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New York
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AII-F.IF.7.e - Graph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.
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- I can graph basic exponential functions from given data and label key features like intercepts and end behavior to help me interpret business growth or decay
- I can also describe, in my own words, how the graph shape connects to whether revenue or debt is increasing or decreasing.
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- I can graph exponential functions accurately and identify intercepts, end behavior, and the growth/decay rate, then connect those graph features to financial context
- I can also explain inverse relationships between exponents and logarithms by translating between an equation in exponential form and its logarithmic form.
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- I can use logarithms (with technology) to solve exponential equations for realistic timing questions, such as when revenue reaches a target or when a loan is repaid
- I can graph the exponential model showing intercepts and end behavior and justify how those features support my solution and decision recommendation in the Brooklyn business scenario.
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- I can develop and defend a complete decision model by graphing exponential functions with correct intercepts and end behavior and using logarithms to solve for meaningful quantities like repayment time or target-reach dates
- I can critically evaluate competing model choices using evidence from the graphs and solutions, explain why my inverse relationship reasoning is valid, and clearly connect the math to the financial planner’s advising decisions.
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New York
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AII-F.BF.5.a - Understand inverse relationships between exponents and logarithms algebraically and graphically.
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- I can explain, using examples and a simple graph, that logarithms undo exponentials (and I can describe this as an inverse relationship)
- I can identify the key features of an exponential/logarithmic graph (intercept and general end behavior) when the relationship is clearly shown
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- I can use algebra to show how exponent and logarithm equations are inverses (for example, rewriting between exponential and logarithmic forms)
- I can graph exponential functions and interpret intercepts and end behavior to describe whether a situation represents growth or decay
- I can connect the graph shape to the inverse relationship between exponentials and logarithms when making a financial prediction.
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- I can solve inverse exponential/logarithmic problems by converting between forms and using logarithms to find an unknown exponent or time variable
- I can use logarithms with technology to accurately determine when a modeled revenue target is reached or when a loan balance is repaid
- I can interpret how changes in the model parameters affect timing and justify my conclusions using graph evidence (intercepts and end behavior).
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- I can develop and apply an advanced inverse-modeling explanation that links algebraic transformations and graph behavior to real financial decisions
- I can evaluate multiple candidate models for revenue growth/repayment and choose the one that best fits a case, using logarithmic solutions and evidence from intercepts and end behavior
- I can critique and revise my forecasting approach based on feedback, clearly explaining how the inverse relationship supports stronger, more reliable decision-making.
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New York
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AII-F.LE.4 - Use logarithms to solve exponential equations, such as $(ab^{c})^{t} = d$ (where a, b, c, and d are real numbers and b > 0) and evaluate the logarithm using technology.
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- I can use technology to evaluate logarithms and substitute values into an equation written in exponential form to find an unknown variable, with support
- I can describe in my own words how the logarithm is helping me reverse the exponential growth/decay in the context of a Brooklyn business forecast.
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- I can solve exponential equations with logarithms using the structure of expressions like
(ab^c)^t = d (with b>0), showing the algebraic steps that lead to t
- I can use technology to evaluate logarithms accurately and explain how the solved timing relates to repayment or reaching a revenue target for a small business, connecting my result to the model’s behavior.
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- I can solve and justify logarithmic solutions to exponential equations, transforming (ab^c)^t = d into a solvable logarithmic form and checking that t makes the original equation true
- I can interpret what the solution means for decision-making by connecting the computed timing to how changing growth/interest rates affects a business’s ability to repay debt or hit revenue goals.
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- I can independently create and solve logarithmic equations from a realistic business scenario, including selecting appropriate variables/expressions from the data and ensuring constraints like b>0 are satisfied
- I can evaluate and verify logarithm-based solutions using technology, explain discrepancies if they arise, and use critical reasoning to connect the mathematical result to credible forecasting choices and repayment options for Brooklyn small businesses.
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