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### 2 Attributes of functions and their inverses

- 2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.
- A graph the functions f(x)=√x, f(x)=1/x, f(x)=x³, f(x)= ³√x, f(x)=b to the x power, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval;
- B graph and write the inverse of a function using notation such as f-¹ (x);
- C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range; and
- D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

### 3 Systems of equations and inequalities

- 3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.
- A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic;
- B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;
- C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;
- D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;
- E formulate systems of at least two linear inequalities in two variables;
- F solve systems of two or more linear inequalities in two variables; and
- G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

### 4 Quadratic and square root functions, equations, and inequalities

- 4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.
- A write the quadratic function given three specified points in the plane;
- B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;
- C determine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x – c) for specific positive and negative values of a, b, c, and d;
- D transform a quadratic function f(x) = ax² + bx + c to the form f(x) = a(x – h)² + k to identify the different attributes of f(x);
- E formulate quadratic and square root equations using technology given a table of data;
- F solve quadratic and square root equations;
- G identify extraneous solutions of square root equations; and
- H solve quadratic inequalities.

### 5 Exponential and logarithmic functions and equations

- 5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.
- A determine the effects on the key attributes on the graphs of f(x) = b to the x power and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x – c) for specific positive and negative real values of a, c, and d;
- B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;
- C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;
- D solve exponential equations of the form y = ab to the x power where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions; and
- E determine the reasonableness of a solution to a logarithmic equation.

### 6 Cubic, cube root, absolute value and rational functions, equations, and inequalities

- 6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.
- A analyze the effect on the graphs of f(x) = x³ and f(x) = ³√x when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- B solve cube root equations that have real roots;
- C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- D formulate absolute value linear equations;
- E solve absolute value linear equations;
- F solve absolute value linear inequalities;
- G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- H formulate rational equations that model real-world situations;
- I solve rational equations that have real solutions;
- J determine the reasonableness of a solution to a rational equation;
- K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and
- L formulate and solve equations involving inverse variation.

### 7 Number and algebraic methods

- 7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.
- A add, subtract, and multiply complex numbers;
- B add, subtract, and multiply polynomials;
- C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;
- D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;
- E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;
- F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;
- G rewrite radical expressions that contain variables to equivalent forms;
- H solve equations involving rational exponents; and
- I write the domain and range of a function in interval notation, inequalities, and set notation.

### 8 Data

- 8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.
- A analyze data to select the appropriate model from among linear, quadratic, and exponential models;
- B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data; and
- C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.