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Curriculum Standards:


Analyze functions that include absolute value expressions. - HSM.A1.5.1
Graph and apply piecewise-defined functions. - HSM.A1.5.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. - LER.M.A1HS.12
Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. - A1.A.4.3
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. - LER.M.A1HS.13
Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. - A1.A.4.1
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. (e.g., 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. - LER.M.A1HS.11
Graph and apply step functions. - HSM.A1.5.3
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - MAFS.912.F-IF.3.9
Graph and analyze transformations of the absolute value function. - HSM.A1.5.4
Add, subtract, and multiply polynomials. - HSM.A2.3.2
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - MAFS.K12.MP.3.1.a
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - MAFS.912.F-BF.1.2
Prove and use polynomial identities. - HSM.A2.3.3
Predict the behavior of polynomial functions. - HSM.A2.3.1
determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; - EI.A.6.a
write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and - EI.A.6.b
graph linear equations in two variables. - EI.A.6.c
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). - F-LE.2
Interpret the parameters in a linear or exponential function in terms of a context. - F-LE.5
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. - F-BF.3
Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1
(HONORS ONLY) Solve an equation of the form 𝘧(𝘧𝘹) = 𝘧𝘹𝘤 for a simple function 𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘧𝘹𝘤𝘧𝘧(𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - MAFS.912.F-BF.2.4.a
Solve systems of linear equations using the substitution method. - HSM.A1.4.2
Solve systems of linear equations using the elimination method. - HSM.A1.4.3
Describe the effect of the transformations 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥), 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥)+𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘, 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) - A1.FBF.3
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯) gives the number of person-hours it takes to assemble 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯 engines in a factory, then the positive integers would be an appropriate domain for the function. - MAFS.912.F-IF.2.5
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. - MAFS.912.F-IF.2.6
Graph solutions to linear inequalities in two variables. - HSM.A1.4.4
Graph and solve a system of linear inequalities. - HSM.A1.4.5
Find the zeros of quadratic functions. - HSM.A2.2.3
Solve problems with complex numbers. - HSM.A2.2.4
Identify key features of quadratic functions. - HSM.A2.2.1
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. - MAFS.912.F-IF.2.4
Write and graph quadratic functions in standard form. - HSM.A2.2.2
Solve linear-quadratic systems. - HSM.A2.2.7
Solve quadratic equations by completing the square. - HSM.A2.2.5
Solve quadratic equations using the Quadratic Formula. - HSM.A2.2.6
solve multistep linear inequalities in one variable algebraically and represent the solution graphically; - EI.A.5.a
represent the solution of linear inequalities in two variables graphically; - EI.A.5.b
solve practical problems involving inequalities; and - EI.A.5.c
represent the solution to a system of inequalities graphically. - EI.A.5.d
Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) - A1.ACE.1
Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) - A1.ACE.2
Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. - A1.ACE.4
Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. (Limit to linear; quadratic.) - A1.AAPR.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. - A-APR.1
intercepts; - F.A.7.d
values of a function for elements in its domain; and - F.A.7.e
connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. - F.A.7.f
Factor a quadratic trinomial. - HSM.A1.7.5
Factor a quadratic trinomial when a ≠ 1. - HSM.A1.7.6
determining whether a relation is a function; - F.A.7.a
domain and range; - F.A.7.b
Factor special trinomials. - HSM.A1.7.7
Solve systems of linear inequalities with a maximum of two variables; graph and interpret the solutions on a coordinate plane. - A1.A.2.3
zeros; - F.A.7.c
Represent relationships in various contexts with linear inequalities; solve the resulting inequalities, graph on a coordinate plane, and interpret the solutions. - A1.A.2.1
Combine like terms to simplify polynomials. - HSM.A1.7.1
Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. - A1.A.2.2
Multiply two polynomials. - HSM.A1.7.2
Use patterns to multiply binomials. - HSM.A1.7.3
Factor a polynomial. - HSM.A1.7.4
Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. - 9.2.1.6
evaluate algebraic expressions for given replacement values of the variables. - EO.A.1.b
Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods. - 9.2.1.7
Make qualitative statements about the rate of change of a function, based on its graph or table of values - 9.2.1.8
Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations - 9.2.1.9
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 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧 is a function and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹 is an element of its domain, then 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹) denotes the output of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹. The graph of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧 is the graph of the equation 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺 = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹). - MAFS.912.F-IF.1.1
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - MAFS.912.F-IF.1.2
Relate roots and rational exponents and use them to simplify expressions and solve equations. - HSM.A2.5.1
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧(0) = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧(1) = 1, 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯+1) = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 greater than or equal to 1. - MAFS.912.F-IF.1.3
represent verbal quantitative situations algebraically; and - EO.A.1.a
Perform operations on functions to answer real-world questions. - HSM.A2.5.5
Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. - A1.SPID.6
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. - N-RN.3
Graph and transform radical functions. - HSM.A2.5.3
Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain - 9.2.1.1
Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. - A1.SPID.7
Distinguish between functions and other relations defined symbolically, graphically or in tabular form - 9.2.1.2
Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.) - A1.FLQE.2
Find the domain of a function defined symbolically, graphically or in a real-world context. - 9.2.1.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function. - A1.FLQE.3
Obtain information and draw conclusions from graphs of functions and other relations. - 9.2.1.4
Represent the inverse of a relation using tables, graphs, and equations. - HSM.A2.5.6
Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form - 9.2.1.5
Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - A1.FLQE.5
Using technology, compute and interpret the correlation coefficient of a linear fit. - A1.SPID.8
Factor a quadratic expression to reveal the zeros of the function it defines. - A-SSE.3a
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - A-SSE.3b
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. - A1.FIF.8a
Use properties of exponents to solve equations with rational exponents. - HSM.A1.6.1
Recognize that arithmetic sequences are linear using equations, tables, graphs, and verbal descriptions. Use the pattern, find the next term. - A1.A.3.5
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. - LER.M.A1HS.32
Simplify polynomial expressions by adding, subtracting, or multiplying. - A1.A.3.2
Factor common monomial factors from polynomial expressions and factor quadratic expressions with a leading coefficient of 1. - A1.A.3.3
Describe and graph exponential functions. - HSM.A1.6.2
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). Instructional Note: In constructing linear functions, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions. - LER.M.A1HS.30
Use exponential functions to model situations and make predictions. - HSM.A1.6.3
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 exponential and linear models. - LER.M.A1HS.31
Solve equations involving several variables for one variable in terms of the others. - A1.A.3.1
Identify and describe geometric sequences. - HSM.A1.6.4
Perform, analyze, and use transformations of exponential functions. - HSM.A1.6.5
adding, subtracting, multiplying, and dividing polynomials; and - EO.A.2.b
applying the laws of exponents to perform operations on expressions; - EO.A.2.a
Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. - A1.AREI.1
factoring completely first- and second-degree binomials and trinomials in one variable. - EO.A.2.c
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: Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. - LER.M.A1HS.27
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. - LER.M.A1HS.28
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. Example:: For example, compare the growth of two linear functions, or two exponential functions such as y = 3ⁿ and y = 100²ⁿ) - LER.M.A1HS.25
Recognize that geometric sequences are exponential using equations, tables, graphs and verbal descriptions. Given the formula f(x) = a(r)x, find the next term and define the meaning of a and r within the context of the problem. - A1.A.3.6
Solve rational equations and identify extraneous solutions. - HSM.A2.4.5
Find the sum or difference of rational expressions. - HSM.A2.4.4
Solve systems of linear equations using the substitution method. - A1.AREI.6
Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. - A1.AREI.5
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A1.AREI.3
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 exponential functions whose domain is a subset of the integers. The Unit on Quadratic Functions and Modeling in this course and the Algebra II course address other types of functions. - LER.M.A1HS.23
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. - LER.M.A1HS.21
Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. (e.g., 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. - LER.M.A1HS.22
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (e.g., 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. Draw examples from linear functions and exponential functions having integral domains. Draw connection to M.A1HS.27, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. - LER.M.A1HS.20
Recognize 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. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.18
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. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.19
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. - LER.M.A1HS.16
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. - LER.M.A1HS.17
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. - LER.M.A1HS.14
Interpret parts of an expression, such as terms, factors, and coefficients. - A-SSE.1a
Write and solve equations with a variable on both sides to solve problems. - HSM.A1.1.3
Rewrite and use literal equations to solve problems. - HSM.A1.1.4
Solve and graph inequalities. - HSM.A1.1.5
Write and solve compound inequalities. - HSM.A1.1.6
Reason about operations with real numbers. - HSM.A1.1.1
Create and solve linear equations with one variable. - HSM.A1.1.2
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. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - MAFS.912.A-CED.1.3
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - MAFS.912.A-SSE.2.3.b
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - MAFS.912.A-CED.1.2
Factor a quadratic expression to reveal the zeros of the function it defines. - MAFS.912.A-SSE.2.3.a
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝 = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙 to highlight resistance 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙. - MAFS.912.A-CED.1.4
Write and solve absolute-value equations and inequalities - HSM.A1.1.7
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. - MAFS.912.A-CED.1.1
Interpret parts of an expression, such as terms, factors, and coefficients. - RQ.M.A1HS.4.a
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. - RQ.M.A1HS.5
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. - RQ.M.A1HS.6
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 and master linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Algebra II. - RQ.M.A1HS.9
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. - RQ.M.A1HS.7
The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of linear and quadratic functions. - S.A.9
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. - RQ.M.A1HS.8
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, quadratic, absolute value, and exponential functions. - A-REI.11
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. - A-REI.12
Fit a linear function for a scatter plot that suggests a linear association. - MAFS.912.S-ID.2.6.c
Informally assess the fit of a function by plotting and analyzing residuals. - MAFS.912.S-ID.2.6.b
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. - MAFS.912.S-ID.2.6.a
Use the structure of an expression to identify ways to rewrite it. Instructional Note: Focus on quadratic and exponential expressions. Example:: 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). - EE.M.A1HS.42
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 Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. - EE.M.A1HS.47
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 Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.46
Fit a linear function for a scatter plot that suggests a linear association. - S-ID.6c
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 Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.45
Informally assess the fit of a function by plotting and analyzing residuals. - S-ID.6b
Recognize 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. - EE.M.A1HS.44
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. - S-ID.6a
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: 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. - EE.M.A1HS.49
Solve absolute value equations and interpret the solutions in the original context. - A1.A.1.2
Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. - A1.A.1.3
Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. - A1.A.1.1
Distinguish between correlation and causation. - S-ID.9
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - S-ID.7
Compute (using technology) and interpret the correlation coefficient of a linear fit. - S-ID.8
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. - S-ID.5
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). - S-ID.3
Represent and analyze data with plots on the real number line (dot plots, histograms, and box plots). - S-ID.1
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. - S-ID.2
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. - MAFS.912.A-APR.1.1
Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - HSF-LE.B.5
Determine whether a function is a relation. - HSM.A1.3.1
Identify, evaluate, and graph linear functions. - HSM.A1.3.2
Transform linear equations - HSM.A1.3.3
Identify and describe arithmetic sequences. - HSM.A1.3.4
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by ??(0) = ??(1) = 1, ??(??+1) = ??(??) + ??(??-1) for ?? greater than or equal to 1. - HSF-IF.A.3
Interpret the parameters in a linear or exponential function in terms of a context. - MAFS.912.F-LE.2.5
Use a scatter plot to describe the relationship between two data sets. - HSM.A1.3.5
Find the line of best fit for a data set and evaluate its goodness of fit. - HSM.A1.3.6
Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A1.ASE.2
Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) - A1.ASE.1
Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. - HSM.A2.1.1
Interpret arithmetic sequences and series. - HSM.A2.1.4
Use graphs and tables to approximate solutions to algebraic equations and inequalities. - HSM.A2.1.5
Apply transformations to graph functions and write equations. - HSM.A2.1.2
Graph and interpret piecewise-defined functions. - HSM.A2.1.3
Use a variety of tools to solve systems of linear equations and inequalities. - HSM.A2.1.6
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - MAFS.912.A-REI.2.3
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). - MAFS.912.F-LE.1.2
Write and graph linear equations using point-slope form. - HSM.A1.2.2
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 equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. - EE.M.A1HS.48.b
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. - MAFS.912.F-LE.1.3
Write and graph linear equations using standard form. - HSM.A1.2.3
Write equations of parallel lines and perpendicular lines. - HSM.A1.2.4
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 solutions. Derive the quadratic formula from this form. - EE.M.A1HS.48.a
Write and graph linear equations using slope-intercept form. - HSM.A1.2.1
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. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. - MAFS.912.N-RN.1.1
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - MAFS.912.N-RN.1.2
Identify the effect on the graph of replacing 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹) by 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹) + 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬, 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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. - MAFS.912.F-BF.2.3
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. - MAFS.912.A-REI.3.5
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - MAFS.912.A-REI.3.6
Distinguish between relations and functions. - A1.F.1.1
Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts. - A1.F.1.2
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. - MAFS.912.N-RN.2.3
Distinguish between correlation and causation. - MAFS.912.S-ID.3.9
Compute (using technology) and interpret the correlation coefficient of a linear fit. - MAFS.912.S-ID.3.8
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - MAFS.912.S-ID.3.7
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. - MAFS.912.F-IF.3.7.e
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. - MAFS.912.F-IF.3.7.c
(HONORS ONLY) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - MAFS.912.F-IF.3.7.d
Graph linear and quadratic functions and show intercepts, maxima, and minima. - MAFS.912.F-IF.3.7.a
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - MAFS.912.F-IF.3.7.b
Write linear functions, using function notation, to model real-world and mathematical situations. - A1.F.1.3
Interpret parts of an expression, such as terms, factors, and coefficients. - MAFS.912.A-SSE.1.1.a
Use inverse functions to solve problems. - HSM.A1.10.7
Add, subtract, and multiply functions. - HSM.A1.10.6
Change functions to compress or stretch their graphs. - HSM.A1.10.5
Graph and analyze transformations of functions. - HSM.A1.10.4
Identify the function family when given an equation or graph. - HSM.A1.10.3
Identify the key features of the cube root function. - HSM.A1.10.2
Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹⁴ – 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺⁴ as (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹²)² – (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹² – 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - A-SSE.2
Describe the key features of the square root function. - HSM.A1.10.1
Identify different types of symmetry in two-dimensional figures. - HSM.G.3.5
Use the relationship between conditional probabilities and relative frequencies in contingency tables. - 9.4.3.9
Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. - 9.4.3.2
Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. - 9.4.3.5
Solve quadratic equations by inspection, 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 complex solutions and write them as 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎+𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖 for real numbers 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎 and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏. (Limit to non-complex roots.) - A1.AREI.4b
Use the method of completing the square to transform any quadratic equation in 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥 into an equation of the form (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥−ℎ)2=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘 that has the same solutions. Derive the quadratic formula from this form. - A1.AREI.4a
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. - MAFS.912.F-IF.3.8.a
Organize data in two-way frequency tables and use them to make inferences and generalizations. - HSM.A1.11.5
Quantify and analyze the spread of data. - HSM.A1.11.4
Interpret shapes of data displays representing different types of data distributions. - HSM.A1.11.3
Use measures of center and spread to compare data sets. - HSM.A1.11.2
(HONORS ONLY) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. - MAFS.912.A-REI.1.2
Organize and understand data using dot plots, histograms, and box plots. - HSM.A1.11.1
Combine standard function types using arithmetic operations. Example:: 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. - QFM.M.A1HS.57.b
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. - MAFS.912.A-REI.1.1
Solve quadratic equations by inspection (e.g., for 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹² = 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 complex solutions and write them as 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢 ± 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪 for real numbers 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢 and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣. - MAFS.912.A-REI.2.4.b
Use the method of completing the square to transform any quadratic equation in 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹 into an equation of the form (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹 – 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱)² = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲 that has the same solutions. Derive the quadratic formula from this form. - MAFS.912.A-REI.2.4.a
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - EE.M.A1HS.43.b
Factor a quadratic expression to reveal the zeros of the function it defines. - EE.M.A1HS.43.a
Identify and generate equivalent representations of linear equations, graphs, tables, and real-world situations. - A1.F.3.1
Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. - A1.F.3.2
Add, subtract, and multiply functions using function notation. - A1.F.3.3
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. - F-IF.8a
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 solutions. Derive the quadratic formula from this form. - A-REI.4a
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 equation. Recognize when the quadratic formula gives complex solutions. - A-REI.4b
Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. - HSM.G.5.5
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). - MAFS.912.S-ID.1.3
Use theorems to compare the sides and angles of a triangle. - HSM.G.5.4
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. - MAFS.912.S-ID.1.2
Represent data with plots on the real number line (dot plots, histograms, and box plots). - MAFS.912.S-ID.1.1
Use perpendicular and angle bisectors to solve problems. - HSM.G.5.1
Graph the solutions to a linear inequality in two variables. - A1.AREI.12
Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. - HSM.G.5.3
Solve an equation of the form 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥)=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥) graphically by identifying the 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥-coordinate(s) of the point(s) of intersection of the graphs of 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥) and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥). (Limit to linear; quadratic; exponential.) - A1.AREI.11
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - F-IF.9
Informally assess the fit of a function by plotting and analyzing residuals. Instructional Note: Focus should be on situations for which linear models are appropriate. - DS.M.A1HS.37.b
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. - DS.M.A1HS.37.a
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. - F-IF.6
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. - F-IF.5
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. - F-IF.4
Fit a linear function for scatter plots that suggest a linear association. - DS.M.A1HS.37.c
Recognize that sequences are functions whose domain is a subset of the integers. - F-IF.3
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - F-IF.2
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). - F-IF.1
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. - MAFS.912.S-ID.2.5
Graph square root and piecewise-defined functions, including absolute value functions. - F-IF.7b
Graph functions (linear and quadratic) and show intercepts, maxima, and minima. - F-IF.7a
Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. - A1.F.2.2
Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹⁴ – 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺⁴ as (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹²)² – (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹² – 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - MAFS.912.A-SSE.1.2
Write equivalent radical expressions. - HSM.A1.9.3
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 quadratic growth. - QFM.M.A1HS.60
Interpret parts of an expression, such as terms, factors, and coefficients. - EE.M.A1HS.41.a
Solve quadratic equations by taking square roots. - HSM.A1.9.4
Use completing the square to solve quadratic equations. - HSM.A1.9.5
Use the quadratic formula to solve quadratic equations. - HSM.A1.9.6
Use tables and graphs to find solutions of quadratic equations. - HSM.A1.9.1
Find the solution of a quadratic equation by factoring. - HSM.A1.9.2
square roots of whole numbers and monomial algebraic expressions; - EO.A.3.a
Represent data with plots on the real number line (dot plots, histograms, and box plots). - DS.M.A1HS.33
Add, subtract, multiply, divide and simplify algebraic fractions. - 9.2.3.4
Check whether a given complex number is a solution of a quadratic equation by substituting it for the variable and evaluating the expression, using arithmetic with complex numbers. - 9.2.3.5
numerical expressions containing square or cube roots. - EO.A.3.c
Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. - 9.2.3.6
Combine standard function types using arithmetic operations. (e.g., 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.) - LER.M.A1HS.26.b
cube roots of integers; and - EO.A.3.b
Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. - 9.2.3.7
Solve a system with linear and quadratic equations. - HSM.A1.9.7
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. - DS.M.A1HS.36
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 – 8, 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. - DS.M.A1HS.35
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 – 8, 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. - DS.M.A1HS.34
Use the unit circle to evaluate the trigonometric functions of any angle. - HSM.A2.7.3
Compute (using technology) and interpret the correlation coefficient of a linear fit. Instructional Note: Build on students’ work with linear relationships in eighth grade 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. - DS.M.A1HS.39
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 in eighth grade 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. - DS.M.A1HS.38
Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. - 9.2.3.1
Create and use graphs of sine and cosine functions. - HSM.A2.7.4
Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. - 9.2.3.3
Find inverse functions. 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. 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. Example:: For example, f(x) = 2 x³ or f(x) = (x+1)/(x-1) for x ≠ 1. - QFM.M.A1HS.59
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. - QFM.M.A1HS.58
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Instructional Note: 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 equation can be factored. Example:: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - QFM.M.A1HS.56
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 studied in the Unit on Linear and Exponential Relationships. - QFM.M.A1HS.53
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. Example:: 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. - QFM.M.A1HS.52
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 studied in the Unit on Linear and Exponential Relationships. - QFM.M.A1HS.51
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). - QFM.M.A1HS.50
Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.) - A1.FIF.9
Distinguish between correlation and causation. Instructional Note: The important distinction between a statistical relationship and a cause-and-effect relationship is the focus. - DS.M.A1HS.40
Rewrite expressions involving simple radicals and rational exponents in different forms. - A1.NRNS.1
Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. - A1.FIF.2
Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. - A1.NRNS.2
Extend previous knowledge of a function to apply to general behavior and features of a function. - A1.FIF.1
Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) - A1.FIF.4
Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.) - A1.FIF.6
Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.) - A1.FIF.5
Explain why the sum or product of 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. - A1.NRNS.3
Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥+𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘.) - A1.FIF.7
Describe a data set using data displays, describe and compare data sets using summary statistics, including measures of central tendency, location, and spread. Know how to use calculators, spreadsheets, or other appropriate technology to display data and calculate summary statistics. - A1.D.1.1
Collect data and use scatterplots to analyze patterns and describe linear relationships between two variables. Using graphing technology, determine regression lines and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. - A1.D.1.2
Interpret graphs as being discrete or continuous. - A1.D.1.3
Use quadratic functions to model real-world situations. - HSM.A1.8.4
Determine whether a linear, exponential, or quadratic function best models a data set. - HSM.A1.8.5
Identify key features of the graph of the quadratic parent function. - HSM.A1.8.1
Graph quadratic functions using the vertex form. - HSM.A1.8.2
Graph quadratic functions using standard form. - HSM.A1.8.3
Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts. - 9.2.2.5
Sketch the graphs of common non-linear functions such as f (x) = √x, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = √x-2+4. Know how to use graphing technology to graph these functions. - 9.2.2.6
Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. - HSM.A2.6.4
Recognize the key features of exponential functions. - HSM.A2.6.1
Write exponential models in different ways to solve problems. - HSM.A2.6.2
Represent and solve problems in various contexts using linear and quadratic functions. - 9.2.2.1
Identify, write, and use geometric sequences and series. - HSM.A2.6.7
Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. - 9.2.2.2
Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. - 9.2.2.3
Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. - 9.2.2.4
Solve exponential and logarithmic equations. - HSM.A2.6.6
Solve systems of linear equations using linear combination. - A1.AREI.6b
Solve systems of linear equations using the substitution method. - A1.AREI.6a
Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. - LER.M.A1HS.24.b
Find the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation. - A1.ASE.3 a
Graph linear and quadratic functions and show intercepts, maxima, and minima. - LER.M.A1HS.24.a
Understand and apply the geometric properties of a parabola. - HSM.A2.9.1
Understand and apply the geometric properties of a hyperbola. - HSM.A2.9.4
Write, graph, and apply the equation of a circle. - HSM.A2.9.2
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. - A1.FIF.1a
Represent a function using function notation and explain that 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥) denotes the output of function 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓 that corresponds to the input 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥. - A1.FIF.1b
Understand that the graph of a function labeled as 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓 is the set of all ordered pairs (𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥,𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦) that satisfy the equation 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦=𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥). - A1.FIF.1c
Graph linear and quadratic functions and show intercepts, maxima, and minima. - QFM.M.A1HS.54.a
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - QFM.M.A1HS.54.b
multistep linear equations in one variable algebraically; - EI.A.4.a
quadratic equations in one variable algebraically; - EI.A.4.b
Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. - 9.4.1.1
literal equations for a specified variable; - EI.A.4.c
Analyze the effects on summary statistics of changes in data sets. - 9.4.1.2
systems of two linear equations in two variables algebraically and graphically; and - EI.A.4.d
Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. - 9.4.1.3
practical problems involving equations and systems of equations. - EI.A.4.e
Identify and describe transformations of two-dimensional figures. - HSM.G.3
Use the relationships between sides, segments, and angles of triangles to solve problems. - HSM.G.5
Use inductive reasoning to make conjectures about mathematical relationships. - HSM.G.1.4
Use properties of segments and angles to find their measures. - HSM.G.1.1
Combine standard function types using arithmetic operations. Example: 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. - MAFS.912.F-BF.1.1.b
Evaluate data distributions. - HSM.A2.11.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A-REI.3
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. - A-REI.1
Solve systems of linear equations algebraically, exactly, and graphically while focusing on pairs of linear equations in two variables. - A-REI.6
Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines. - 9.2.4.4
Given a system of two equations in two variables, show and explain why the sum of equivalent forms of the equations produces the same solution as the original system. - A-REI.5
Solve linear programming problems in two variables using graphical methods. - 9.2.4.5
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. - A-CED.4
Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically. - 9.2.4.6
Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. - 9.2.4.7
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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - A-CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - 9.2.4.8
Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [Note this standard appears in future courses with a slight variation in the standard language.] - A-CED.2
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. - A-CED.1
Explain why the 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹-coordinates of the points where the graphs of the equations 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹𝘺 = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹𝘺𝘧(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹𝘺𝘧𝘹) and 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹𝘺𝘧𝘹𝘺 = 𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦𝑦𝑓𝑥𝘹𝘺𝘧𝘹𝘺𝑔(𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘝𝘭𝘙𝘙𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝑥𝑥𝑘𝘹𝘢𝘣𝘪𝘢𝘣𝘹𝘹𝘱𝘲𝑓𝑥𝑔𝑥𝑥𝑦𝑓𝑥𝑦𝑔𝑥𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑦𝑎𝑥𝑘𝑓𝑥𝑓𝑥𝑓𝑥𝑦�