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Product Name: enVision Algebra 1 Common Core 2018
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Curriculum Standards:


Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. - CCSS.Math.Practice.MP2.a
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
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3
Fit a linear function for a scatter plot that suggests a linear association. - CCSS.Math.Content.HSS-ID.B.6c
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦 – 2)/(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥 – 1)(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥 + 1), (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥 – 1)(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥² + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. - CCSS.Math.Practice.MP8.a
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
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
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
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. - CCSS.Math.Practice.MP6.a
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
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 complicated expressions by viewing one or more of their parts as a single entity. Example: For example, interpret 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗(1+𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳)ⁿ as the product of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗 and a factor not depending on 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗. - CCSS.Math.Content.HSA-SSE.A.1b
Interpret parts of an expression, such as terms, factors, and coefficients. - CCSS.Math.Content.HSA-SSE.A.1a
Understand that two events 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉 are independent if the probability of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉 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
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
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
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
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
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - CCSS.Math.Content.HSN-RN.A.2
Explain why the 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹-coordinates of the points where the graphs of the equations 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹) and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) 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
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
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
Solve an equation of the form 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤 for a simple function 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 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 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 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧(0) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧(1) = 1, 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯+1) = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 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 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧 is a function and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹 is an element of its domain, then 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹) denotes the output of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹. The graph of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧 is the graph of the equation 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹). - CCSS.Math.Content.HSF-IF.A.1
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - CCSS.Math.Content.HSF-BF.A.2
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, 'Does this make sense?' They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. - CCSS.Math.Practice.MP1.a
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 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝 = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙 to highlight resistance 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙. - CCSS.Math.Content.HSA-CED.A.4
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7d
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥² + 9𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥 – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦. - CCSS.Math.Practice.MP7.a
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
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
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
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - CCSS.Math.Content.HSA-REI.C.6
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - CCSS.Math.Content.HSA-REI.C.5
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - CCSS.Math.Practice.MP3.a
Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - CCSS.Math.Content.HSG-CO.A.3
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺 = –3𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹 and the circle 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹² + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺² = 3. - CCSS.Math.Content.HSA-REI.C.7
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. - CCSS.Math.Content.HSG-CO.C.10
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. - CCSS.Math.Content.HSF-TF.A.2
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
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
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
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). - CCSS.Math.Content.HSS-ID.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
Compute (using technology) and interpret the correlation coefficient of a linear fit. - CCSS.Math.Content.HSS-ID.C.8
Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹⁴ – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺⁴ as (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹²)² – (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹² – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - CCSS.Math.Content.HSA-SSE.A.2
Distinguish between correlation and causation. - CCSS.Math.Content.HSS-ID.C.9
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
Factor a quadratic expression to reveal the zeros of the function it defines. - CCSS.Math.Content.HSA-SSE.B.3a
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. - CCSS.Math.Content.HSF-IF.B.6
Solve quadratic equations by inspection (e.g., for 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢 ± 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪 for real numbers 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢 and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣. - CCSS.Math.Content.HSA-REI.B.4b
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯) gives the number of person-hours it takes to assemble 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯 engines in a factory, then the positive integers would be an appropriate domain for the function. - CCSS.Math.Content.HSF-IF.B.5
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - CCSS.Math.Content.HSF-IF.B.4
Use the method of completing the square to transform any quadratic equation in 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹 into an equation of the form (𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹 – 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱)² = 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲 that has the same solutions. Derive the quadratic formula from this form. - CCSS.Math.Content.HSA-REI.B.4a
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
Compose functions. Example: For example, if 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺) is the temperature in the atmosphere as a function of height, and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺𝘩(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺𝘩𝘵) is the height of a weather balloon as a function of time, then 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺𝘩𝘵𝘛(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺𝘩𝘵𝘛𝘩(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘗𝘳𝘗𝘗𝘈𝘉𝘈𝘉𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯𝘹𝘹𝘱𝘲𝘛𝘺𝘩𝘵𝘛𝘩𝘵)) is the temperature at the location of the weather balloon as a function of time. - CCSS.Math.Content.HSF-BF.A.1c
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




List of all Files Validated:


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