**Number and Quantity:**

__Quantities__

- Reason Quantitatively and Use Units to Solve Problems
- 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;
- Define appropriate quantities for the purpose of descriptive modeling.
- Use Complex Numbers in Polynomial Identities and Equations
- Solve quadratic equations with real coefficients that have complex solutions (Unit 3 Aactivity 1 & Unit 6 Activity 2)

**Vector and Matrix Quantities:**

__Represent and Model with Vector Quantities__

- Reason Quantitatively and Use Units to Solve Problems
- (+) Solve problems involving velocity and other quantities that can be represented by vectors.

**Algebra:**

__Seeing Structure in Expressions__

- Interpret the structure of expressions
- Interpret expressions that represent a quantity in terms of its context*
- Interpret parts of an expression, such as terms, factors and coefficients

__Write Expressions in Equivalent Forms to Solve Problems__

- Interpret the structure of expressions
- Choose and produce and equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* (Units 5 & 6)
- a. Factor a quadratic expression to reveal the zeros of the function it defines.
- b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

__Creating Equations__

- Create equations that describe numbers or relationships
- 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; (Unit 2 Activity 3 & Units 3-7)* - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales;
- Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
*For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*; - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
*For example, rearrange Ohmâ€™s law V =IR to highlight resistance R.*(Units 5-6)

__Reasoning with Equations and Inequalities__

- Solve equations and inequalities in one variable
- Solve quadratic equations in one variable.
- a. 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. - b. 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 and write them as a +/- bi for real numbers
*a*and*b*. - Represent and solve equations and inequalities graphically
- 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).
- 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, polynomial, rational, absolute value, exponential, and logarithmic functions.*

**Functions:**

__Interpreting Functions__

- Understand the concept of a function and use function notation
- 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 f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input
*x*. The graph of f is the graph of the equation*y = f(x)*. - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- Interpret functions that arise in applications in terms of the context
- 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.** - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
*For example, 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.** - 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.* (Unit 2 & Units 5-6)
- Analyze functions using different representations
- Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
- a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
- b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- c. Graph
**exponential**and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (Units 4-6)
- 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.
- 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.*

__Building Functions__

- Build a Function that Models a Relationship Between Two Quantities
- Write a function that describes a relationship between two quantities.*
- a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
- b. 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.* - c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
- Build New Functions from Existing Functions
- WIdentify the effect on the graph of replacing f(x) by
*f(x) + k*,*k f(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.*(Units 3 & 4)

__Linear, Quadratic, and Exponential Models*__

- Construct and compare linear, quadratic, and exponential models and solve problems
- Distinguish between situations that can be modeled with linear functions and with exponential functions. (Units 1 & 7)
- a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
- 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). (Unit 7)
- Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (Units 1 & 7)
- Interpret expressions for functions in terms of the situation they model
- Interpret the parameters in a linear or exponential function in terms of a context (Unit 7)