types of Roots Previously, you learned that a zero of a function f(x) is any value c such that f(c) = 0. When the function is graphed, the real zeros of the function are the x-intercepts of the graph.
Example
After 8.5 minutes, the wave could be expected to be felt approximately 4980 kilometers from the epicenter. MENTAL CHECK
1
Mathematical Practices
Would the given equation be valid for negative values of x? Possibly; negative values of x would correspond to years before 1990.
Other
0338_0339_ALG2_T_C05_EXT4_663991.indd
Find the zeros of a polynomial function.
Ask: According to the equation, what was the average price of a gallon of gas in 1990? about $1.37
Step 3 Determine and graph the equation for a curve of best fit. Use a quartic regression for the data. 7
2
The function g(x ) = 1.384x 4 - 0.003x 3 + 0.28x 2 0.078x + 1.365 can be used to model the average price of a gallon of gasoline in a given year if x is the number of years since 1990. To find the average price of gasoline in a specific year, you can use the roots of the related polynomial equation.
Have students read the Why? section of the lesson.
[-0.2, 14.2] scl: 1 by [-1232, 11632] scl: 1000
KEYSTROKES:
Determine the number and type of roots for a polynomial equation.
TORU YAMANAKA/AFP/Getty Images
Ask: In Step 3, a quartic curve will fit the data best. An easy way to verify this is to find quadratic and cubic regression equations and then copy them to the Y= list along with the quartic regression equation. Change the window settings for the x-axis to [-0.2, 18.6]. Turn off the scatter plot and graph all three regression equations on the same screen. Then use TRACE to go to each x value in the first row of the table. While at each x value, use the up/down arrow keys to move between the curves, comparing the y values for each regression curve with the y value given in the table.
1
6 Attend to precision.
Scaffolding Questions
Working in Cooperative Groups
Why?
You used complex numbers to describe solutions of quadratic equations.
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Step 2 Graph the scatter plot. KEYSTROKES:
Now
Content Standards
After Lesson 5-7 Use tools including factoring to transform and solve equations.
13
12 8400
6250
3900
2500
10
7
5
2
11
KEYSTROKES:
Then
Common Core State Standards
Source: University of Arizona
Step 1 Enter time in L1 and distance in L2.
400
Roots and Zeros
N.CN.9 Know the 1 Focus Fundamental VerticalAlignment Theorem of VerticalAlignment Algebra; show Before Lesson 5-7 Usethat complexit is tr to describe the solutions of for numbers quadratic quadratic equations. polynomials
Content Standards C F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Mathematical Practices 5 Use appropriate tools strategically.
Materials Mater ate ials as
Students should clear lists L1 and L2 before entering the data from the table in Step 1. This is a more reliable approach than simply “overwriting” old data with new data.
Seeing Structure in Expressions
Content Standards
Common Core State Standards C
Teaching Tip
CN
Common Core State Standards
Common Core State Standards C
TI-83/84 TI 83/84 Plus or other graphing calculator
Th l N b SSystem t The C Complex Number
Roots and Zeros Using Probability to Make Decisions MD
There are numerous tools for implementing the Common Core State Standards available throughout the program, including: Standards at point-of-use in the Chapter Planner and in each lesson of the Teacher Edition,
Abbreviations
5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
Study Guide and Intervention, pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45 Enrichment, p. 46 5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
Differentiated Instruction, pp. 361, 365 Practice, p. 44 Word Problem Practice, p. 45 Enrichment, p. 46
English Learners ELL
Differentiated Instruction, pp. 361, 365
5-Minute Check 5-7 Study Notebook
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Differentiated Instruction, p. 361
Study Guide and Intervention, pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45
5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
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Common Core State Standards Common Core State Standards, Traditional Algebra II Pathway, Correlated to Glencoe Algebra 2, Common Core Edition Lessons in which the standard is the primary focus are indicated in bold. Student Edition Lesson(s)
Standards
Student Edition Page(s)
Number and Quantity The Complex Number System N-CN Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bii with a and b real.
4-4
246–252
2. Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
4-4
246–252
Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions.
4-5, Extend 4-5, 4-6
256–262, 263, 264–272
8. (+) Extend polynomial identities to the complex numbers.
4-4, 4-6
246–252, 264–272
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
5-7
358–365
Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
(+) Advanced Mathematics Standards
T12
Mathematical Modeling Standards
1-1, 4-1
5–10, 219–227
1-1, 1-4, 2-2, 2-4, 2-6, 2-7, 4-1, 4-6, 4-7, 5-4, 9-2, 9-3, 9-4, 9-5, 9-6, 10-7
5–10, 27–32, 69–74, 83–89, 101–107, 109–116, 219–227, 264–272, 275–280, 330–337, 599–605, 607–613, 615–622, 624–631, 632–636, 705–709
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
2. Use the structure of an expression to identify ways to rewrite it.
1-2, 4-3, 4-5, 6-4, 6-5, 7-2, 7-3, 7-4, 7-7, 7-8
11–17, 238–245, 256–262, 407–411, 415–421, 461–467, 468–475, 478–483, 501–507, 509–515
Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
10-3
674–680
Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
4-3, 5-1
238–245, 303–309
Understand the relationship between zeros and factors of polynomials. 2. Know and apply the Remainder Theorem: For a polynomial p(x ) and a number a, the remainder on division by x - a is p(a ), so p(a ) = 0 if and only if (x - aa) is a factor of p (x ).
5-6
352–357
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
5-7
358–365
Use polynomial identities to solve problems. 4. Prove polynomial identities and use them to describe numerical relationships.
4-3, 4-5, 4-6, Extend 5-5
238–245, 256–262, 264–272, 366
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y ) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
10-6, Extend 10-6, 10-7, 11-4
699–703, 704, 705–709, 752–759
Rewrite rational expressions. a (x ) 6. Rewrite simple rational expressions in different forms; write _ in b (x ) r (x ) the form q (x ) + _, where a (x ), b (x ), q (x ), and r (x ) are
5-2, Extend 5-2
311–317, 318–319
8-1, 8-2, 8-4, 8-6
529–537, 538–544, 553–560, 570–578
b (x )
polynomials with the degree of r (x ) less than the degree of b (x ), using inspection, long division, or, for the more complicated examples, a computer algebra system. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
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Common Core State Standards
Continued
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Creating Equations A-CED Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems.
1-3, 1-4, 1-5, 1-6, 4-3, 4-5, 4-6, 4-8, 5-5, 5-6, 5-7, 7-2, 7-4, 7-5, 7-6, 7-8, 8-6
18–25, 27–32, 33–39, 41–48, 238–245, 256–262, 264–272, 282–288, 342–349, 352–357, 358–365, 461–467, 478–483, 485–491, 492–498, 508–515, 570–578
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Throughout the text; for example, 2-4, 3-1, 4-2, 6-3, 9-3, 12-7
Throughout the text; for example, 83–89, 136–145, 229–236, 400–406, 607–613, 837–843
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
1-4, 1-5, 1-6, 2-6, 2-7, 2-8, 3-1, 3-2, 3-3, 3-4, 3-7, 3-8, 4-8, 7-8, 8-6
27–32, 33–39, 41–48, 101– 107, 109–116, 117–121, 136–145, 146–152, 154–160, 189–197, 198–204, 282–288, 509–515, 570–578
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
4-6, 9-1, 9-3, 10-2
264–272, 593–598, 607–613, 666–673
Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
6-7, Extend 6-7, 8-6, Extend 8-6
429–435, 436–437, 570–578, 579–580
Represent and solve equations and inequalities graphically. 11. Explain why the xx-coordinates of the points where the graphs of the equations y = f (x ) and y = g (x ) intersect are the solutions of the equation f (x ) = g (x ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f (x ) and/or g (x ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
3-1, Extend 4-2, Extend 5-7, Extend 6-7, Explore 7-2, Extend 7-6, Extend 8-6, 9-7
136–145, 237, 350–351, 436–437, 459–460, 499–500, 579–580, 640–645
Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (+) Advanced Mathematics Standards
T14
Mathematical Modeling Standards
Throughout the text; for example, 2-1, Extend 2-1, Extend 2-2, 2-6, 5-3, 8-4, 10-1, 12-6
Throughout the text; for example, 61–67, 68, 75, 101– 107, 322–329, 553–560, 659–665, 830–836
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
2-1, 2-6, 4-1, 5-3, 6-2, 6-3, 7-1, 7-3, 8-3, 8-4, 12-7
61–67, 101–107, 219–227, 322–329, 393–398, 400–406, 451–458, 468–475, 545–551, 553–560, 837–843
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
2-3, Extend 4-7
76–82, 281
Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
2-6, 6-3, Extend 6-4
101–107, 400–406, 413
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
5-3, 5-4, Extend 5-4, 5-6, 5-7
322–329, 330–337, 338–339, 352–357, 358–365
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
7-1, 7-3, 12-7, 12-8
451–458, 468–475, 837–843, 845–852
4-3, 4-5, 4-7
238–245, 256–262, 275–280
7-1, 7-8
451–458, 509–515
2-2, 2-7, 4-1, 5-3, 6-1, 6-3, 7-1, 8-4, 9-6
69–74, 109–116, 219–227, 322–329, 385–392, 400–406, 451–458, 553–560, 632–636
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Building Functions F-BF 1. Build a function that models a relationship between two quantities. b. Combine standard function types using arithmetic operations.
6-1, Extend 7-8, 12-8
385–392, 516, 845–852
Build new functions from existing functions. 3. Identify the effect on the graph of replacing f (x ) by f (x ) + k, k k f (x ), f (kx ), and f (x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Explore 2-7, 2-7, Explore 4-7, 4-7, 6-3, Extend 6-4, 7-1, 7-3, 8-3, Explore 12-8, 12-8
108, 109–116, 273–274, 275–280, 400–406, 416, 451–458, 468–475, 545–551, 844, 845–852
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Common Core State Standards
Continued
Student Edition Lesson(s)
Standards 4. Find inverse functions. a. Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse.
6-2
Student Edition Page(s) 393–398
Linear, Quadratic, and Exponential Models F-LE Construct and compare linear and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
7-2, 7-8
461–467, 509–515
Trigonometric Functions F-TF Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
12-2, 12-6
799–805, 830–836
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
12-6
830–836
Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
12-7, 12-8
837–843, 845–852
Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to calculate trigonometric ratios.
13-1, 13-2, 13-3, 13-4, 13-5
873–879, 880–885, 886–891, 893–899, 901–907
Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. (+) Advanced Mathematics Standards
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Mathematical Modeling Standards
11-5, Extend 11-5
760–766, 767–768
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Making Inferences and Justifying Conclusions S-IC Understand and evaluate random processes underlying statistical experiments 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
11-2, 11-6
733–741, 769–776
2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
11-1, Extend 11-1
723–730, 731–732
Make inferences and justify conclusions from sample surveys, experiments, and observational studies 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
11-1
723–730
4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
Extend 11-1, 11-6
731–732, 769–775
5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
11-1
723–730
6. Evaluate reports based on data.
Extend 11-1
731–732
Using Probability to Make Decisions S-MD 6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
11-4
752–759
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
11-3, 11-4, 11-6
742–750, 752–759, 769–776
co connectED.mcgraw-hill.com
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Trusted Content
Common Core State Standards Common Core State Standards for Mathematical Practice, Correlated to Glencoe Algebra 2, Common Core Edition 1. Make sense of problems and persevere in solving them. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-1, 2-8, 3-2, 4-1, 5-3, 6-5, 7-8, 8-5, 9-2, 10-7, 11-2, 12-7, and 13-2. 2. Reason abstractly and quantitatively. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-4, 3-1, 4-3, 5-1, 6-1, 7-2, 8-3, 9-1, 10-1, 11-3, 12-2, and 13-1. 3. Construct viable arguments and critique the reasoning of others. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-2, 3-4, 4-2, 5-4, 6-3, 7-1, 8-2, 9-6, 10-7, 11-1, 12-4, and 13-3. 4. Model with mathematics. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-5, 2-5, 3-3, Extend 4-8, 5-5, 6-7, Extend 7-3, 8-5, 9-3, 10-6, 11-4, 12-8, and 13-5. 5. Use appropriate tools strategically. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-6, 2-5, 3-8, Extend 4-1, Extend 5-4, Extend 6-4, Explore 7-8, Extend 8-6, Explore 9-4, Extend 10-5, Extend 11-1, Explore 12-8, and Explore 13-5. 6. Attend to precision. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-4, 2-7, 3-1, 4-4, 5-2, 6-4, 7-3, 8-6, 9-5, 10-4, 11-5, 12-3, and 13-5. 7. Look for and make use of structure. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-1, 3-7, 4-7, 5-6, 6-2, 7-7, 8-4, 9-4, 10-1, 11-6, 12-6, and 13-1. 8. Look for and express regularity in repeated reasoning. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-3, 3-3, 4-6, 5-8, 6-2, 7-5, 8-1, 9-6, 10-2, 11-5, Explore 12-1, and 13-2.
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Common Core State Standards
W
ith American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. — Common Core State Standards Initiative
What is the goal of the Common Core State Standards? The mission of the Common Core State Standards is to provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that students need for success in college and careers.
Who wrote the standards? The National Governors Association Center for Best Practices and the Council of Chief State School Officers worked with representatives from participating states, a wide range of educators, content experts, researchers, national organizations, and community groups.
uadratic Functions Functions uadratic nd Relations Relations nd
At the high school level the Common Core State Standards are organized by conceptual category. To ease implementation four model course pathways were created: traditional, integrated, accelerated traditional and accelerated integrated. Glencoe Algebra 1, Glencoe Geometry, and Glencoe Algebra 2 follow the traditional pathway.
What are the major points of the standards? The standards seek to develop both students’ mathematical understanding and their procedural skill. The Standards for Mathematical Practice describe varieties of expertise that mathematics teachers at all levels should seek to develop in their students. The Standards for Mathematical Content define what students should understand and be able to do at each level in their study of mathematics.
Mathematical Content Unit 2
|
Quadratic, Polynomial, and Radical FFunctions unctio and Relations
Quadratic Quadr Q uadrrat Functions Functions Quadratic and and nd Relations R Re Relations el and
A.SSE.1.a, F.IF.9 F IF 4
4-1 Graphing Quadratic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Extend: Graphing Technology Lab Modeling Real-World Data . . . . . . . . . . . . . . . . . . . 228
4-2 Solving Quadratic Equations by Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Extend: Graphing Technology Lab Solving Quadratic Equations by Graphing . . . . . . . . 237
How do I implement the standards?
A.SSE.1.a, F.IF.9 F.IF.4 A.CED.2, F.IF.4 A.REI.11
4-3 Solving Quadratic Equations by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.SSE.2, F.IF.8.a
4-4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
N.CN.1, N.CN.2
Extend: Algebra Lab The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4-5 Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Extend: Graphing Technology Lab Solving Quadratic Equations. . . . . . . . . . . . . . . . . . 263
4-6 The Quadratic Formula and the Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . 264 Explore: Graphing Technology Lab Families of Parabolas . . . . . . . . . . . . . . . . . . . . . . 273
4-7 Transformations of Quadratic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Extend: Algebra Lab Quadratics and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . 281
4-8 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Black 100/Allsport Concepts/Getty Images
The Common Core State Standards are shared goals and expectations for what knowledge and skills your students need to succeed. You as a teacher, in partnership with your colleagues, principals, superintendents, decide how the standards are to be met. Glencoe Algebra 2 is designed to help you devise lesson plans and tailor instruction to the individual needs of the students in your classroom as you meet the Common Core State Standards.
Mathematical Content
Get Ready for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
N.CN.7, F.IF.8.a N.CN.7 N.CN.7, A.SSE.1.b F.IF.4, F.BF.3 F.IF.8.a, F.BF.3 F.IF.4, F.IF.6 A.CED.1, A.CED.3
Extend: Graphing Technology Lab Modeling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Assessment Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Preparing for Standardized Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Standardized Test Practice, Chapters 1–4 . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Virtual Manipulatives
Graphing Calculator
pp. 137, 219
pp. 205, 237
Foldables pp. 134, 218
Self-Check Practice pp. 183, 295
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Domain Names
How do I decode the standards? This diagram provides clarity for decoding the standard identifiers.
A.REI.2 Conceptual Category N = Number and Quantity A = Algebra F = Functions S = Statistics and Probability
Domain Standard
Graphing Technology Lab coverage in Glencoe Algebra 2 ensures Complete standards
Modeling Data that you have allUsing the content you need to teach the standards, Polynomial Functions C Content Standards F.IF.4 For a function that models a relations Extend and tables in5-4terms of the quantities, and sk Graphing Technology Lab of the relationship. Modeling Data Using F.IF.7.c Graph polynomial functions, identify 1 Focus Polynomial Functions showing Objective end behavior. Use a graphing calculator to You can use a TI-83/84 Plus graphing calculator model data whose curve of best fit is a to model data dat points when a curve of best fit is a Mathematical Practices polynomial function. fu polynomial function. 5 Use appropriate tools strategically.
SSE
Arithmetic with Polynomials and Rational Expressions
APR
Creating Equations
CED
Reasoning with Equations and Inequalities
REI
Interpreting Functions
IF
Building Functions
BF
Linear, Quadratic, and Exponential Models
LE
Trigonometric Functions
TF
Interpreting Categorical and Quantitative Data
ID
Making Inferences and Justifying Conclusions
IC
Correlations that show at a glance where each standard is addressed in Glencoe Algebra 2.
You can also visit connectED.mcgraw-hill.com to learn more about the Common Core State Standards. There you can choose from an extensive collection of resources to use when planning instruction.
Lesson 5-7
Lesson 5-7 Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.
Example The table shows the distance a seismic wave produced by an earthquake travels from the epicenter. Draw a scatter plot and a curve of best fit to show how the distance is related to time. Then determine approximately how far away from the epicenter a seismic wave will be felt 8.5 minutes after an earthquake occurs.
Travel Time (min)
1
2
5
7
10
12
13
Distance (km) 400 800 2500 3900 6250 8400 10,000
800
2 Teach
[STAT PLOT] 1
10000
2 Teach
9
Put students in pairs, mixing abilities. Then have pairs complete all steps in the Example.
Practice Have students complete Exercises 1–3.
5
1
screen. If rounded, the regression The equation is shown in the equation shown on the calculator can be written as the algebraic equation y = 0.7x 4 - 17x 3 + 161x 2 - 21x + 293.
Step 4 Use the [CALC] feature to find the value of the function for x = 8.5. KEYSTROKES:
ConceptSummary Zeros, Factors, Roots, and Intercepts Let P (x ) = a nx n + … + a 1x + a 0 be a polynomial function. Then the following statements are equivalent.
Words
• c is a zero of P (x ). • c is a root or solution of P (x ) = 0. • x - c is a factor of a nx n + … + a 1x + a 0. • If c is a real number, then (c, 0) is an x-intercept of the graph of P (x ). Consider the polynomial function P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12. The zeros of P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12 are -3, -2, 1, and 2.
P(x) 12
The roots of x 4 + 2x 3 - 7x 2 - 8x + 12 = 0 are -3, -2, 1, and 2. 4
3
8
2
The factors of x + 2x - 7x - 8x + 12 are (x + 3), (x + 2), (x - 1), and (x - 2).
4 −4
The x-intercepts of the graph of P (x ) = x 4 + 2x 3 - 7x 2 - 8x + 12 are (-3, 0), (-2, 0), (1, 0), and (2, 0).
−2
O
2
4x
When solving a polynomial equation with degree greater than zero, there may be one or more real roots or no real roots (the roots are imaginary numbers). Since real numbers and imaginary numbers both belong to the set of complex numbers, all polynomial equations with degree greater than zero will have at least one root in the set of complex numbers. This is the Fundamental Theorem of Algebra.
[CALC] 1 8.5
The table gives the distance for 7 minutes as 3900 and the distance for 10 minutes as 6250. Since 8.5 is halfway between 7 and 10 a reasonable estimate for the distance is halfway between 3900 and 6250.
KeyConcept Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. (continued on the next page)
338 | Extend 5-4 | Graphing Technology Lab: Modeling Data Using Polynomial Functions
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Lesson 5-7 Resources OL
BL
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Extension The nature of polynomial functions makes them perfect for inventing funny stories about how things change over time. Challenge students to create stories about something that is growing or shrinking (or both) and display the growth graphically. Extend the problem by having them create scales for the graphs that match the context of their stories.
Resource Teacher Edition
Approaching Level AL
Chapter Resource Masters
Study Guide and Intervention,pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45
338 | Extend 5-4 | Graphing Technology Lab: Modeling Data Using Polynomial Functions
On Level OL358
Beyond Level BL
0358_0365_ALG2_S_C05_L07_663990.indd
Differentiated Instruction, p. 361
338
Synthetic Types of Roots Previously, you learned that a zero of a function f(x) is any value c such that f(c) = 0. When the function is graphed, the real zeros of the function are the x-intercepts of the graph.
Example
After 8.5 minutes, the wave could be expected to be felt approximately 4980 kilometers from the epicenter. MENTAL CHECK
1
Mathematical Practices
Would the given equation be valid for negative values of x? Possibly; negative values of x would correspond to years before 1990.
Other
0338_0339_ALG2_T_C05_EXT4_663991.indd
Find the zeros of a polynomial function.
Ask: According to the equation, what was the average price of a gallon of gas in 1990? about $1.37
Step 3 Determine and graph the equation for a curve of best fit. Use a quartic regression for the data. 7
2
The function g(x ) = 1.384x 4 - 0.003x 3 + 0.28x 2 0.078x + 1.365 can be used to model the average price of a gallon of gasoline in a given year if x is the number of years since 1990. To find the average price of gasoline in a specific year, you can use the roots of the related polynomial equation.
Have students read the Why? section of the lesson.
[-0.2, 14.2] scl: 1 by [-1232, 11632] scl: 1000
KEYSTROKES:
Determine the number and type of roots for a polynomial equation.
TORU YAMANAKA/AFP/Getty Images
Ask: In Step 3, a quartic curve will fit the data best. An easy way to verify this is to find quadratic and cubic regression equations and then copy them to the Y= list along with the quartic regression equation. Change the window settings for the x-axis to [-0.2, 18.6]. Turn off the scatter plot and graph all three regression equations on the same screen. Then use TRACE to go to each x value in the first row of the table. While at each x value, use the up/down arrow keys to move between the curves, comparing the y values for each regression curve with the y value given in the table.
1
6 Attend to precision.
Scaffolding Questions
Working in Cooperative Groups
Why?
You used complex numbers to describe solutions of quadratic equations.
N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Step 2 Graph the scatter plot. KEYSTROKES:
Now
Content Standards
After Lesson 5-7 Use tools including factoring to transform and solve equations.
13
12 8400
6250
3900
2500
10
7
5
2
11
KEYSTROKES:
Then
Common Core State Standards
Source: University of Arizona
Step 1 Enter time in L1 and distance in L2.
400
Roots and Zeros
N.CN.9 Know the 1 Focus Fundamental VerticalAlignment Theorem of VerticalAlignment Algebra; show Before Lesson 5-7 Usethat complexit is tr to describe the solutions of for numbers quadratic quadratic equations. polynomials
Content Standards C F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Mathematical Practices 5 Use appropriate tools strategically.
Materials Mater ate ials as
Students should clear lists L1 and L2 before entering the data from the table in Step 1. This is a more reliable approach than simply “overwriting” old data with new data.
Seeing Structure in Expressions
Content Standards
Common Core State Standards C
Teaching Tip
CN
Common Core State Standards
Common Core State Standards C
TI-83/84 TI 83/84 Plus or other graphing calculator
Th l N b SSystem t The C Complex Number
Roots and Zeros Using Probability to Make Decisions MD
There are numerous tools for implementing the Common Core State Standards available throughout the program, including: Standards at point-of-use in the Chapter Planner and in each lesson of the Teacher Edition,
Abbreviations
5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
Study Guide and Intervention, pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45 Enrichment, p. 46 5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
Differentiated Instruction, pp. 361, 365 Practice, p. 44 Word Problem Practice, p. 45 Enrichment, p. 46
English Learners ELL
Differentiated Instruction, pp. 361, 365
5-Minute Check 5-7 Study Notebook
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4:50 PM
Differentiated Instruction, p. 361
Study Guide and Intervention, pp. 41–42 Skills Practice, p. 43 Practice, p. 44 Word Problem Practice, p. 45
5-Minute Check 5-7 Study Notebook Teaching Algebra with Manipulatives
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Common Core State Standards Common Core State Standards, Traditional Algebra II Pathway, Correlated to Glencoe Algebra 2, Common Core Edition Lessons in which the standard is the primary focus are indicated in bold. Student Edition Lesson(s)
Standards
Student Edition Page(s)
Number and Quantity The Complex Number System N-CN Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bii with a and b real.
4-4
246–252
2. Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
4-4
246–252
Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions.
4-5, Extend 4-5, 4-6
256–262, 263, 264–272
8. (+) Extend polynomial identities to the complex numbers.
4-4, 4-6
246–252, 264–272
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
5-7
358–365
Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
(+) Advanced Mathematics Standards
T12
Mathematical Modeling Standards
1-1, 4-1
5–10, 219–227
1-1, 1-4, 2-2, 2-4, 2-6, 2-7, 4-1, 4-6, 4-7, 5-4, 9-2, 9-3, 9-4, 9-5, 9-6, 10-7
5–10, 27–32, 69–74, 83–89, 101–107, 109–116, 219–227, 264–272, 275–280, 330–337, 599–605, 607–613, 615–622, 624–631, 632–636, 705–709
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
2. Use the structure of an expression to identify ways to rewrite it.
1-2, 4-3, 4-5, 6-4, 6-5, 7-2, 7-3, 7-4, 7-7, 7-8
11–17, 238–245, 256–262, 407–411, 415–421, 461–467, 468–475, 478–483, 501–507, 509–515
Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
10-3
674–680
Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
4-3, 5-1
238–245, 303–309
Understand the relationship between zeros and factors of polynomials. 2. Know and apply the Remainder Theorem: For a polynomial p(x ) and a number a, the remainder on division by x - a is p(a ), so p(a ) = 0 if and only if (x - aa) is a factor of p (x ).
5-6
352–357
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
5-7
358–365
Use polynomial identities to solve problems. 4. Prove polynomial identities and use them to describe numerical relationships.
4-3, 4-5, 4-6, Extend 5-5
238–245, 256–262, 264–272, 366
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y ) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
10-6, Extend 10-6, 10-7, 11-4
699–703, 704, 705–709, 752–759
Rewrite rational expressions. a (x ) 6. Rewrite simple rational expressions in different forms; write _ in b (x ) r (x ) the form q (x ) + _, where a (x ), b (x ), q (x ), and r (x ) are
5-2, Extend 5-2
311–317, 318–319
8-1, 8-2, 8-4, 8-6
529–537, 538–544, 553–560, 570–578
b (x )
polynomials with the degree of r (x ) less than the degree of b (x ), using inspection, long division, or, for the more complicated examples, a computer algebra system. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
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Common Core State Standards
Continued
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Creating Equations A-CED Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems.
1-3, 1-4, 1-5, 1-6, 4-3, 4-5, 4-6, 4-8, 5-5, 5-6, 5-7, 7-2, 7-4, 7-5, 7-6, 7-8, 8-6
18–25, 27–32, 33–39, 41–48, 238–245, 256–262, 264–272, 282–288, 342–349, 352–357, 358–365, 461–467, 478–483, 485–491, 492–498, 508–515, 570–578
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Throughout the text; for example, 2-4, 3-1, 4-2, 6-3, 9-3, 12-7
Throughout the text; for example, 83–89, 136–145, 229–236, 400–406, 607–613, 837–843
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
1-4, 1-5, 1-6, 2-6, 2-7, 2-8, 3-1, 3-2, 3-3, 3-4, 3-7, 3-8, 4-8, 7-8, 8-6
27–32, 33–39, 41–48, 101– 107, 109–116, 117–121, 136–145, 146–152, 154–160, 189–197, 198–204, 282–288, 509–515, 570–578
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
4-6, 9-1, 9-3, 10-2
264–272, 593–598, 607–613, 666–673
Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
6-7, Extend 6-7, 8-6, Extend 8-6
429–435, 436–437, 570–578, 579–580
Represent and solve equations and inequalities graphically. 11. Explain why the xx-coordinates of the points where the graphs of the equations y = f (x ) and y = g (x ) intersect are the solutions of the equation f (x ) = g (x ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f (x ) and/or g (x ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
3-1, Extend 4-2, Extend 5-7, Extend 6-7, Explore 7-2, Extend 7-6, Extend 8-6, 9-7
136–145, 237, 350–351, 436–437, 459–460, 499–500, 579–580, 640–645
Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (+) Advanced Mathematics Standards
T14
Mathematical Modeling Standards
Throughout the text; for example, 2-1, Extend 2-1, Extend 2-2, 2-6, 5-3, 8-4, 10-1, 12-6
Throughout the text; for example, 61–67, 68, 75, 101– 107, 322–329, 553–560, 659–665, 830–836
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
2-1, 2-6, 4-1, 5-3, 6-2, 6-3, 7-1, 7-3, 8-3, 8-4, 12-7
61–67, 101–107, 219–227, 322–329, 393–398, 400–406, 451–458, 468–475, 545–551, 553–560, 837–843
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
2-3, Extend 4-7
76–82, 281
Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
2-6, 6-3, Extend 6-4
101–107, 400–406, 413
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
5-3, 5-4, Extend 5-4, 5-6, 5-7
322–329, 330–337, 338–339, 352–357, 358–365
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
7-1, 7-3, 12-7, 12-8
451–458, 468–475, 837–843, 845–852
4-3, 4-5, 4-7
238–245, 256–262, 275–280
7-1, 7-8
451–458, 509–515
2-2, 2-7, 4-1, 5-3, 6-1, 6-3, 7-1, 8-4, 9-6
69–74, 109–116, 219–227, 322–329, 385–392, 400–406, 451–458, 553–560, 632–636
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Building Functions F-BF 1. Build a function that models a relationship between two quantities. b. Combine standard function types using arithmetic operations.
6-1, Extend 7-8, 12-8
385–392, 516, 845–852
Build new functions from existing functions. 3. Identify the effect on the graph of replacing f (x ) by f (x ) + k, k k f (x ), f (kx ), and f (x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Explore 2-7, 2-7, Explore 4-7, 4-7, 6-3, Extend 6-4, 7-1, 7-3, 8-3, Explore 12-8, 12-8
108, 109–116, 273–274, 275–280, 400–406, 416, 451–458, 468–475, 545–551, 844, 845–852
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T15
SECTION 1
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Trusted Content
Common Core State Standards
Continued
Student Edition Lesson(s)
Standards 4. Find inverse functions. a. Solve an equation of the form f (x ) = c for a simple function f that has an inverse and write an expression for the inverse.
6-2
Student Edition Page(s) 393–398
Linear, Quadratic, and Exponential Models F-LE Construct and compare linear and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
7-2, 7-8
461–467, 509–515
Trigonometric Functions F-TF Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
12-2, 12-6
799–805, 830–836
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
12-6
830–836
Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
12-7, 12-8
837–843, 845–852
Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to calculate trigonometric ratios.
13-1, 13-2, 13-3, 13-4, 13-5
873–879, 880–885, 886–891, 893–899, 901–907
Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. (+) Advanced Mathematics Standards
T16
Mathematical Modeling Standards
11-5, Extend 11-5
760–766, 767–768
Correlation
Student Edition Lesson(s)
Standards
Student Edition Page(s)
Making Inferences and Justifying Conclusions S-IC Understand and evaluate random processes underlying statistical experiments 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
11-2, 11-6
733–741, 769–776
2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
11-1, Extend 11-1
723–730, 731–732
Make inferences and justify conclusions from sample surveys, experiments, and observational studies 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
11-1
723–730
4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
Extend 11-1, 11-6
731–732, 769–775
5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
11-1
723–730
6. Evaluate reports based on data.
Extend 11-1
731–732
Using Probability to Make Decisions S-MD 6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
11-4
752–759
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
11-3, 11-4, 11-6
742–750, 752–759, 769–776
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T17
SECTION 1
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Trusted Content
Common Core State Standards Common Core State Standards for Mathematical Practice, Correlated to Glencoe Algebra 2, Common Core Edition 1. Make sense of problems and persevere in solving them. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-1, 2-8, 3-2, 4-1, 5-3, 6-5, 7-8, 8-5, 9-2, 10-7, 11-2, 12-7, and 13-2. 2. Reason abstractly and quantitatively. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-4, 3-1, 4-3, 5-1, 6-1, 7-2, 8-3, 9-1, 10-1, 11-3, 12-2, and 13-1. 3. Construct viable arguments and critique the reasoning of others. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-2, 3-4, 4-2, 5-4, 6-3, 7-1, 8-2, 9-6, 10-7, 11-1, 12-4, and 13-3. 4. Model with mathematics. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-5, 2-5, 3-3, Extend 4-8, 5-5, 6-7, Extend 7-3, 8-5, 9-3, 10-6, 11-4, 12-8, and 13-5. 5. Use appropriate tools strategically. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-6, 2-5, 3-8, Extend 4-1, Extend 5-4, Extend 6-4, Explore 7-8, Extend 8-6, Explore 9-4, Extend 10-5, Extend 11-1, Explore 12-8, and Explore 13-5. 6. Attend to precision. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-4, 2-7, 3-1, 4-4, 5-2, 6-4, 7-3, 8-6, 9-5, 10-4, 11-5, 12-3, and 13-5. 7. Look for and make use of structure. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-2, 2-1, 3-7, 4-7, 5-6, 6-2, 7-7, 8-4, 9-4, 10-1, 11-6, 12-6, and 13-1. 8. Look for and express regularity in repeated reasoning. Glencoe Algebra 2 exhibits these practices throughout the entire program. Some specific lessons for review are: Lessons 1-3, 2-3, 3-3, 4-6, 5-8, 6-2, 7-5, 8-1, 9-6, 10-2, 11-5, Explore 12-1, and 13-2.
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