# Equations of Linear. Functions

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1 SECTION Trusted Content Common Core State 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 Initiative What is the goal of the Common Core State? The mission of the Common Core State is to provide a consistent, clear understanding of what students are epected 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 eperts, researchers, national organizations, and community groups. uations of Linear ear ctions Functions What are the major points of the standards? The standards seek to develop both students mathematical understanding and their procedural skill. The for Mathematical Practice describe varieties of epertise that mathematics teachers at all levels should seek to develop in their students. The for Mathematical Content define what students should understand and be able to do at each level in their study of mathematics. How do I implement the standards? The Common Core State are shared goals and epectations 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 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. At the high school level the Common Core State are organized by conceptual category. To ease implementation four model course pathways were created: traditional, integrated, accelerated traditional and accelerated integrated. Glencoe Algebra, Glencoe Geometry, and Glencoe Algebra 2 follow the traditional pathway. Mathematical Unit 2 Linear Relationships Content Lester Lefkowitz/Photographer s Choice/Getty Images Equations of Linear Functions Linear Functions Get Ready for Chapter Eplore: Graphing Technology Lab Investigating Slope-Intercept Form Graphing Equations in Slope-Intercept Form F.IF.7a, S.ID.7 Etend: Graphing Technology Lab The Family of Linear Graphs F.BF.3, S.ID Writing Equations in Slope-Intercept Form Writing Equations in Point-Slope Form Parallel and Perpendicular Lines Mid-Chapter Quiz Scatter Plots and Lines of Fit S.ID.6a, S.ID.6c Etend: Algebra Lab Correlation and Causation S.ID Regression and Median-Fit Lines Inverse Linear Functions A.CED.2, F.BF.4a Etend: Algebra Lab Drawing Inverses F.BF.4a ASSESSMENT Study Guide and Review Practice Test Preparing for Standardized Tests Standardized Test Practice, Chapters F.BF., F.LE.2 F.IF.2, F.LE.2 F.LE.2, S.ID.7 S.ID.6, S.ID.8 Mathematical Content Virtual Manipulatives pp. 58, 248 Graphing Calculator pp. 69, 25 Foldables pp. 52, 24 Self-Check Practice pp. 207, 23 v T0 _iv_alg_s_fm_toc_ indd i-v_alg_t_fm_toc_ indd v v 5/7/2 9:45 AM :9

3 SECTION Trusted Content Common Core State Common Core State, Traditional Algebra I Pathway, Correlated to Glencoe Algebra, Common Core Edition Lessons in which the standard is the primary focus are indicated in bold. Number and Quantity The Real Number System N-RN Etend the properties of eponents to rational eponents.. Eplain how the definition of the meaning of rational eponents follows from etending the properties of integer eponents to those values, allowing for a notation for radicals in terms of rational eponents. 2. Rewrite epressions involving radicals and rational eponents using the properties of eponents. Use properties of rational and irrational numbers. 3. Eplain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational , 0-3, Etend 0-3, , , , Etend Quantities N-Q Reason quantitatively and use units to solve problems.. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Throughout the tet; for eample, 2-6, 2-7, 2-8, 2-9, Etend 3-2, 4-5, 7-5 Throughout the tet; for eample, 7, 9 24, 26 3, 32 38, 69 70, , Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Etend Etend Mathematical Modeling T2

4 Correlation Algebra Seeing Structure in Epressions A-SSE Interpret the structure of epressions. Interpret epressions that represent a quantity in terms of its contet. a. Interpret parts of an epression, such as terms, factors, and coefficients. b. Interpret complicated epressions by viewing one or more of their parts as a single entity., -4, 8, 9 5 9, 25 3, , , -3, , 6 22, Use the structure of an epression to identify ways to rewrite it., -2, -3, -4, 7, 7-2, 7-3, 7-4, Eplore 8-5, 8-5, Eplore 8-6, 8-6, 8-7, 8-8, , 0 5, 6 22, 25 3, , , , , 493, , , , 50 55, 56 52, Write epressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an epression to reveal and eplain properties of the quantity represented by the epression. a. Factor a quadratic epression to reveal the zeros of the function it defines. b. Complete the square in a quadratic epression to reveal the maimum or minimum value of the function it defines. c. Use the properties of eponents to transform epressions for eponential functions. 8-5, 8-6, 8-7, 8-8, , , 50 55, 56 52, , 9-4, Etend , , Etend Arithmetic with Polynomials and Rational Epressions A-APR Perform arithmetic operations on polynomials.. 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. Eplore 8, 8, 8-2, Eplore 8-3, 8-3, , , , , , Creating Equations A-CED Create equations that describe numbers or relationships.. Create equations and inequalities in one variable and use them to solve problems. -5, 2, 2-2, 2-3, 2-4, 2-5, 2-9, 3-2, 5, 5-2, 5-3, 5-4, 5-5, 7-6, 8-5, 8-6, 8-7, 9-4, 9-5, 0-4, , 75 80, 83 89, 9 96, 97 02, 03 09, 32 38, 63 68, , , , 306 3, 32 36, , , , 50 55, , , , T3

5 SECTION Trusted Content Common Core State Continued 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. Etend -7, 3, 3-4, 3-5, 3-6, 4, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 6, Etend 6, 6-2, 6-3, 6-4, 6-5, 7-5, 7-6, 8-6, 8-7, 8-8, 9, 9-2, 9-4, 9-5, 0, Etend 0, 0-4, -2, -8 55, 55 62, 82 88, 89 95, , , , , , , , , , , , , , , , , , 50 55, 56 52, , , , , , 627, , , Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling contet. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. 4-2, 5-6, 6, , , , , 2-9, , 32 38, Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and eplain the reasoning.. Eplain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. -5, 2-2, 2-3, 2-4, 2-5, 2-6, 2-9, 8-6, 8-7, , Eplore 2-2, 2-2, Eplore 2-3, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 5, Eplore 5-2, 5-2, 5-3, 5-4, 5-5, , 83 89, 9 96, 97 02, 02 09, 7, 32 38, , 50 55, , 8 82, 83 89, 90, 9 96, 97 02, 03 09, 7, 9 24, 26 3, 32 38, , 29, , , 306 3, 32 36, Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( p) 2 = q that has the same solutions. Derive the quadratic formula from this form. 9-4, 9-5, , , b. Solve quadratic equations by inspection (e.g., for 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. Mathematical Modeling 8-6, 8-7, 8-8, 9-2, 9-4, Etend 9-4, , 50 55, 56 52, , , , T4

6 Correlation Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions Solve systems of linear equations eactly and approimately (e.g., with graphs), focusing on pairs of linear equations in two variables. 6, Etend 6, 6-2, 6-3, 6-4, 6-5, Etend , , , , , , Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Etend Represent and solve equations and inequalities graphically. 0. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).. Eplain why the -coordinates of the points where the graphs of the equations y = f ( ) and y = g ( ) intersect are the solutions of the equation f ( ) = g ( ); find the solutions approimately, e.g., using technology to graph the functions, make tables of values, or find successive approimations. Include cases where f ( ) and/or g ( ) are linear, polynomial, rational, absolute value, eponential, and logarithmic functions. 2. Graph the solutions to a linear inequality in two variables as a halfplane (ecluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. -6, -7, 3, 3-2, 3-4, 7-5, 9, 0 Etend 6, Etend 7-5, Etend 9-3, 9-7, Etend , Etend 5-6, 6-6, Etend , 47 54, 55 62, 63 68, 82 88, , , , , , , , 323, , 377 Functions Interpreting Functions F-IF Understand the concept of a function and use function notation.. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain eactly one element of the range. If f is a function and is an element of its domain, then f ( ) denotes the output of f corresponding to the input. The graph of f is the graph of the equation y = f ( ). -6, , Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a contet. -7, 3-6, 4-3, 7-5, 7-6, 9, , , , , , , Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 3-5, 7-7, , , T5

7 SECTION Trusted Content Common Core State Continued Interpret functions that arise in applications in terms of the contet. 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. -8, Eplore 3, 3, Etend 4, 7-5, 9, 9-7, , 53 54, 55 62, , , , , Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. -7, 7-5, 7-6, 9, , , , , 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. Analyze functions using different representations. 7. Graph functions epressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Eplore 3-3, 3-3, Etend 7-7, Etend 9, 9-6 3, 3-2, Etend 3-2, 3-4, 4, Etend 4, 9, 9-2, Eplore 9-3, 9-3, Etend , Etend 9-7, 0, Etend 0 7, 72 80, 444, 554, , 63 68, 69 70, 82 88, , , , , , , , 606, , 627 e. Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an epression in different but equivalent forms to reveal and eplain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, etreme values, and symmetry of the graph, and interpret these in terms of a contet. Eplore 7-5, , , 9-3, 9-4, Etend , , , b. Use the properties of eponents to interpret epressions for eponential functions. 7, 7-2, 7-5, 7-6, Etend , , , , Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). -7, 3-6, 4-3, 7-8, 9, , , , , , Building Functions F-BF Build a function that models a relationship between two quantities.. Write a function that describes a relationship between two quantities. a. Determine an eplicit epression, a recursive process, or steps for calculation from a contet. -7, 3, 3-4, 3-6, 4, 4-2, 4-3, 4-4, 4-5, 4-6, 4-7, 7-6, , 55 62, 82 88, , , , , , , , , , T6 b. Combine standard function types using arithmetic operations. 4-2, 7-6, , , Mathematical Modeling

8 Correlation 2. Write arithmetic and geometric sequences both recursively and with an eplicit formula, use them to model situations, and translate between the two forms. 3-5, 7-7, , , Build new functions from eisting functions. 3. Identify the effect on the graph of replacing f ( ) by f ( ) + k, kf ( ), f (k ), and f ( + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph using technology. Etend 4, Eplore 7-5, Etend 7-6, Eplore 9-3, 9-3, 0, Etend , , 437, , , , Find inverse functions. a. Solve an equation of the form f ( ) = c for a simple function f that has an inverse and write an epression for the inverse. 4-7, Etend 4-7, Eplore , 27, Linear, Quadratic, and Eponential Models F-LE Construct and compare linear, quadratic, and eponential models and solve problems.. Distinguish between situations that can be modeled with linear functions and with eponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that eponential functions grow by equal factors over equal intervals. Eplore 3-3, 3-3, 3-5, 7-7, 9-6 7, 72 80, 89 95, , b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 3-5, 3-6, 7-6, 7-7, , , , , , 7-7, , , Construct linear and eponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 3-5, 3-6, 4-2, 4-3, 4-4, 4-5, 4-6, 7-5, 7-6, 7-7, 9-6, Etend , , , , , , , , , , , Observe using graphs and tables that a quantity increasing eponentially eventually eceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 9-6, Etend , Interpret epressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or eponential function in terms of a contet. 3-4, Eplore 4, 4, Etend 4, 4-5, 7-5, 7-6, 7-7, , 25, , , , , , , T7

9 SECTION Trusted Content Common Core State Continued Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable.. Represent data with plots on the real number line (dot plots, histograms, and bo plots). 03, 2-3, 2-4 P40 P46, , Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 2-2, 2-3, 2-4, Etend , , , Interpret differences in shape, center, and spread in the contet of the data sets, accounting for possible effects of etreme data points (outliers). Summarize, represent, and interpret data on two categorical and quantitative variables. 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the contet of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the contet of the data. Use given functions or choose a function suggested by the contet. Emphasize linear, quadratic, and eponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret linear models. 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the contet of the data. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. 2-3, , Etend , 4-6, Etend , , , , , Etend 4, 4-4, , , , Distinguish between correlation and causation. Etend Mathematical Modeling T8

10 Correlation Common Core State for Mathematical Practice, Correlated to Glencoe Algebra, Common Core Edition. Make sense of problems and persevere in solving them. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons 0, -8, 2-4, 3-4, 4-5, 5-4, 6-4, 7-5, 8-8, 9-3, 0-5,, and Reason abstractly and quantitatively. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons -3, 2, 3-3, 4, 5, 6-5, 7-2, 8-5, 9, 0-3, -8, and Construct viable arguments and critique the reasoning of others. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons -3, 2-5, Etend 3-5, 4-2, 5-5, 6, 7-4, 8, 9-2, 0-4, -2, and Etend Model with mathematics. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons, 2-9, 3-2, 4-5, 5, 6-5, 7-6, 8-7, 9-7, 0-4, -7, and Use appropriate tools strategically. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Etend -7, 2-4, Etend 3-2, 4-4, Etend 5-6, Etend 6, Etend 7-5, 8-2, Etend 9-6, 0-6, Etend -8, and Attend to precision. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Etend -3, 2-8, 3-4, 4-2, 5-2, 6-6, 7-4, 8-9, 9-5, 0, -6, and Look for and make use of structure. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons -2, 2-5, 3-6, Etend 4, 5-5, 6-3, 7-7, 8-6, 9-6, 0-2, -2 and Look for and epress regularity in repeated reasoning. Glencoe Algebra ehibits these practices throughout the entire program. Some specific lessons for review are: Lessons -4, 2-7, 3, 4, 5-4, 6, 7, 8-4, 9-3, 0-2, -5 and 2-6. T9

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