Organization: Pearson Product Name: enVision Algebra 2 Common Core 2018 Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-22597625-81ad-3a54-a44f-25e4fb3a83f1 Timestamp: Tuesday, December 10, 2019 10:34 AM EST Status: VALID! Conformant: true ----- VALID! ----- Resource Validation Results The document is valid. ----- VALID! ----- Schema Location Results Schema locations are valid. ----- VALID! ----- Schema Validation Results The document is valid. ----- VALID! ----- Schematron Validation Results The document is valid. Curriculum Standards: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. - CCSS.Math.Content.HSF-BF.B.5 Identify the effect on the graph of replacing 𝘧(𝘧𝘹) by 𝘧𝘹𝘧(𝘧𝘹𝘧𝘹) + 𝘧𝘹𝘧𝘹𝘬, 𝘧𝘹𝘧𝘹𝘬𝘬 𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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. - CCSS.Math.Content.HSF-BF.B.3 Fit a linear function for a scatter plot that suggests a linear association. - CCSS.Math.Content.HSS-ID.B.6c 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. - CCSS.Math.Content.HSS-ID.B.6a Informally assess the fit of a function by plotting and analyzing residuals. - CCSS.Math.Content.HSS-ID.B.6b For exponential models, express as a logarithm the solution to 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣 to the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵 power = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥 where 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤, and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥 are numbers and the base 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣 is 2, 10, or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦; evaluate the logarithm using technology. - CCSS.Math.Content.HSF-LE.A.4 Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷ₓ, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷 subscript 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺) = (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷ₓ, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷 subscript 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺). - CCSS.Math.Content.HSN-VM.B.5a 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). - CCSS.Math.Content.HSF-LE.A.2 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. - CCSS.Math.Content.HSS-IC.B.4 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. - CCSS.Math.Content.HSS-IC.B.5 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. - CCSS.Math.Content.HSG-CO.B.6 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. - CCSS.Math.Content.HSS-IC.B.3 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - CCSS.Math.Content.HSG-CO.B.8 Evaluate reports based on data. - CCSS.Math.Content.HSS-IC.B.6 Rewrite simple rational expressions in different forms; write 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹)/𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹) in the form 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹) + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹)/𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹), where 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹), 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹), 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹), and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹) are polynomials with the degree of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹) less than the degree of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹), using inspection, long division, or, for the more complicated examples, a computer algebra system. - CCSS.Math.Content.HSA-APR.D.6 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. - CCSS.Math.Content.HSA-APR.D.7 Use the properties of exponents to interpret expressions for exponential functions. Example: For example, identify percent rate of change in functions such as y = (1.02) to the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵 power, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺 = (0.97) to the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵 power, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺 = (1.01) to the 12𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵 power, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺 = (1.2) to the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵/10 power, and classify them as representing exponential growth or decay. - CCSS.Math.Content.HSF-IF.C.8b 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. - CCSS.Math.Content.HSF-IF.C.8a 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. - CCSS.Math.Content.HSA-REI.D.12 Explain why the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹-coordinates of the points where the graphs of the equations 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹) and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - CCSS.Math.Content.HSA-REI.D.11 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). - CCSS.Math.Content.HSA-REI.D.10 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. Example: For example, calculate mortgage payments. - CCSS.Math.Content.HSA-SSE.B.4 Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. - CCSS.Math.Content.HSN-VM.B.4a 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. - CCSS.Math.Content.HSF-IF.C.9 Understand vector subtraction 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫 – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬 as 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫 + (–𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬), where –𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬 is the additive inverse of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬, with the same magnitude as 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬 and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. - CCSS.Math.Content.HSN-VM.B.4c Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. - CCSS.Math.Content.HSN-VM.B.4b Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - CCSS.Math.Content.HSG-SRT.C.8 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. - CCSS.Math.Content.HSG-SRT.C.6 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. - CCSS.Math.Content.HSF-TF.C.8 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. - CCSS.Math.Content.HSF-TF.C.9 Graph linear and quadratic functions and show intercepts, maxima, and minima. - CCSS.Math.Content.HSF-IF.C.7a Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - CCSS.Math.Content.HSF-IF.C.7e Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙 to highlight resistance 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙. - CCSS.Math.Content.HSA-CED.A.4 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. - CCSS.Math.Content.HSN-CN.C.9 Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7d 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. - CCSS.Math.Content.HSA-CED.A.3 Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7c Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. - CCSS.Math.Content.HSA-REI.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - CCSS.Math.Content.HSA-CED.A.2 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - CCSS.Math.Content.HSF-IF.C.7b 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. - CCSS.Math.Content.HSA-CED.A.1 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. - CCSS.Math.Content.HSA-REI.A.1 Extend polynomial identities to the complex numbers. Example: For example, rewrite 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹² + 4 as (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹 + 2𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪)(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹 – 2𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪). - CCSS.Math.Content.HSN-CN.C.8 Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5 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. - CCSS.Math.Practice.MP5.a Apply the Addition Rule, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈 or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈) + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.7 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. - CCSS.Math.Content.HSS-IC.A.1 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? - CCSS.Math.Content.HSS-IC.A.2 Apply the general Multiplication Rule in a uniform probability model, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈)𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉|𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉)𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈|𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.8 Use permutations and combinations to compute probabilities of compound events and solve problems. - CCSS.Math.Content.HSS-CP.B.9 Evaluate and compare strategies on the basis of expected values. Example: For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. - CCSS.Math.Content.HSS-MD.B.5b Find the expected payoff for a game of chance. Example: For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. - CCSS.Math.Content.HSS-MD.B.5a Find the conditional probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈 given 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 as the fraction of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉’s outcomes that also belong to 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.6 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. - CCSS.Math.Content.HSG-CO.C.9 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - CCSS.Math.Content.HSG-GPE.B.7 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. - CCSS.Math.Content.HSG-C.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). - CCSS.Math.Content.HSG-GPE.B.5 Construct a tangent line from a point outside a given circle to the circle. - CCSS.Math.Content.HSG-C.A.4 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. - CCSS.Math.Content.HSG-C.A.2 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - CCSS.Math.Content.HSG-C.A.3 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. - CCSS.Math.Content.HSA-APR.A.1 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - CCSS.Math.Content.HSS-ID.C.7 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫, |𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫|, ||𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫||, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷). - CCSS.Math.Content.HSN-VM.A.1 Compute (using technology) and interpret the correlation coefficient of a linear fit. - CCSS.Math.Content.HSS-ID.C.8 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. - CCSS.Math.Content.HSN-RN.B.3 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. - CCSS.Math.Content.HSN-CN.B.6 Factor a quadratic expression to reveal the zeros of the function it defines. - CCSS.Math.Content.HSA-SSE.B.3a Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. - CCSS.Math.Content.HSN-CN.B.4 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. Example: For example, (-1 + √3𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪)³ = 8 because (-1 + √3𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪) has modulus 2 and argument 120°. - CCSS.Math.Content.HSN-CN.B.5 Use the properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵 power can be rewritten as ((1.15 to the 1/12 power) to the 12𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵 power) is approximately equal to (1.012 to the 12𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵 power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - CCSS.Math.Content.HSA-SSE.B.3c Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - CCSS.Math.Content.HSA-SSE.B.3b Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - CCSS.Math.Content.HSG-SRT.B.5 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. - CCSS.Math.Content.HSN-Q.A.1 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. - CCSS.Math.Content.HSG-SRT.B.4 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3 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. - CCSS.Math.Practice.MP4.a Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. - CCSS.Math.Content.HSG-GPE.A.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. - CCSS.Math.Content.HSG-GPE.A.1 Derive the equation of a parabola given a focus and directrix. - CCSS.Math.Content.HSG-GPE.A.2 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. - CCSS.Math.Content.HSS-CP.A.4 Interpret parts of an expression, such as terms, factors, and coefficients. - CCSS.Math.Content.HSA-SSE.A.1a Understand that two events 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉 are independent if the probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - CCSS.Math.Content.HSS-CP.A.2 Understand the conditional probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈 given 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉 as 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉)/𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - CCSS.Math.Content.HSS-CP.A.3 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. - CCSS.Math.Content.HSF-TF.B.7 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. - CCSS.Math.Content.HSF-TF.B.5 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. - CCSS.Math.Content.HSF-TF.B.6 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. - CCSS.Math.Content.HSG-SRT.A.2 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - CCSS.Math.Content.HSG-GMD.A.3 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. - CCSS.Math.Content.HSA-APR.B.3 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. - CCSS.Math.Content.HSS-ID.B.5 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - CCSS.Math.Content.HSN-RN.A.2 Know and apply the Remainder Theorem: For a polynomial 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹) and a number 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢, the remainder on division by 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹 – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢 is 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢), so 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢) = 0 if and only if (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹 – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢) is a factor of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹). - CCSS.Math.Content.HSA-APR.B.2 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. - CCSS.Math.Content.HSN-RN.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. - CCSS.Math.Content.HSG-GMD.A.1 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. - CCSS.Math.Content.HSN-CN.A.3 Produce an invertible function from a non-invertible function by restricting the domain. - CCSS.Math.Content.HSF-BF.B.4d Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? - CCSS.Math.Content.HSS-MD.A.4 Read values of an inverse function from a graph or a table, given that the function has an inverse. - CCSS.Math.Content.HSF-BF.B.4c Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. - CCSS.Math.Content.HSS-MD.A.3 Know there is a complex number 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪 such that 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪² = –1, and every complex number has the form 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢 + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪 with 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣 real. - CCSS.Math.Content.HSN-CN.A.1 Verify by composition that one function is the inverse of another. - CCSS.Math.Content.HSF-BF.B.4b Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. - CCSS.Math.Content.HSS-MD.A.2 Use the relation 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - CCSS.Math.Content.HSN-CN.A.2 Solve an equation of the form 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤 for a simple function 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - CCSS.Math.Content.HSF-BF.B.4a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. - CCSS.Math.Content.HSF-LE.A.1a Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. - CCSS.Math.Content.HSF-LE.A.1b Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. - CCSS.Math.Content.HSF-LE.A.1c 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 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧(0) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧(1) = 1, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯+1) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 greater than or equal to 1. - CCSS.Math.Content.HSF-IF.A.3 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - CCSS.Math.Content.HSF-IF.A.2 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧 is a function and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹 is an element of its domain, then 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹) denotes the output of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘤𝘷𝘷𝘺𝘤𝘷𝘤𝘷𝘺𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝙫𝙬𝙫𝙬𝙬𝙬𝙬𝘝𝘭𝘙𝘙𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝙫𝙫𝙫𝘷𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧�