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Product Name: Florida Algebra 1
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Curriculum Standards:


Choose and produce equivalent forms of a quadratic expression or equations to reveal and explain properties. a. Find the zeros of a quadratic function by rewriting it in factored form. b. Find the maximum or minimum value of a quadratic function by completing the square. - A1.SSE.A.2
Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely. - MAFS.912.A-SSE.1.AP.2b
Solve quadratic equations by using the quadratic formula. - MAFS.912.A-REI.2.AP.4b
Solve quadratic equations by factoring. - MAFS.912.A-REI.2.AP.4c
Simplify expressions including combining like terms, using the distributive property, and other operations with polynomials. - MAFS.912.A-SSE.1.AP.2c
Analyze functions that include absolute value expressions. - HSM.A1.5.1
Graph and apply piecewise-defined functions. - HSM.A1.5.2
Solve quadratic equations by completing the square. - MAFS.912.A-REI.2.AP.4a
Rewrite algebraic expressions in different equivalent forms, such as factoring or combining like terms. - MAFS.912.A-SSE.1.AP.2a
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
Translate between a graph and a situation described qualitatively. - A1.A.4.4
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
Graph and analyze transformations of the absolute value function. - HSM.A1.5.4
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
(HONORS ONLY) Rewrite simple rational expressions in different forms; write 𝘢(𝘢𝘹)/𝘢𝘹𝘣(𝘢𝘹𝘣𝘹) in the form 𝘢𝘹𝘣𝘹𝘲(𝘢𝘹𝘣𝘹𝘲𝘹) + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹)/𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹), where 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹), 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹), 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹), and 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹) are polynomials with the degree of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹) less than the degree of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹), using inspection, long division, or, for the more complicated examples, a computer algebra system. - MAFS.912.A-APR.4.6
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
Prove and use polynomial identities. - HSM.A2.3.3
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
Predict the behavior of polynomial functions. - HSM.A2.3.1
Use roots of a polynomial equation to find other roots. - HSM.A2.3.6
Identify symmetry in and transform polynomial functions. - HSM.A2.3.7
Divide polynomials. - HSM.A2.3.4
Model and solve problems using the zeros of a polynomial function. - HSM.A2.3.5
Solve multi-variable formulas or literal equations for a specific variable. - MAFS.912.A-CED.1.AP.4a
Solve linear equations in one variable, including coefficients represented by letters. - MAFS.912.A-REI.2.AP.3a
Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f-_, and the range of the function f is the domain of the inverse function f-_. - A2.F.2.3
Solve linear inequalities in one variable, including coefficients represented by letters. - MAFS.912.A-REI.2.AP.3b
Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function. - NC.M1.F-IF.3
Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums. - NC.M1.F-IF.4
end behavior; - F.AII.7.h
Build an understanding 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 by recognizing that: 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). - NC.M1.F-IF.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
composition of functions algebraically and graphically. - F.AII.7.k
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
Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - NC.M1.F-IF.2
inverse of a function; and - F.AII.7.j
graph linear equations in two variables. - EI.A.6.c
Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior. - NC.M1.F-IF.7
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
intercepts; - F.AII.7.e
Use equivalent expressions to reveal and explain different properties of a function. - NC.M1.F-IF.8
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. - ELD.K12.ELL.MA.1
Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes. - NC.M1.F-IF.5
connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; - F.AII.7.g
Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically. - NC.M1.F-IF.6
Interpret the parameters in a linear or exponential function in terms of a context. - F-LE.5
extrema; - F.AII.7.c
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 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹) = 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤 for a simple function 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - MAFS.912.F-BF.2.4.a
Solve systems of linear equations using the substitution method. - HSM.A1.4.2
(HONORS ONLY) Verify by composition that one function is the inverse of another. - MAFS.912.F-BF.2.4.b
Solve systems of linear equations using the elimination method. - HSM.A1.4.3
(HONORS ONLY) Read values of an inverse function from a graph or a table, given that the function has an inverse. - MAFS.912.F-BF.2.4.c
(HONORS ONLY) Produce an invertible function from a non-invertible function by restricting the domain. - MAFS.912.F-BF.2.4.d
Describe the effect of the transformations 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥), 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥)+𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘, 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 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 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯) gives the number of person-hours it takes to assemble 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯 engines in a factory, then the positive integers would be an appropriate domain for the function. - MAFS.912.F-IF.2.5
Use function notation to evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - A1.IF.A.2
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
Graph equations in two or more variables on coordinate axes with labels and scales. - MAFS.912.A-CED.1.AP.2a
Find the zeros of quadratic functions. - HSM.A2.2.3
Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other. - NC.M2.G-CO.6
Solve problems with complex numbers. - HSM.A2.2.4
Identify key features of quadratic functions. - HSM.A2.2.1
Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - NC.M2.G-CO.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 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
Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image. - NC.M2.G-CO.5
Solve linear-quadratic systems. - HSM.A2.2.7
Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry. - NC.M2.G-CO.3
Solve quadratic equations by completing the square. - HSM.A2.2.5
Solve quadratic equations using the Quadratic Formula. - HSM.A2.2.6
Given a correlation in a real-world scenario, determine if there is causation. - MAFS.912.S-ID.3.AP.9a
Identify the different parts of the expression and explain their meaning within the context of a problem. - MAFS.912.A-SSE.1.AP.1a
Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts. - MAFS.912.A-SSE.1.AP.1b
Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. - A2.F.1.2
Describe the meaning of the factors and intercepts on linear and exponential functions. - MAFS.912.F-LE.2.AP.5a
Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2
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
Determine an explicit expression or steps for calculation from a context. - F-BF.1a
represent the solution to a system of inequalities graphically. - EI.A.5.d
Extend the knowledge of factoring to include factors with complex coefficients. - A2.APR.A.1
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
Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational 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). - PRR.M.A2HS.7
Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. - A1.ACE.4
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: Extend beyond the quadratic polynomials found in Algebra I. - PRR.M.A2HS.9
Identify and interpret the solution of a system of linear equations from a real-world context that has been graphed. - MAFS.912.A-CED.1.AP.3a
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
Understand that the graph of a function labeled 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓 is the set of all ordered pairs (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥, y) that satisfy the equation 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦=f (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥). - A1.IF.A.1b
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial (limit to 1st- and 2nd degree polynomials). - A-APR.3
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
determining whether a relation is a function; - F.A.7.a
Factor a quadratic trinomial when a ≠ 1. - HSM.A1.7.6
Solve systems of linear inequalities with a maximum of two variables; graph and interpret the solutions on a coordinate plane. - A1.A.2.3
domain and range; - F.A.7.b
Factor special trinomials. - HSM.A1.7.7
zeros; - F.A.7.c
Combine like terms to simplify polynomials. - HSM.A1.7.1
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
Multiply two polynomials. - HSM.A1.7.2
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
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
Compare the properties of two functions. - MAFS.912.F-IF.3.AP.9a
Make qualitative statements about the rate of change of a function, based on its graph or table of values - 9.2.1.8
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 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧 is a function and 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹 is an element of its domain, then 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹) denotes the output of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹. The graph of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧 is the graph of the equation 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺 = 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹). - MAFS.912.F-IF.1.1
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
Find the maximum or minimum value of a quadratic function by completing the square. - A1.SSE.A.3b
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 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧(0) = 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧(1) = 1, 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯+1) = 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 greater than or equal to 1. - MAFS.912.F-IF.1.3
Find the zeros of a quadratic function by rewriting it in factored form. - A1.SSE.A.3a
represent verbal quantitative situations algebraically; and - EO.A.1.a
Solve radical equations and inequalities. - HSM.A2.5.4
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
Use properties of exponents and radicals to simplify radical expressions. - HSM.A2.5.2
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. - N-RN.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
Graph and transform radical functions. - HSM.A2.5.3
Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. - A1.FLQE.1
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
Distinguish between functions and other relations defined symbolically, graphically or in tabular form - 9.2.1.2
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
Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - A1.FLQE.5
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
Using technology, compute and interpret the correlation coefficient of a linear fit. - A1.SPID.8
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - N-RN.2
(HONORS ONLY) Know and apply the Remainder Theorem: For a polynomial 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹) and a number 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢, the remainder on division by 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹 – 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢 is 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢), so 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢) = 0 if and only if (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹 – 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢) is a factor of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹). - MAFS.912.A-APR.2.2
Extend the properties of exponents to justify that (51/2)2=5. - MAFS.912.N-RN.1.AP.1b
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. - MAFS.912.A-APR.2.3
Understand that the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand (e.g., 51/2 = 5). - MAFS.912.N-RN.1.AP.1a
Convert from radical representation to using rational exponents and vice versa. - MAFS.912.N-RN.1.AP.2a
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
Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. - A2.AAPR.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. - A1.FIF.8a
Use properties of exponents to solve equations with rational exponents. - HSM.A1.6.1
Rewrite expressions involving radicals and rational exponents using the properties of exponents. Limit to rational exponents with a numerator of 1. - A1.NQ.A.2
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
Compare key features of two functions (linear, quadratic, or exponential) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions). - NC.M1.F-IF.9
Explain how the meaning of rational exponents extends from the properties of integer exponents. - A1.NQ.A.1
Factor common monomial factors from polynomial expressions and factor quadratic expressions with a leading coefficient of 1. - A1.A.3.3
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). 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
Describe and graph exponential functions. - HSM.A1.6.2
Interpret expressions composed of multiple parts by viewing one or more of their parts as a single entity to give meaning in terms of a context. - NC.M3.A-SSE.1b
Solve equations involving several variables for one variable in terms of the others. - A1.A.3.1
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
Identify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including terms, factors, coefficients, and exponents. - NC.M3.A-SSE.1a
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
Describe the effects of transformations algebraically and graphically, creating vertical and horizontal translations, vertical and horizontal reflections and dilations (expansions/compressions) for linear, quadratic, cubic, square and cube root, absolute value, exponential and logarithmic functions. - A2.BF.A.3
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
Derive inverses of functions, and compose the inverse with the original function to show that the functions are inverses. - A2.BF.A.2
factoring completely first- and second-degree binomials and trinomials in one variable. - EO.A.2.c
add, subtract, multiply, divide, and simplify rational algebraic expressions; - EO.AII.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. Instructional Note: Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. - LER.M.A1HS.27
Use inverse variation and graph translations of the reciprocal function. - HSM.A2.4.1
Locate the key features of linear and quadratic equations. - MAFS.912.F-IF.3.AP.7b
add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and - EO.AII.1.b
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
Graph rational functions. - HSM.A2.4.2
(HONORS ONLY) Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹² + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺²)² = (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹² – 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺²)² + (2𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺)² can be used to generate Pythagorean triples. - MAFS.912.A-APR.3.4
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
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
Write arguments focused on discipline-specific content. a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience’s knowledge level and concerns. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. - LAFS.910.WHST.1.1
Select a graph of a function that displays its symbolic representation (e.g., f(x) = 3x + 5). - MAFS.912.F-IF.3.AP.7a
Solve rational equations and identify extraneous solutions. - HSM.A2.4.5
Find the product and the quotient of rational expressions. - HSM.A2.4.3
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
Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. - A1.FLQE.1a
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
Write or select an equivalent form of a function [e.g., y = mx + b, f(x) = y, y - y1 = m(x - x1), Ax + By = C]. - MAFS.912.F-IF.3.AP.8a
Recognize 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. - LER.M.A1HS.15
Describe the properties of a function (e.g., rate of change, maximum, minimum, etc.). - MAFS.912.F-IF.3.AP.8b
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
Calculate, interpret, and apply expected value. - HSM.G.12.5
Reason about operations with real numbers. - HSM.A1.1.1
Create and solve linear equations with one variable. - HSM.A1.1.2
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
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
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 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝 = 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙 to highlight resistance 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙. - MAFS.912.A-CED.1.4
Write and solve absolute-value equations and inequalities - HSM.A1.1.7
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, absolute, and exponential functions. - MAFS.912.A-CED.1.1
Add and subtract two rational expressions, a(x) and b(x), where the denominators of both a(x) and b(x) are linear expressions. - NC.M3.A-APR.7a
Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics. - LAFS.910.RST.2.4
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
Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. - LAFS.910.WHST.2.4
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
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
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
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
The student, given a data set or practical situation, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically. - S.A.8
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). - A-REI.10
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
Rewrite algebraic expressions with integer exponents using the properties of exponents. - NC.M1.N-RN.2
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
Identify and graph the solutions (ordered pairs) on a graph of an equation in two variables. - MAFS.912.A-REI.4.AP.10a
Fit a linear function for a scatter plot that suggests a linear association. - S-ID.6c
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
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
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. - MAFS.K12.MP.5.1.a
Solve equations with one or two variables and explain the process. - MAFS.912.A-REI.1.AP.1a
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
Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. - LAFS.910.RST.3.7
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
Add, subtract, multiply, divide, and simplify polynomial and rational expressions. - A2.A.2.2
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
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - A2.A.2.4
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
Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. - A2.A.2.1
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. - MAFS.912.A-APR.1.1
Distinguish between association and causation. - NC.M1.S-ID.9
Analyze patterns and describe relationships between two variables in context. Using technology, determine the correlation coefficient of bivariate data and interpret it as a measure of the strength and direction of a linear relationship. Use a scatter plot, correlation coefficient, and a residual plot to determine the appropriateness of using a linear function to model a relationship between two variables. - NC.M1.S-ID.8
Solve simple rational and radical equations in one variable. - MAFS.912.A-REI.1.AP.2a
Use technology to represent data with plots on the real number line (histograms, and box plots). - NC.M1.S-ID.1
Create equations and inequalities in one variable that represent linear, exponential, and quadratic relationships and use them to solve problems. - NC.M1.A-CED.1
Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities. - NC.M1.A-CED.2
Create systems of linear equations and inequalities to model situations in context. - NC.M1.A-CED.3
Examine the effects of extreme data points (outliers) on shape, center, and/or spread. - NC.M1.S-ID.3
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. Interpret differences in shape, center, and spread in the context of the data sets. - NC.M1.S-ID.2
Solve for a quantity of interest in formulas used in science and mathematics using the same reasoning as in solving equations. - NC.M1.A-CED.4
Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - HSF-LE.B.5
Interpret in context the rate of change and the intercept of a linear model. Use the linear model to interpolate and extrapolate predicted values. Assess the validity of a predicted value. - NC.M1.S-ID.7
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. - NC.M1.S-ID.6
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
Write or select the graph that represents a defined change in the function (e.g., recognize the effect of changing k on the corresponding graph). - MAFS.912.F-BF.2.AP.3a
Recognize exponential situations in which a quantity grows or decays by a constant percent rate per unit interval. - A1.LQE.A.1b
Use a scatter plot to describe the relationship between two data sets. - HSM.A1.3.5
Determine the average rate of change of a function over a specified interval and interpret the meaning. - A1.IF.B.5
Find the line of best fit for a data set and evaluate its goodness of fit. - HSM.A1.3.6
Interpret the parameters of a linear or exponential function in terms of the context. - A1.IF.B.6
Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A1.ASE.2
Using tables, graphs and verbal descriptions, interpret key characteristics of a function that models the relationship between two quantities. - A1.IF.B.3
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
Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. - A1.IF.B.4
Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents. - NC.M1.A-SSE.1a
Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression. - NC.M1.A-SSE.1b
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
Solve linear equations and inequalities in one variable. - NC.M1.A-REI.3
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
Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning. - NC.M1.A-REI.1
Graph and interpret piecewise-defined functions. - HSM.A2.1.3
Create a multiple of a linear equation showing that they are equivalent (e.g., x + y = 6 is equivalent to 2x + 2y = 12). - MAFS.912.A-REI.3.AP.5a
Use a variety of tools to solve systems of linear equations and inequalities. - HSM.A2.1.6
Solve systems of equations using matrices. - HSM.A2.1.7
Find the sum of two equations. - MAFS.912.A-REI.3.AP.5b
Explain how each step taken when solving an equation or inequality in one variable creates an equivalent equation or inequality that has the same solution(s) as the original. - A1.REI.A.1
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. - MAFS.K12.MP.4.1.a
Find an inverse function. - NC.M3.F-BF.4
Use tables, graphs, or algebraic methods (substitution and elimination) to find approximate or exact solutions to systems of linear equations and interpret solutions in terms of a context. - NC.M1.A-REI.6
Explain why replacing one equation in a system of linear equations by the sum of that equation and a multiple of the other produces a system with the same solutions. - NC.M1.A-REI.5
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - MAFS.912.A-REI.2.3
Describe the effect of the transformations 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥), 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥)+𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘, 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3
Create linear, quadratic, rational, and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems. - MAFS.912.A-CED.1.AP.1a
Given a graph, describe or select the solution to a system of linear equations. - MAFS.912.A-REI.3.AP.6a
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
Translate between different but equivalent forms of a function to reveal and explain properties of the function and interpret these in terms of a context. - A1.IF.C.8
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
Compare the properties of two functions given different representations. - A1.IF.C.9
Write equations of parallel lines and perpendicular lines. - HSM.A1.2.4
Graph functions expressed symbolically and identify and interpret key features of the graph. - A1.IF.C.7
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
Add, subtract, multiply and divide radical expressions. - A2.NQ.A.3
Write and graph linear equations using slope-intercept form. - HSM.A1.2.1
Extend the system of powers and roots to include rational exponents. - A2.NQ.A.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
Analyze the effect of translations and scale changes on functions. - A1.BF.A.1
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - A1.BF.A.2
Identify the effect on the graph of replacing 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹) by 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹) + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬, 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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
Understand the solution to a system of two linear equations in two variables corresponds to a point(s) of an intersection of their graphs, because the point(s) of intersection satisfies both equations simultaneously. - MAFS.912.A-REI.4.AP.11a
Write an equivalent form of an exponential expression by using the properties of exponents to transform expressions to reveal rates based on different intervals of the domain. - NC.M3.A-SSE.3
Use the structure of an expression to identify ways to write equivalent expressions. - NC.M3.A-SSE.2
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
(HONORS ONLY) Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺 = –3𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺𝘹 and the circle 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺𝘹𝘹² + 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺𝘹𝘹𝘺² = 3. - MAFS.912.A-REI.3.7
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
Select a function that describes a relationship between two quantities (e.g., relationship between inches and centimeters, Celsius Fahrenheit, distance = rate x time, recipe for peanut butter and jelly- relationship of peanut butter to jelly f(x)=2x, where x is the quantity of jelly, and f(x) is peanut butter). - MAFS.912.F-BF.1.AP.1a
Graph a linear inequality in two variables using at least two coordinate pairs that are solutions. - MAFS.912.A-REI.4.AP.12a
Graph a system of linear inequalities in two variables using at least two coordinate pairs for each inequality. - MAFS.912.A-REI.4.AP.12b
Solve a system of linear equations algebraically and/or graphically. - A1.REI.B.3
Justify that the technique of linear combination produces an equivalent system of equations. - A1.REI.B.5
Solve a system consisting of a linear equation and a quadratic equation algebraically and/or graphically. - A1.REI.B.4
Use the method of completing the square to create an equivalent quadratic equation. - A1.REI.A.2a
Determine and interpret the correlation coefficient for a linear association. - A1.DS.A.7
Interpret the slope (rate of change) and the y-intercept (constant term) of a linear model in the context of the data. - A1.DS.A.6
Derive the quadratic formula. - A1.REI.A.2b
Recognize that the domain of a sequence is a subset of the integers. - MAFS.912.F-IF.1.AP.3a
Distinguish between relations and functions. - A1.F.1.1
Distinguish between correlation and causation. - A1.DS.A.8
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
Analyze different methods of solving quadratic equations. - A1.REI.A.2c
Analyze and interpret graphical displays of data. - A1.DS.A.1
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
Interpret differences in shape, center and spreads in the context of the data sets, accounting for possible effects of outliers. - A1.DS.A.3
Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets. - A1.DS.A.2
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
Summarize data in two-way frequency tables. a. Interpret relative frequencies in the context of the data. b. Recognize possible associations and trends in the data. - A1.DS.A.4
Use the zeros of a function to sketch a graph of the function. - MAFS.912.A-APR.2.AP.3b
Create and graph linear, quadratic and exponential equations in two variables. - A1.CED.A.2
Find the zeros of a polynomial when the polynomial is factored (e.g., If given the polynomial equation y = x ² + 5x + 6, factor the polynomial as y = (x + 3)(x + 2). Then find the zeros of y by setting each factor equal to zero and solving. x = -2 and x = -3 are the two zeroes of y.). - MAFS.912.A-APR.2.AP.3a
Create equations and inequalities in one variable and use them to model and/or solve problems. - A1.CED.A.1
Solve literal equations and formulas for a specified variable that highlights a quantity of interest. - A1.CED.A.4
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
Represent constraints by equations or inequalities and by systems of equations or inequalities, and interpret the data points as a solution or non-solution in a modeling context. - A1.CED.A.3
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
Complete a graph given the data, using dot plots, histograms or box plots. - MAFS.912.S-ID.1.AP.1a
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
Construct an exponential function to model bivariate data represented on a scatter plot that minimizes residuals. - A1.DS.A.5b
Describe the accuracy of measurement when reporting quantities (you can lessen your limitations by measuring precisely). - MAFS.912.N-Q.1.AP.3a
Construct a linear function to model bivariate data represented on a scatter plot that minimizes residuals. - A1.DS.A.5a
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
Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺𝘹𝘹𝘺𝘹⁴ – 𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘩𝘯𝘯𝑓𝑥𝑦𝑥𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘝𝘭𝘙𝘙𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘺𝘹𝘹𝘺𝘹𝘺⁴ as (𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝑘𝑓�