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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
Use the change of base formula. - MAFS.912.F-BF.2.a
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ - A.SSE.4 (+)
Apply the Addition Rule, 𝘗(𝘗𝘈 or 𝘗𝘈𝘉) = 𝘗𝘈𝘉𝘗(𝘗𝘈𝘉𝘗𝘈) + 𝘗𝘈𝘉𝘗𝘈𝘗(𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - S-CP.7
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
Emphasize the selection of a type of function for a model based on behavior of data and context. (A2, M3) - F.IF.5c
Understand the conditional probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈 given 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 as 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉)/𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - S-CP.3
Rewrite simple rational expressions in different forms; write 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹)/𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹) in the form 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹) + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹)/𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹), where 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹), 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹), 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹), and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹) are polynomials with the degree of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹) less than the degree of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹), using inspection, long division, or, for the more complicated examples, a computer algebra system. - MAFS.912.A-APR.4.6
(HONORS ONLY) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. - MAFS.912.A-APR.4.7
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Example: For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. - S-CP.4
Find the conditional probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈 given 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉 as the fraction of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉’s outcomes that also belong to 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - S-CP.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
Understand that two events 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉 are independent if the probability of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - S-CP.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
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
Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. - A2.F.2.4
Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. - A2.F.2.1
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - F-BF.2
Combine functions by composition and recognize that g(x) = f-_(x), the inverse function of f(x), if and only if f(g(x))= g(f(x)) = x. - A2.F.2.2
vertical and horizontal asymptotes; - F.AII.7.i
end behavior; - F.AII.7.h
composition of functions algebraically and graphically. - F.AII.7.k
inverse of a function; and - F.AII.7.j
intercepts; - F.AII.7.e
Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to calculate trigonometric ratios. - MAFS.912.F-TF.3.8
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
zeros; - F.AII.7.d
For exponential models, express as a logarithm the solution to 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣 to the 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵 power = 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥 where 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢, 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤, and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥 are numbers and the base 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣 is 2, 10, or 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦; evaluate the logarithm using technology. - F-LE.4
connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; - F.AII.7.g
values of a function for elements in its domain; - F.AII.7.f
Interpret the parameters in a linear or exponential function in terms of a context. - F-LE.5
domain, range, and continuity; - F.AII.7.a
extrema; - F.AII.7.c
Understand and apply theorems about circles: understand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles; understand and apply theorems about relationships with line segments and circles including, radii, diameter, secants, tangents and chords. - NC.M3.G-C.2
intervals in which a function is increasing or decreasing; - F.AII.7.b
Identify the effect on the graph of replacing 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹) by 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹) + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬, 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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
Using similarity, demonstrate that the length of an arc, s, for a given central angle is proportional to the radius, r, of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the circle, s/r. Find arc lengths and areas of sectors of circles. - NC.M3.G-C.5
Graph exponential functions, indicating intercepts and end behavior, and trigonometric functions, showing period, midlineG, and amplitude. (A2, M3) - F.IF.7f
Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1
Solve an equation of the form 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹) = 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤 for a simple function 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - MAFS.912.F-BF.2.4.a
Solve systems of linear equations using the substitution method. - HSM.A1.4.2
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
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
Produce an invertible function from a non-invertible function by restricting the domain. - MAFS.912.F-BF.2.4.d
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯) gives the number of person-hours it takes to assemble 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯 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
Find the additive and multiplicative inverses of square matrices, using technology as appropriate. - A2.5.a
Explain the role of the determinant in determining if a square matrix has a multiplicative inverse. - A2.5.b
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (A2, M3) - F.IF.7c
Graph polynomial functions, identifying zeros, when factoring is reasonable, and indicating end behavior. (A2, M3) - F.IF.7d
Know there is a complex number 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖 such that 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖2=−1, and every complex number has the form 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎+𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖 with 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏 real. - A2.NCNS.1
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
Write and graph quadratic functions in standard form. - HSM.A2.2.2
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
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
Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. - A2.F.1.4
Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease. - A2.F.1.5
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
Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. - A2.F.1.3
Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. - A2.ASE.1
Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. - A2.F.1.8
Graph a rational function and identify the x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology. (Excluding slant or oblique asymptotes and holes.) - A2.F.1.6
Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2
Graph a radical function (square root and cube root only) and identify the x- and y-intercepts using various methods and tools that may include a graphing calculator or other appropriate technology. - A2.F.1.7
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. (A2, M3) - F.IF.8a
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t , and y = (0.97)^t and classify them as representing exponential growth or decay. (A2, M3) - F.IF.8b
Determine an explicit expression, a recursive process, or steps for calculation from a context. - F-BF.1a
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. - F-BF.1b
Add, subtract, multiply and divide rational expressions. - A2.APR.A.4
Find the least common multiple of two or more polynomials. - A2.APR.A.3
Understand the Remainder Theorem and use it to solve problems. - A2.APR.A.2
Extend the knowledge of factoring to include factors with complex coefficients. - A2.APR.A.1
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Instructional Note: Consider extending this standard to infinite geometric series in curricular implementations of this course description. Example:: For example, calculate mortgage payments. - PRR.M.A2HS.8
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
Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems. - NC.M3.G-CO.14
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
Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). - PRR.M.A2HS.4
Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain. - A2.F.1.1
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Instructional Note: Limit to polynomials with real coefficients. - PRR.M.A2HS.5
Know and apply the Remainder Theorem: For a polynomial 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹) and a number 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢, the remainder on division by 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹 – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢 is 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢), so 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢) = 0 if and only if (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹 – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢) is a factor of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹). - A-APR.2
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. - A-APR.3
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹² + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺²)² = (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹² – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺²)² + (2𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺)² can be used to generate Pythagorean triples. - A-APR.4
Rewrite simple rational expressions in different forms; write 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹)/𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹) in the form 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹) + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹)/𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹), where 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹), 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹), 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹), and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹) are polynomials with the degree of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹) less than the degree of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹), using inspection, long division, or, for the more complicated examples, a computer algebra system. - A-APR.6
Factor a quadratic trinomial. - HSM.A1.7.5
Factor a quadratic trinomial when a ≠ 1. - HSM.A1.7.6
Factor special trinomials. - HSM.A1.7.7
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to sketch the function defined by the polynomial. - A2.APR.A.5
Combine like terms to simplify polynomials. - HSM.A1.7.1
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
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. ★ - S.ID.4
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
The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. - EO.AII.2
Relate roots and rational exponents and use them to simplify expressions and solve equations. - HSM.A2.5.1
Solve radical equations and inequalities. - HSM.A2.5.4
Perform operations on functions to answer real-world questions. - HSM.A2.5.5
Use properties of exponents and radicals to simplify radical expressions. - HSM.A2.5.2
Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain. - 9.2.1.1
Graph and transform radical functions. - HSM.A2.5.3
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
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 f(x) = a(x – h)2 + k , or in factored form. - 9.2.1.5
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. - N-RN.1
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - N-RN.2
Know and apply the Remainder Theorem: For a polynomial 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹) and a number 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢, the remainder on division by 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹 – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢 is 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢), so 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢) = 0 if and only if (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹 – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢) is a factor of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹). - MAFS.912.A-APR.2.2
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
Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph. (Limit to polynomials with degrees 3 or less.) - A2.AAPR.3
Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. - A2.AAPR.1
Use the properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵 power can be rewritten as ((1.15 to the 1/12 power) to the 12𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵 power) is approximately equal to (1.012 to the 12𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵 power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - A-SSE.3c
Use properties of exponents to solve equations with rational exponents. - HSM.A1.6.1
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Instructional Note: Extend to simple rational and radical equations. - PRR.M.A2HS.16
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. Instructional Note: Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Instructional Note: Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions. - PRR.M.A2HS.17
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Instructional Note: Relate this standard to the relationship between zeros of quadratic functions and their factored forms. - PRR.M.A2HS.18
Prove polynomial identities and use them to describe numerical relationships. Instructional Note: This cluster has many possibilities for optional enrichment, such as relating the example in M.A2HS.10 to the solution of the system u² + v² = 1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x + y)ⁿ+¹ = (x + y)(x + y)ⁿ, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. Example:: For example, the polynomial identity (x² + y²)² = (x² – y²)² + (2xy)² can be used to generate Pythagorean triples. - PRR.M.A2HS.12
Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. Instructional Note: This cluster has many possibilities for optional enrichment, such as relating the example in M.A2HS.10 to the solution of the system u² + v² = 1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x + y)ⁿ+¹ = (x + y)(x + y)ⁿ, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. - PRR.M.A2HS.13
Translate between equivalent forms of functions. - A2.IF.A.2
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Instructional Note: The limitations on rational functions apply to the rational expressions. - PRR.M.A2HS.14
Identify and interpret key characteristics of functions represented graphically, with tables and with algebraic symbolism to solve problems. - A2.IF.A.1
Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Instructional Note: This standard requires the general division algorithm for polynomials. - PRR.M.A2HS.15
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
Describe and graph exponential functions. - HSM.A1.6.2
Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning. - NC.M2.A-REI.1
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
Use exponential functions to model situations and make predictions. - HSM.A1.6.3
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. - NC.M3.G-GPE.1
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). - PRR.M.A2HS.10
Identify and describe geometric sequences. - HSM.A1.6.4
Perform, analyze, and use transformations of exponential functions. - HSM.A1.6.5
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. - PRR.M.A2HS.11
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
Determine the maximum or minimum value of a quadratic function by completing the square. - A2.ASE.3b
Derive inverses of functions, and compose the inverse with the original function to show that the functions are inverses. - A2.BF.A.2
add, subtract, multiply, divide, and simplify rational algebraic expressions; - EO.AII.1.a
add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and - EO.AII.1.b
Use inverse variation and graph translations of the reciprocal function. - HSM.A2.4.1
Graph rational functions. - HSM.A2.4.2
factor polynomials completely in one or two variables. - EO.AII.1.c
Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹² + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺²)² = (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹² – 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺²)² + (2𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺)² can be used to generate Pythagorean triples. - MAFS.912.A-APR.3.4
(HONORS ONLY) Know and apply the Binomial Theorem for the expansion of (𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹 + 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺)ⁿ in powers of 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹 and y for a positive integer 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯, where 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺 are any numbers, with coefficients determined for example by Pascal’s Triangle. - MAFS.912.A-APR.3.5
Read values of an inverse function from a graph or a table, given that the function has an inverse. (A2, M3) - F.BF.4b (+)
Extend to include solving systems of linear equations in three variables, but only algebraically. (A2, M3) - A.REI.6b
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
Use the properties of exponents to transform expressions for exponential functions. - A2.ASE.3c
Create new functions by applying the four arithmetic operations and composition of functions (modifying the domain and range as necessary). - A2.BF.A.1
Create new functions by applying the four arithmetic operations and composition of functions (modifying the domain and range as necessary). - A2.FM.A.1
use knowledge of transformations to convert between equations and the corresponding graphs of functions. - F.AII.6.b
recognize the general shape of function families; and - F.AII.6.a
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 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎+𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖 for real numbers 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎 and 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏. - A2.AREI.4b
Find the probability of an event given that another event has occurred. - HSM.G.12.2
Write and solve equations with a variable on both sides to solve problems. - HSM.A1.1.3
Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.G.12.3
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. Example: For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? - MAFS.912.S-IC.1.2
Rewrite and use literal equations to solve problems. - HSM.A1.1.4
Evaluate reports based on data. - A2.DS.A.7
Add, subtract, multiply and divide complex numbers. - A2.NQ.B.6
Solve and graph inequalities. - HSM.A1.1.5
Know and apply the Fundamental Theorem of Algebra. - A2.NQ.B.7
Use probability to make decisions. - HSM.G.12.6
Represent complex numbers. - A2.NQ.B.5
Understand statistics as a process for making inferences about population parameters based on a random sample from that population. - MAFS.912.S-IC.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
Analyze how random sampling could be used to make inferences about population parameters. - A2.DS.A.1
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
Determine whether a specified model is consistent with a given data set. - A2.DS.A.2
Use the properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵 power can be rewritten as ((1.15 to the 1/12 power) to the 12𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵 power) is approximately equal to (1.012 to the 12𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵 power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - MAFS.912.A-SSE.2.3.c
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝 = 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘙 to highlight resistance 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘙𝘙. - MAFS.912.A-CED.1.4
Describe and explain how the relative sizes of a sample and the population affect the margin of error of predictions. - A2.DS.A.5
Write and solve absolute-value equations and inequalities - HSM.A1.1.7
Analyze decisions and strategies using probability concepts. - A2.DS.A.6
Describe and explain the purposes, relationship to randomization and differences, among sample surveys, experiments and observational studies. - A2.DS.A.3
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
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
Use relationships among events to find probabilities. - HSM.G.12.1
Use data from a sample to estimate characteristics of the population and recognize the meaning of the margin of error in these estimates. - A2.DS.A.4
Multiply and divide two rational expressions. - NC.M3.A-APR.7b
Solve one-variable rational equations and check for extraneous solutions. - A2.A.1.3
Solve polynomial equations with real roots using various methods and tools that may include factoring, polynomial division, synthetic division, graphing calculators or other appropriate technology. - A2.A.1.4
Solve square root equations with one variable and check for extraneous solutions. - A2.A.1.5
Solve common and natural logarithmic equations using the properties of logarithms. - A2.A.1.6
Create and solve systems of equations that may include non- linear equations and inequalities. - A2.REI.B.3
Explain why the 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘙𝘙𝘹-coordinates of the points where the graphs of the equations 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘙𝘙𝘹𝘺 = 𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘩𝘯𝘯𝑖𝑖𝑎𝑏𝑖𝑎𝑏𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘵𝘵𝘵𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘵𝘵𝘵𝘝𝘭𝘙𝘙𝘹𝘺𝘧(𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘈𝘉𝘉𝘈𝘈𝘉𝘈𝘉𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧�