Graphs as Solution Sets and Intersection Methods

Identify and distinguish the structural parts of algebraic expressions, including terms, factors, and coefficients.

Interpret the parts and structure of algebraic expressions in real-world situations, including the role of grouping symbols and exponents.

Write algebraic expressions that model real-world, geometric, and multi-part situations, and analyze the structure of complex expressions.

Identify common algebraic structures—common factors, difference of squares, perfect square trinomials, sum/difference of cubes—and use substitution and grouping to rewrite expressions in factored form.

Recognize hidden structure—quadratic forms, higher-degree polynomials, and rational expressions—and combine multiple structural patterns to rewrite complex expressions equivalently.

Choose and produce equivalent forms of quadratic expressions (factored, vertex, standard) to reveal properties such as zeros, maximums, and minimums.

Use exponent properties and strategic rewriting to transform exponential expressions, revealing equivalent growth rates and enabling comparison across different time units.

Derive the formula for the sum of a finite geometric series and use it to solve real-world problems.

Classify polynomials, understand closure, and add and subtract polynomials in horizontal and vertical formats.

Multiply monomials and binomials using multiple strategies, including the distributive property, FOIL, the area model, and special product patterns.

Multiply trinomials by binomials and trinomials, simplify expressions with multiple operations, and apply polynomial operations to real-world problems.

Understand and apply the Remainder Theorem, Factor Theorem, and Rational Root Theorem to find zeros and write polynomial equations from roots.

Use zeros, end behavior, and multiplicity to sketch polynomial graphs, and combine algebraic techniques to find all real and complex zeros.

Prove polynomial identities, use them to describe numerical relationships, and apply the Binomial Theorem to expand binomials.

Simplify, multiply, divide, add, and subtract rational expressions, and simplify complex fractions involving rational expressions.

Use polynomial long division, synthetic division, and inspection to rewrite rational expressions in quotient-plus-remainder form, and solve rational equations.

Translate real-world scenarios into one-variable equations and inequalities across linear, quadratic, rational, and exponential function families.

Apply equation-writing skills to specific modeling contexts such as distance-rate-time, cost-revenue, area, projectile motion, and exponential growth/decay, and synthesize across model types for complex problems.

Create equations in two or more variables to represent relationships and graph them on coordinate axes.

Represent constraints by equations, inequalities, and systems, and interpret solutions as viable or nonviable options.

Rearrange formulas to isolate a quantity of interest using the same reasoning as solving equations.

Explain each step in solving an equation as following from properties of equality and construct viable arguments for solution methods.

Solve a full range of linear equations in one variable, including those with fractions, decimals, and letter coefficients.

Solve one-step, multi-step, and compound linear inequalities, and graph their solution sets on a number line.

Analyze the solution sets of linear equations, including finding ordered pairs, identifying equations with no or infinitely many solutions, and solving one-step equations.

Solve simple rational and radical equations in one variable and identify extraneous solutions.

Solve quadratic equations by taking square roots and by factoring, including identifying rational and irrational roots.

Solve quadratic equations with no real solutions by completing the square, and derive and apply the quadratic formula.

Choose the best method to solve quadratics, use the discriminant, and apply these skills to word problems and mixed practice.

Recognize when a quadratic equation has complex solutions and express them in the form a ± bi.

Solve systems of two and three linear equations using graphing, substitution, elimination, and technology, and prove why elimination works.

Solve systems consisting of a linear equation and a quadratic equation both algebraically and graphically, including cases with no real solutions.

Represent systems of linear equations as matrix equations and use matrix inverses to solve them.

Understand that graphs represent all solutions to equations and find intersections of two functions using graphical, tabular, and numerical approximation methods.

Find intersections of linear, polynomial, absolute value, exponential, and logarithmic functions, and use graphing both sides of an equation to solve for unknowns.

Graph single linear inequalities as half-planes and graph systems of linear inequalities as intersections of half-planes.