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 \(d\);

formulate absolute value linear equations;

solve absolute value linear equations;

solve absolute value linear inequalities;

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 \(d\);

formulate rational equations that model real-world situations;

solve rational equations that have real solutions;

determine the reasonableness of a solution to a rational equation;

determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and

formulate and solve equations involving inverse variation.

Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:

add, subtract, and multiply complex numbers;

add, subtract, and multiply polynomials;

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;

determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;

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;

determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;

rewrite radical expressions that contain variables to equivalent forms;

solve equations involving rational exponents; and

write the domain and range of a function in interval notation, inequalities, and set notation.

Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:

analyze data to select the appropriate model from among linear, quadratic, and exponential models;

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

predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

apply mathematics to problems arising in everyday life, society, and the workplace;

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

create and use representations to organize, record, and communicate mathematical ideas;

analyze mathematical relationships to connect and communicate mathematical ideas; and

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to:

determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint;

derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines; and

determine an equation of a line parallel or perpendicular to a given line that passes through a given point.

Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to:

describe and perform transformations of figures in a plane using coordinate notation;

determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane;

identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane; and

identify and distinguish between reflectional and rotational symmetry in a plane figure.

Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:

distinguish between undefined terms, definitions, postulates, conjectures, and theorems;

identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse;

verify that a conjecture is false using a counterexample; and

compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:

investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools;

construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge;

use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships; and