### FR: Functional Relationships

#### FR.1.BTAII: Interpret the structure of expressions, write expressions in equivalent forms to solve problems, perform arithmetic operations on functions, and understand the relationship between zeros and factors of polynomials.

FR.1.BTAII.1: Interpret expressions that represent a quantity in terms of its context:

FR.1.BTAII.1.1: interpret parts of an expression, such as terms, factors, and coefficients

FR.1.BTAII.1.2: interpret complicated expressions by viewing one or more of their parts as a single entity [e.g., interpret P (1 + r)ⁿ as the product of P and a factor not depending on P]

FR.1.BTAII.2: Use the structure of an expression to identify ways to rewrite it [e.g., see x to the 4th power - y to the 4th power as (x²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² = y²)]

FR.1.BTAII.3: Understand that polynomials form a system analogous to the integers and exhibit closure under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials

FR.1.BTAII.4: Use various methods to factor quadratic polynomials; understand the relationship between the factored form of a quadratic polynomial and the zeros of a function

FR.1.BTAII.5: Identify zeros of linear and quadratic polynomials when suitable factorizations are available; use the zeros to construct a rough graph of the function defined by the polynomial

FR.1.BTAII.6: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters

FR.1.BTAII.7: 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

FR.1.BTAII.8: Interpret the parameters in a linear or exponential function in terms of a context

### RF: Representing Functions

#### RF.2.BTAII: Represent and solve equations and inequalities graphically and analyze functions using different representations.

RF.2.BTAII.1: Explain why the x-coordinates of the points where the graphs of the equations 𝑦 = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately [e.g., using technology to graph the functions, make tables of values or find successive approximations; include cases where f(x) and/or g(x) are linear, quadratic, absolute value, and exponential functions]

RF.2.BTAII.2: Graph polynomial functions identifying real zeros from the factored form; show end behavior by hand in simple cases and by technology in more complicated cases

RF.2.BTAII.5: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally, as a polynomial function

RF.2.BTAII.6: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines

RF.2.BTAII.7: Solve quadratic equations in one variable:

RF.2.BTAII.7.1: use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions; derive the quadratic formula from this form

RF.2.BTAII.7.2: solve quadratic equations by inspection (e.g., for x² = 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

### FM: Function Modeling

#### FM.3.BTAII: Create equations that describe numbers or relationships, interpret functions that arise in applications in terms of a context, analyze functions using different representations, build a function that models a relationship between two quantities, and build new functions from existing functions.

FM.3.BTAII.1: Create equations and inequalities in one variable and use them to solve problems, including equations arising from linear, quadratic, and exponential functions

FM.3.BTAII.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

FM.3.BTAII.3: 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 (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)

FM.3.BTAII.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm’s law V = IR to highlight resistance R)

FM.3.BTAII.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; maximums and minimums; symmetries; and end behavior

FM.3.BTAII.6: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function ℎ(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]

FM.3.BTAII.7: 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

FM.3.BTAII.8: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and by technology in more complicated cases:

3.1.8.1: graph exponential functions, showing intercepts and end behavior

3.1.8.2: graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions

FM.3.BTAII.9: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function:

FM.3.BTAII.9.2: use the properties of exponents to interpret expressions for exponential functions

FM.3.BTAII.11: Write a function that describes a relationship between two quantities:

FM.3.BTAII.11.1: combine standard function types using arithmetic operations (e.g., 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)

FM.3.BTAII.11.2: determine an explicit expression, a recursive process, or steps for calculation from a context

FM.3.BTAII.12: Identify the effect on the graph of replacing f(x) by f(x) + k, k f (x), 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

FM.3.BTAII.14: Define appropriate quantities for the purpose of descriptive modeling

FM.3.BTAII.15: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

FM.3.BTAII.16: Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary in the case of a strict inequality; graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes

FM.3.BTAII.19: Construct linear and exponential functions, including arithmetic sequences and geometric sequences, given a graph, a description of a relationship, or two input-output pairs; read linear and exponential functions from a table

FM.3.BTAII.20: Use the properties of exponents to transform expressions for exponential functions

### SP: Statistics and Probability

#### SP.4.BTAII: Summarize, represent, and interpret data on a single count or a measurement variable and use probability to evaluate outcomes of decisions.

SP.4.BTAII.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

SP.4.BTAII.2: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)

SP.4.BTAII.3: Represent data on two quantitative variables on a scatter plot and describe how the variables are related:

SP.4.BTAII.3.1: 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 and exponential models

SP.4.BTAII.3.2: informally assess the fit of a function by plotting and analyzing residuals

SP.4.BTAII.4: Compute and interpret the correlation coefficient of a linear fit using technology

Correlation last revised: 5/8/2018

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.