factor, if possible, trinomials with real factors in the form \(ax^2 + bx + c\), including perfect square trinomials of degree two; and

decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.

Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:

simplify numerical radical expressions involving square roots; and

simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:

decide whether relations represented verbally, tabularly, graphically, and symbolically define a function;

evaluate functions, expressed in function notation, given one or more elements in their domains;

identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes;

write a formula for the \(n^{th}\) term of arithmetic and geometric sequences, given the value of several of their terms; and

solve mathematic and scientific formulas, and other literal equations, for a specified variable.

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

Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:

graph the functions \(f(x)=\sqrt{x}\), \(f(x)=1/x\), \(f(x)=x^3\), \(f(x)= \sqrt[3]{x}\), \(f(x)=b^x\), \(f(x)=|x|\), and \(f(x)=log_b (x)\) 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;

graph and write the inverse of a function using notation such as \(f^{-1} (x)\);

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

use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:

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;

solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;

solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;

determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;

formulate systems of at least two linear inequalities in two variables;

solve systems of two or more linear inequalities in two variables; and

determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

Quadratic and square root functions, equations, and inequalities. 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. The student is expected to:

write the quadratic function given three specified points in the plane;

write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;

determine the effect on the graph of \(f(x) = \sqrt{x}\) 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 \(d\);

transform a quadratic function \(f(x) = ax^2+ bx + c\) to the form \(f(x) = a(x - h)^2+ k\) to identify the different attributes of \(f(x)\);

formulate quadratic and square root equations using technology given a table of data;

solve quadratic and square root equations;

identify extraneous solutions of square root equations; and

solve quadratic inequalities.

Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:

determine the effects on the key attributes on the graphs of \(f(x) = b^x\) and \(f(x) = log_b (x)\) 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 \(d\);

formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;

rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;

solve exponential equations of the form \(y = ab^x\) 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

determine the reasonableness of a solution to a logarithmic equation.

Cubic, cube root, absolute value and rational functions, equations, and inequalities. 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. The student is expected to:

analyze the effect on the graphs of \(f(x) = x^3\) and \(f(x) = \sqrt[3]{x}\) 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 \(d\);

solve cube root equations that have real roots;