All West Virginia teachers are responsible for classroom instruction that integrates content standards and mathematical habits of mind. Students in this course will focus on five critical units that deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. Students are introduced to methods for analyzing and using quadratic functions, including manipulating expressions for them, and solving quadratic equations. Students in 8th grade High School Algebra understand and apply the Pythagorean theorem, and use quadratic functions to model and solve problems. Mathematical habits of mind, which should be integrated in these content areas, include: making sense of problems and persevering in solving them, reasoning abstractly and quantitatively; constructing viable arguments and critiquing the reasoning of others; modeling with mathematics; using appropriate tools strategically; attending to precision, looking for and making use of structure; and looking for and expressing regularity in repeated reasoning. Students will continue developing mathematical proficiency in a developmentally-appropriate progressions of standards.

### Relationships between Quantities

#### Standards

M.A18.1

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.

M.A18.2

Define appropriate quantities for the purpose of descriptive modeling.  Instructional Note:  Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.

M.A18.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

#### High School Algebra I for 8th Grade

M.A18.1

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.

M.A18.2

Define appropriate quantities for the purpose of descriptive modeling.  Instructional Note:  Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.

M.A18.3

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

#### Standards

M.A18.4

Interpret expressions that represent a quantity in terms of its context.

1. Interpret parts of an expression, such as terms, factors, and coefficients.
2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Instructional Note:  Limit to linear expressions and to exponential expressions with integer exponents.

#### High School Algebra I for 8th Grade

M.A18.4

Interpret expressions that represent a quantity in terms of its context.

1. Interpret parts of an expression, such as terms, factors, and coefficients.
2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Instructional Note:  Limit to linear expressions and to exponential expressions with integer exponents.

#### Standards

M.A18.5

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.  Instructional Note:  Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

M.A18.6

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.  Instructional Note:  Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

M.A18.7

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. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.) Instructional Note:  Limit to linear equations and inequalities.

M.A18.8

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)  Instructional Note:  Limit to formulas with a linear focus.

#### High School Algebra I for 8th Grade

M.A18.5

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.  Instructional Note:  Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

M.A18.6

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.  Instructional Note:  Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

M.A18.7

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. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.) Instructional Note:  Limit to linear equations and inequalities.

M.A18.8

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)  Instructional Note:  Limit to formulas with a linear focus.

#### Standards

M.A18.9

Explain 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.  Instructional Note:  Students should focus on linear equations and be able to extend and apply their reasoning to other types of equations in future units and courses.  Students will solve exponential equations in Algebra II.

#### High School Algebra I for 8th Grade

M.A18.9

Explain 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.  Instructional Note:  Students should focus on linear equations and be able to extend and apply their reasoning to other types of equations in future units and courses.  Students will solve exponential equations in Algebra II.

#### Standards

M.A18.10

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.  Instructional Note:  Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1 /16.

#### High School Algebra I for 8th Grade

M.A18.10

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.  Instructional Note:  Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1 /16.

### Linear and Exponential Relationships

#### Standards

M.A18.11

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.  Instructional Note:  Address this standard before discussing exponential functions with continuous domains.

M.A18.12

Rewrite expressions involving radicals and rational exponents using the properties of exponents.  Instructional Note:  Address this standard before discussing exponential functions with continuous domains.

#### High School Algebra I for 8th Grade

M.A18.11

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.  Instructional Note:  Address this standard before discussing exponential functions with continuous domains.

M.A18.12

Rewrite expressions involving radicals and rational exponents using the properties of exponents.  Instructional Note:  Address this standard before discussing exponential functions with continuous domains.

#### Standards

M.A18.13

Analyze and solve pairs of simultaneous linear equations.

1. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
2. Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
3. Solve real-world and mathematical problems leading to two linear equations in two variables. (e.g., Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.)

Instructional Note:  While this content is likely subsumed by M.A18.10, 14, and 15, it could be used for scaffolding instruction to the more sophisticated content found there.

#### High School Algebra I for 8th Grade

M.A18.13

Analyze and solve pairs of simultaneous linear equations.

1. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
2. Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
3. Solve real-world and mathematical problems leading to two linear equations in two variables. (e.g., Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.)

Instructional Note:  While this content is likely subsumed by M.A18.10, 14, and 15, it could be used for scaffolding instruction to the more sophisticated content found there.

#### Standards

M.A18.14

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.  Instructional Note:  Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

M.A18.15

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.  Instructional Note:  Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

#### High School Algebra I for 8th Grade

M.A18.14

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.  Instructional Note:  Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

M.A18.15

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.  Instructional Note:  Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

#### Standards

M.A18.16

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).  Instructional Note:  Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.

M.A18.17

Explain why the x-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.  Instructional Note: Focus on cases where f(x) and g(x) are linear or exponential.

M.A18.18

Graph the solutions to a linear inequality in two variables as a half-plane (excluding 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.

#### High School Algebra I for 8th Grade

M.A18.16

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).  Instructional Note:  Focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses.

M.A18.17

Explain why the x-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.  Instructional Note: Focus on cases where f(x) and g(x) are linear or exponential.

M.A18.18

Graph the solutions to a linear inequality in two variables as a half-plane (excluding 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.

#### Standards

M.A18.19

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.20

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.)

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.21

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. (e.g., The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.)

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

#### High School Algebra I for 8th Grade

M.A18.19

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.20

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.)

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.21

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. (e.g., The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.)

Instructional Note:  While this content is likely subsumed by M.A18.22-24 and M.A18.30a it could be used for scaffolding instruction to the more sophisticated content found there.

#### Standards

M.A18.22

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains.

M.A18.23

Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains.

M.A18.24

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains. Draw connection to M.A18.33, which requires students to write arithmetic and geometric sequences.)

#### High School Algebra I for 8th Grade

M.A18.22

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains.

M.A18.23

Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains.

M.A18.24

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.  Instructional Note:  Students should experience a variety of types of situations modeled by functions.  Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.  Constrain examples to linear functions and exponential functions having integral domains. Draw connection to M.A18.33, which requires students to write arithmetic and geometric sequences.)

#### Standards

M.A18.25

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Instructional Note:  While this content is likely subsumed by M.A18.27and M.A18.32a, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.26

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Instructional Note:  While this content is likely subsumed by M.A18.27and M.A18.32a, it could be used for scaffolding instruction to the more sophisticated content found there.

#### High School Algebra I for 8th Grade

M.A18.25

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Instructional Note:  While this content is likely subsumed by M.A18.27and M.A18.32a, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.26

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Instructional Note:  While this content is likely subsumed by M.A18.27and M.A18.32a, it could be used for scaffolding instruction to the more sophisticated content found there.

#### Standards

M.A18.27

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.  Instructional Note:  Focus on linear and exponential functions.

M.A18.28

Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.  Instructional Note:  Focus on linear and exponential functions.

M.A18.29

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.  Instructional Note:  Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers. The Quadratic Functions and Modeling unit of this course and Algebra II course will address other function types.

#### High School Algebra I for 8th Grade

M.A18.27

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.  Instructional Note:  Focus on linear and exponential functions.

M.A18.28

Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.  Instructional Note:  Focus on linear and exponential functions.

M.A18.29

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.  Instructional Note:  Focus on linear functions and intervals for exponential functions whose domain is a subset of the integers. The Quadratic Functions and Modeling unit of this course and Algebra II course will address other function types.

#### Standards

M.A18.30

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

1. Graph linear and quadratic functions and show intercepts, maxima, and mini
2. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude.

Instructional Note:  Focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y = 3n and y = 100·2n.

M.A18.31

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)  Instructional Note:  Focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y = 3n and y = 100·2n.

#### High School Algebra I for 8th Grade

M.A18.30

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

1. Graph linear and quadratic functions and show intercepts, maxima, and mini
2. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude.

Instructional Note:  Focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y = 3n and y = 100·2n.

M.A18.31

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)  Instructional Note:  Focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y = 3n and y = 100·2n.

#### Standards

M.A18.32

Write a function that describes a relationship between two quantities.

1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example,  build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the

Instructional Note:  Limit to linear and exponential functions.

M.A18.33

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.  Instructional Note:  Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

#### High School Algebra I for 8th Grade

M.A18.32

Write a function that describes a relationship between two quantities.

1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example,  build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the

Instructional Note:  Limit to linear and exponential functions.

M.A18.33

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.  Instructional Note:  Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

#### Standards

M.A18.34

Identify the effect on the graph of replacing f(x) by f(x) + 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.  Instructional Note:  Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard.

#### High School Algebra I for 8th Grade

M.A18.34

Identify the effect on the graph of replacing f(x) by f(x) + 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.  Instructional Note:  Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard.

#### Standards

M.A18.35

Distinguish between situations that can be modeled with linear functions and with exponential functions.

1. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal inte
2. Recognize situations in which one quantity changes at a constant rate per unit interval relative to anothe
3. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

M.A18.36

Construct linear and exponential 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).

M.A18.37

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.  Instructional Note:  Limit to comparisons between linear and exponential models.

#### High School Algebra I for 8th Grade

M.A18.35

Distinguish between situations that can be modeled with linear functions and with exponential functions.

1. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal inte
2. Recognize situations in which one quantity changes at a constant rate per unit interval relative to anothe
3. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

M.A18.36

Construct linear and exponential 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).

M.A18.37

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.  Instructional Note:  Limit to comparisons between linear and exponential models.

#### Standards

M.A18.38

Interpret the parameters in a linear or exponential function in terms of a context.  Instructional Note:  Limit exponential functions to those of the form f(x) = bx + k.

#### High School Algebra I for 8th Grade

M.A18.38

Interpret the parameters in a linear or exponential function in terms of a context.  Instructional Note:  Limit exponential functions to those of the form f(x) = bx + k.

### Descriptive Statistics

#### Standards

M.A18.39

Represent data with plots on the real number line (dot plots, histograms, and box plots).

M.A18.40

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.  Instructional Note:  In grades 6 – 7, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

M.A18.41

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).  Instructional Note:  In grades 6 – 7, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

#### High School Algebra I for 8th Grade

M.A18.39

Represent data with plots on the real number line (dot plots, histograms, and box plots).

M.A18.40

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.  Instructional Note:  In grades 6 – 7, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

M.A18.41

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).  Instructional Note:  In grades 6 – 7, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

#### Standards

M.A18.42

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.43

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.44

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (e.g., In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.)

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.45

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. (e.g., Collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?)

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

#### High School Algebra I for 8th Grade

M.A18.42

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.43

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.44

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (e.g., In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.)

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

M.A18.45

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. (e.g., Collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?)

Instructional Note:  While this content is likely subsumed by M.A18.47-50, it could be used for scaffolding instruction to the more sophisticated content found there.

#### Standards

M.A18.46

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.

M.A18.47

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

1. Fit a function to the data; use functions fitted to data to solve problems in the context of the Use given functions or choose a function suggested by the context. Instructional Note: Emphasize linear and exponential models.
2. Informally assess the fit of a function by plotting and analyzing residual
Instructional Note: Focus should be on situations for which linear models are appropriate, but may be used to preview quadratic functions in the Quadratic Functions and Modeling Unit.
3. Fit a linear function for scatter plots that suggest a linear associat

Instructional Note:  Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals.

#### High School Algebra I for 8th Grade

M.A18.46

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.

M.A18.47

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

1. Fit a function to the data; use functions fitted to data to solve problems in the context of the Use given functions or choose a function suggested by the context. Instructional Note: Emphasize linear and exponential models.
2. Informally assess the fit of a function by plotting and analyzing residual
Instructional Note: Focus should be on situations for which linear models are appropriate, but may be used to preview quadratic functions in the Quadratic Functions and Modeling Unit.
3. Fit a linear function for scatter plots that suggest a linear associat

Instructional Note:  Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals.

#### Standards

M.A18.48

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.  Instructional Note:  Build on students’ work with linear relationships and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.

M.A18.49

Compute (using technology) and interpret the correlation coefficient of a linear fit.  Instructional Note:  Build on students’ work with linear relationships and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.

M.A18.50

Distinguish between correlation and causation. Instructional Note:  The important distinction between a statistical relationship and a cause-and-effect relationship arises here.

#### High School Algebra I for 8th Grade

M.A18.48

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.  Instructional Note:  Build on students’ work with linear relationships and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.

M.A18.49

Compute (using technology) and interpret the correlation coefficient of a linear fit.  Instructional Note:  Build on students’ work with linear relationships and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship.

M.A18.50

Distinguish between correlation and causation. Instructional Note:  The important distinction between a statistical relationship and a cause-and-effect relationship arises here.

### Expressions and Equations

#### Standards

M.A18.51

Interpret expressions that represent a quantity in terms of its context.

1. Interpret parts of an expression, such as terms, factors, and coefficie
2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Instructional Note:  Focus on quadratic and exponential expressions. For M.A18.51b, exponents are extended from integer found in the unit on Relationships between Quantities to rational exponents focusing on those that represent square roots and cube roots.

M.A18.52

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

#### High School Algebra I for 8th Grade

M.A18.51

Interpret expressions that represent a quantity in terms of its context.

1. Interpret parts of an expression, such as terms, factors, and coefficie
2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Instructional Note:  Focus on quadratic and exponential expressions. For M.A18.51b, exponents are extended from integer found in the unit on Relationships between Quantities to rational exponents focusing on those that represent square roots and cube roots.

M.A18.52

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

#### Standards

M.A18.53

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

1. Factor a quadratic expression to reveal the zeros of the function it de
2. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
3. Use the properties of exponents to transform expressions for exponential functi For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.

Instructional Note:  It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.

#### High School Algebra I for 8th Grade

M.A18.53

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

1. Factor a quadratic expression to reveal the zeros of the function it de
2. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
3. Use the properties of exponents to transform expressions for exponential functi For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the
annual rate is 15%.

Instructional Note:  It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.

#### Standards

M.A18.54

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.  Instructional Note:  Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.

#### High School Algebra I for 8th Grade

M.A18.54

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.  Instructional Note:  Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.

#### Standards

M.A18.55

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations.

M.A18.56

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations.

M.A18.57

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations. Extend M.A18.57 to formulas involving squared variables.

#### High School Algebra I for 8th Grade

M.A18.55

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations.

M.A18.56

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations.

M.A18.57

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note:  Extend work on linear and exponential equations in the unit on Relationships between Quantities to include quadratic equations. Extend M.A18.57 to formulas involving squared variables.

#### Standards

M.A18.58

Solve quadratic equations in one variable.

1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solut Derive the quadratic formula from this form.
2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equat Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Instructional Note:  Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.

#### High School Algebra I for 8th Grade

M.A18.58

Solve quadratic equations in one variable.

1. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solut Derive the quadratic formula from this form.
2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equat Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Instructional Note:  Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.

#### Standards

M.A18.59

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (e.g., Find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.)  Instructional Note:  Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2 + y2 = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32 + 42 = 52.

#### High School Algebra I for 8th Grade

M.A18.59

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (e.g., Find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.)  Instructional Note:  Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2 + y2 = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32 + 42 = 52.

#### Standards

M.A18.60

Explain 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.  Instructional Note:  Connect to physical situations (e.g., finding the perimeter of a square of area 2).

#### High School Algebra I for 8th Grade

M.A18.60

Explain 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.  Instructional Note:  Connect to physical situations (e.g., finding the perimeter of a square of area 2).

#### Standards

M.A18.61

Explain a proof of the Pythagorean Theorem and its converse.

M.A18.62

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.  Instructional Note:  Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

M.A18.63

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Instructional Note:  Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

#### High School Algebra I for 8th Grade

M.A18.61

Explain a proof of the Pythagorean Theorem and its converse.

M.A18.62

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.  Instructional Note:  Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

M.A18.63

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Instructional Note:  Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

#### Standards

M.A18.64

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

M.A18.65

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

M.A18.66

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. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

#### High School Algebra I for 8th Grade

M.A18.64

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

M.A18.65

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

M.A18.66

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. Instructional Note:  Focus on quadratic functions; compare with linear and exponential functions studies in the unit on Linear and Exponential Functions.

#### Standards

M.A18.67

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Instructional Note:  Compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.

M.A18.68

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

1. 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.
2. Use the properties of exponents to interpret expressions for exponential functi For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

Instructional Note:  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.  This unit, and in particular in M.A18.68b, extends the work begun in Unit 2 on exponential functions with integral exponents.

M.A18.69

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)  Instructional Note:  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.

#### High School Algebra I for 8th Grade

M.A18.67

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions

Instructional Note:  Compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.

M.A18.68

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

1. 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.
2. Use the properties of exponents to interpret expressions for exponential functi For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

Instructional Note:  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.  This unit, and in particular in M.A18.68b, extends the work begun in Unit 2 on exponential functions with integral exponents.

M.A18.69

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)  Instructional Note:  Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic function can be factored.

#### Standards

M.A18.70

Write a function that describes a relationship between two quantities.

1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Instructional Note:  Focus on situations that exhibit a quadratic relationship.

#### High School Algebra I for 8th Grade

M.A18.70

Write a function that describes a relationship between two quantities.

1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Instructional Note:  Focus on situations that exhibit a quadratic relationship.

#### Standards

M.A18.71

Identify the effect on the graph of replacing f(x) by f(x) + 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.  Instructional Note:  Focus on quadratic functions, and consider including absolute value functions.

M.A18.72

Find inverse functions.

1. 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. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

Instructional Note:  Focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x > 0.

#### High School Algebra I for 8th Grade

M.A18.71

Identify the effect on the graph of replacing f(x) by f(x) + 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.  Instructional Note:  Focus on quadratic functions, and consider including absolute value functions.

M.A18.72

Find inverse functions.

1. 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. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

Instructional Note:  Focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x > 0.

#### Standards

M.A18.73

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.  Instructional Note:  Compare linear and exponential growth to growth of quadratic growth.

#### High School Algebra I for 8th Grade

M.A18.73

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.  Instructional Note:  Compare linear and exponential growth to growth of quadratic growth.