**SEEING STRUCTURE IN EXPRESSIONS (A.SSE)**

__INTERPRET THE STRUCTURE OF EXPRESSIONS__**MCC9-12.A.SSE.1**Interpret expressions that represent a quantity in terms of its context.★

**MCC9-12.A.SSE.1a**Interpret parts of an expression, such as terms, factors, and coefficients.★

**MCC9-12.A.SSE.1b**Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.★

**MCC9-12.A.SSE.2**Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2 , thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).

__WRITE EXPRESSIONS IN EQUIVALENT FORMS TO SOLVE PROBLEMS__**MCC9-12.A.SSE.3**Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

**MCC9-12.A.SSE.3a**Factor a quadratic expression to reveal the zeros of the function it defines.★

**MCC9-12.A.SSE.3b**Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.★

**ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS ****(A.APR)**

__PERFORM ARTIHMETIC OPERATIONS ON POLYNOMIALS__**MCC9-12.A.APR.1**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.

- Students will simplify and operate with polynomials.
- Division of polynomials is optional

**CREATING EQUATIONS (A.CED)**

__CREATE EQUATIONS THAT DESCRIBE NUMBERS OR RELATIONSHIPS__**MCC9-12.A.CED.1**Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions. ★

- Use all available types of functions to create equations, but constrain to simple cases.

**MCC9-12.A.CED.2**Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★

**MCC9-12.A.CED.4**Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

*For example, rearrange Ohm’s law V = IR to highlight resistance R.*★

- Extend to include linear, exponential and quadratic.

**REASONING WITH EQUATIONS AND INEQUALITIES (A.REI)**

__SOLVE EQUATIONS AND INEQUALITIES IN ONE VARIABLE__**MCC9-12.A.REI.4**Solve quadratic equations in one variable.

**MCC9-12.A.REI.4a**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.

**MCC9-12.A.REI.4b**Solve quadratic equations by inspection (e.g., for x^2 = 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.

__SOLEV SYSTEMS OF EQUATIONS__**MCC9-12.A.REI.7**Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

*For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.*

- Include systems that lead to work with fractions

**INTERPRETING FUNCTIONS (F.IF)**

__INTERPRET FUNCTIONS THAT ARISE IN APPLICATIONS IN TERMS OF THE CONTEXT__**MCC9-12.F.IF.4**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. ★

- Students should be able to interpret the intercepts; intervals where the function is increasing, decreasing, positive, or negative; and end behavior.
- Compare and graph characteristics of a function represented in a variety of ways. Characteristics include domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals of increase and decrease, and rates of change.

**MCC9-12.F.IF.5**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.★*

**MCC9-12.F.IF.6**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.★

__ANALYZE FUNCTIONS USING DIFFERENT REPRESENTATIONS__**MCC9-12.F.IF.7**Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

**MCC9-12.F.IF.7a**Graph quadratic functions and show intercepts, maxima, and minima.★

**MCC9-12.F.IF.8**Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

**MCC9-12.F.IF.8a**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.

**MCC9-12.F.IF.9**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 (F-BF)**

**BUILD A FUNCTION THAT MODELS A RELATIONSHIP BETWEEN TWO QUANTITIES****MCC9-12.F.BF.1**Write a function that describes a relationship between two quantities

**MCC9-12.F.BF.1a**Determine an explicit expression, a recursive process, or steps for calculation from a context.

**MCC9-12.F.BF.1b**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.*

__BUILD NEW FUNCTIONS FROM EXISTING FUNCTIONS__**MCC9-12.F.BF.3**Identify 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.

- Note the effect of multiple transformations on a single graph and the common effect of each transformation across function types.

**LINEAR, QUADRATIC, AND EXPONENTIAL MODELS (F.LE)**

__CONSTRUCT AND COMPARE LINEAR, QUADRATIC, AND EXPONENTIAL MODELS AND SOLVE PROBLEMS__**MCC9-12.F.LE.3**Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.★

- Explore rates of change, comparing rates of change.