Organization: SAVVAS Product Name: envision Integrated Mathematics III 2019 Common Core Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-e930be78-56fe-3576-af6b-5b0c150e8b66 Timestamp: Monday, December 28, 2020 11:21 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: Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. - G.GGMD.2 Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. - G.GGMD.3 Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. - G.GGMD.1 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 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 (+) 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. (A2, M3) - S.ID.6a 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 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 Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. - A2.ACE.4 Evaluate reports based on data. - CCSS.Math.Content.HSS-IC.B.6 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. - A2.ACE.2 Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) - A2.ACE.3 Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. - A2.ACE.1 Extend to include more complicated function situations with the option to solve with technology. (A2, M3) - A.CED.1c Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. - ETD.M.GHS.26 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 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - MAFS.912.G-GMD.1.3 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. For example by using coefficients determined for by Pascal’s Triangle. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. - A.APR.5 (+) Graph exponential functions, indicating intercepts and end behavior, and trigonometric functions, showing period, midlineG, and amplitude. (A2, M3) - F.IF.7f Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. ★ - F.TF.5 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 - F.TF.8 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘝 = 𝘝𝘭𝘝𝘭𝘙 to highlight resistance 𝘝𝘭𝘙𝘙. - CCSS.Math.Content.HSA-CED.A.4 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 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 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 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (A2, M3) - F.IF.7c Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (A2, M3) - F.IF.9 Graph polynomial functions, identifying zeros, when factoring is reasonable, and indicating end behavior. (A2, M3) - F.IF.7d 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 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. ★ (A2, M3) - F.IF.6 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 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 the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★ (A2, M3) - F.IF.4 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 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 Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. - CCSS.Math.Content.HSA-APR.A.1 Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. - 9.3.4.7 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 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., 𝘝𝘭𝘙𝘙𝙫, |𝘝𝘭𝘙𝘙𝙫𝙫|, ||𝘝𝘭𝘙𝘙𝙫𝙫𝙫||, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷). - CCSS.Math.Content.HSN-VM.A.1 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. - F.TF.2 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 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. - F.TF.1 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ (A2, M3) - G.SRT.8b (+) 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 Interpret parts of an expression, such as terms, factors, and coefficients. - A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. - CCSS.Math.Content.HSA-SSE.A.1a 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 Know and apply the Remainder Theorem: For a polynomial 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹) and a number 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢, the remainder on division by 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹 – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢 is 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢), so 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢) = 0 if and only if (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹 – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢) is a factor of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹). - CCSS.Math.Content.HSA-APR.B.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - CCSS.Math.Content.HSG-GMD.A.2 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 Use the relation 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - CCSS.Math.Content.HSN-CN.A.2 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 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. - CCSS.Math.Content.HSG-CO.A.1 Determine the maximum or minimum value of a quadratic function by completing the square. - A2.ASE.3b Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - CCSS.Math.Content.HSG-CO.A.4 Read values of an inverse function from a graph or a table, given that the function has an inverse. (A2, M3) - F.BF.4b (+) Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1 Extend to include solving systems of linear equations in three variables, but only algebraically. (A2, M3) - A.REI.6b Use the properties of exponents to transform expressions for exponential functions. - A2.ASE.3c Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - CCSS.Math.Content.HSG-GMD.B.4 Represent data with plots on the real number line (dot plots, histograms, and box plots). - CCSS.Math.Content.HSS-ID.A.1 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - CCSS.Math.Content.HSS-ID.A.2 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. - CCSS.Math.Content.HSS-ID.A.4 Know and apply the Binomial Theorem for the expansion of (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹 + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺)ⁿ in powers of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹 and y for a positive integer 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯, where 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺 are any numbers, with coefficients determined for example by Pascal’s Triangle. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. - CCSS.Math.Content.HSA-APR.C.5 Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺²)² = (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺²)² + (2𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺)² can be used to generate Pythagorean triples. - CCSS.Math.Content.HSA-APR.C.4 Use coordinates to prove simple geometric theorems algebraically. - G.GGPE.4 Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. - G.GGPE.5 Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. - G.GGPE.6 Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. - G.GGPE.7 Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹⁴ – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺⁴ as (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹²)² – (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - CCSS.Math.Content.HSA-SSE.A.2 Compose functions. Example: For example, if 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺) is the temperature in the atmosphere as a function of height, and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵) is the height of a weather balloon as a function of time, then 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵)) is the temperature at the location of the weather balloon as a function of time. - CCSS.Math.Content.HSF-BF.A.1c Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). - CCSS.Math.Content.HSG-SRT.D.11 Determine an explicit expression, a recursive process, or steps for calculation from a context. - CCSS.Math.Content.HSF-BF.A.1a Prove the Laws of Sines and Cosines and use them to solve problems. - CCSS.Math.Content.HSG-SRT.D.10 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. - CCSS.Math.Content.HSF-BF.A.1b 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 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎+𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖 for real numbers 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏. - A2.AREI.4b 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 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹) by 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹) + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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 For exponential models, express as a logarithm the solution to 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣 to the 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵 power = 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥 where 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤, and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥 are numbers and the base 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣 is 2, 10, or 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦; evaluate the logarithm using technology. - CCSS.Math.Content.HSF-LE.A.4 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. - CCSS.Math.Content.HSF-LE.A.3 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. - N.CN.9 (+) 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 Rewrite simple rational expressions in different forms; write 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹)/𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹) in the form 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹) + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹)/𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹), where 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹), 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹), 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹), and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹) are polynomials with the degree of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹) less than the degree of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹), using inspection, long division, or, for the more complicated examples, a computer algebra system. - CCSS.Math.Content.HSA-APR.D.6 Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. - 9.3.1.1 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. - 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 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵 power, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺 = (0.97) to the 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵 power, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺 = (1.01) to the 12𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵 power, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺 = (1.2) to the 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵/10 power, and classify them as representing exponential growth or decay. - CCSS.Math.Content.HSF-IF.C.8b 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 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹-coordinates of the points where the graphs of the equations 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺 = 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹) and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺 = 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) 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 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - CCSS.Math.Content.HSG-CO.D.13 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 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. - CCSS.Math.Content.HSG-CO.D.12 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 Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. - A2.FLQE.1b Use the volumes of right and oblique pyramids and cones to solve problems. - HSM.G.11.3 Calculate the volume of a sphere and solve problems involving the volumes of spheres. - HSM.G.11.4 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 determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; - TDF.G.14.b 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 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 Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7c Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. - G.GSRT.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 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - CCSS.Math.Content.HSF-IF.C.7b Extend polynomial identities to the complex numbers. Example: For example, rewrite 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹² + 4 as (𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹 + 2𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪)(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹 – 2𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪). - CCSS.Math.Content.HSN-CN.C.8 Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. - G.GSRT.8 Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. - A2.FIF.9 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5 Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5 Use the properties of prisms and cylinders to calculate their volumes. - HSM.G.11.2 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between sample statistics are statistically significant. ★ - S.IC.5 Apply the Addition Rule, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈 or 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉) = 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈) + 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.7 Evaluate reports based on data. ★ - S.IC.6 Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. - A2.FIF.6 Use permutations and combinations to compute probabilities of compound events and solve problems. - CCSS.Math.Content.HSS-CP.B.9 Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. - A2.FIF.5 Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. - A2.FIF.4 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. - A.APR.7 (+) Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. - A2.FIF.3 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. ★ - S.IC.1 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 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. 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?★ - S.IC.2 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. ★ - S.IC.3 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. ★ - S.IC.4 Find the conditional probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 as the fraction of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉’s outcomes that also belong to 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.6 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5. - N.RN.1 Verify by composition that one function is the inverse of another. (A2, M3) - F.BF.4c (+) Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. - A.REI.2 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - CCSS.Math.Content.HSG-GPE.B.6 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 Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). - CCSS.Math.Content.HSG-GPE.B.4 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 The student will use surface area and volume of three-dimensional objects to solve practical problems. - TDF.G.13 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - N.RN.2 Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic. (A2, M3) - A.APR.1b Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. - G.GCI.2 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 Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. - G.GCI.5 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 + √3𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪)³ = 8 because (-1 + √3𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪) 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 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵 power can be rewritten as ((1.15 to the 1/12 power) to the 12𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵 power) is approximately equal to (1.012 to the 12𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵 power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - CCSS.Math.Content.HSA-SSE.B.3c Graph rational functions, identifying zeros and asymptotes when factoring is reasonable, and indicating end behavior. (A2, M3) - F.IF.7g (+) 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 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 While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations. (A2, M3) - A.CED.4d Explain why the x-coordinates of the points where the graphs of the equation 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, making tables of values, or finding successive approximations. - A.REI.11 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3 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 Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. - G.3D.1.1 Derive the equation of a parabola given a focus and directrix. - CCSS.Math.Content.HSG-GPE.A.2 Graph logarithmic functions, indicating intercepts and end behavior. - F.IF.7h (+) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ - G.GMD.3 Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. - A2.FLQE.2 Understand that two events 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉 are independent if the probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉 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 Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. - A2.FLQE.1 Understand the conditional probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉 as 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉)/𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - 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 Interpret the parameters in a linear or exponential function in terms of the context. - A2.FLQE.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 Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (A2, M3) - F.BF.1b Rewrite expressions involving radicals and rational exponents using the properties of exponents. - CCSS.Math.Content.HSN-RN.A.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 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (A2, M3) - F.BF.3 Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems. - NC.M3.G-GMD.3 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - A2.FBF.2 Describe the effect of the transformations 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥), 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥)+𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘, 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 𝘝𝘭𝘙𝘙𝙫𝙫𝙫𝘷𝘱𝘹𝘢𝘹𝘢𝘱𝘢𝘱𝘢𝘹𝘢𝘱𝘹𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘛𝘺𝘩𝘵𝘛𝘩𝘵𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘣𝘹𝘢𝘹𝘣𝘹𝘲𝘹𝘳𝘹𝘳𝘹𝘣𝘹𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘪𝘪𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3 Produce an invertible function from a non-invertible function by restricting the domain. - CCSS.Math.Content.HSF-BF.B.4d 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 probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the numb