Algebra I
Algebra I is a foundational mathematics course covering the language and methods of symbolic reasoning, from solving linear equations and inequalities to analyzing quadratic functions and their graphs. Students develop fluency with polynomial arithmetic, factoring strategies, rational expressions, and radicals while building the problem-solving toolkit required for all subsequent mathematics.
Who Should Take This
This course is ideal for students entering high school mathematics or anyone seeking a rigorous review of core algebraic concepts before advancing to Geometry or Algebra II. Learners should have comfort with arithmetic operations on integers and fractions and should aim to build procedural fluency and conceptual understanding in equal measure.
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Linear Equations 7 topics
Describe the properties of equality including addition, subtraction, multiplication, and division properties and explain how each property justifies a step when solving one-variable linear equations
Solve one-variable linear equations including equations with variables on both sides, parentheses requiring distribution, and fractional or decimal coefficients, checking solutions by substitution
Solve literal equations and formulas for a specified variable, applying inverse operations and the properties of equality to isolate the target variable in multi-variable expressions
Identify the slope and y-intercept of a linear equation written in slope-intercept form y equals mx plus b and state their geometric and contextual meanings in a given problem scenario
Write linear equations in slope-intercept, point-slope, and standard form given various combinations of slope, intercepts, or two coordinate points on the line
Analyze when a linear equation has one solution, no solution, or infinitely many solutions by examining coefficient relationships and constant terms on both sides of the equation
Calculate the slope of a line using the rise-over-run formula between two coordinate points and distinguish positive, negative, zero, and undefined slopes from tables, graphs, and equations
2Inequalities 6 topics
Describe the rules for solving linear inequalities including the direction-reversal rule when multiplying or dividing by a negative number, and represent solutions on a number line
Solve one-variable linear inequalities with one or two operations and graph the solution set on a number line using open and closed circles to indicate strict versus non-strict inequality
Solve compound inequalities involving 'and' (intersection) and 'or' (union) and express the solution set using interval notation and number-line graphs
Solve absolute value equations and inequalities by converting to equivalent compound linear equations or inequalities and interpreting the solution as a distance on the number line
Analyze real-world constraint scenarios to set up, solve, and interpret linear inequalities and compound inequalities, explaining whether endpoint values satisfy the problem context
Graph linear inequalities in two variables on the coordinate plane, determining whether the boundary line is solid or dashed and which half-plane to shade based on a test point
3Systems of Equations and Inequalities 7 topics
Identify the three possible outcomes of a two-variable linear system (one solution, no solution, infinitely many) and describe how the relative positions of lines in the plane correspond to each case
Solve a two-variable linear system by graphing both equations on the coordinate plane and identifying the intersection point, verifying the solution satisfies both equations
Solve a two-variable linear system by the substitution method, isolating one variable in one equation and substituting into the other, then solving for both unknowns
Solve a two-variable linear system by the elimination method, multiplying equations by constants to create opposite coefficients, then adding equations to eliminate one variable
Solve systems of linear inequalities by graphing each inequality, shading the appropriate half-plane, and identifying the feasible region satisfying all constraints simultaneously
Analyze a real-world mixture, rate, or investment problem by defining variables, writing a system of two linear equations, solving with an appropriate method, and interpreting the solution in context
Compare the efficiency of substitution, elimination, and graphing methods for solving a given system and justify the most appropriate method based on the structure of the equations
4Polynomials 6 topics
Identify polynomial terminology including term, coefficient, degree of a term, degree of a polynomial, leading coefficient, monomial, binomial, trinomial, and standard form
Add and subtract polynomials by identifying and combining like terms, applying the distributive property when subtracting, and writing the result in standard form
Multiply polynomials using the distributive property and FOIL method for binomials, extending to products of a polynomial by a monomial and products of two trinomials
Apply special product patterns including perfect-square trinomials (a plus b squared, a minus b squared) and difference of two squares (a plus b)(a minus b) to expand binomial products efficiently
Divide a polynomial by a monomial by dividing each term individually and applying the quotient rule for exponents, simplifying the result completely
Analyze the degree and leading coefficient of a polynomial to determine the general end behavior of its graph and predict how the number of terms affects the graph's shape
5Factoring 6 topics
Identify the greatest common factor of a polynomial's terms and factor it out completely, verifying the result by distributing back through the factored expression
Factor trinomials of the form x squared plus bx plus c by finding two integers whose product equals c and whose sum equals b, checking factorability with the discriminant
Factor trinomials of the form ax squared plus bx plus c with a not equal to one using the ac method or trial-and-error, always checking for a GCF first before applying advanced techniques
Apply the difference of two squares pattern to factor binomials of the form a squared minus b squared and identify when this pattern does and does not apply
Solve quadratic equations by factoring, applying the zero-product property to set each factor equal to zero and solve for both roots, then verifying solutions in the original equation
Analyze which factoring strategy to apply (GCF, difference of squares, trinomial, grouping) based on the number of terms and coefficient patterns, executing a complete factoring algorithm
6Rational Expressions 6 topics
Identify values excluded from the domain of a rational expression by setting the denominator equal to zero and solving, listing all restrictions on the variable
Simplify rational expressions by factoring the numerator and denominator completely and canceling common factors, stating any excluded values for the simplified form
Multiply and divide rational expressions by factoring all numerators and denominators, converting division to multiplication by the reciprocal, and canceling common factors before multiplying
Add and subtract rational expressions with like and unlike denominators by finding the least common denominator, rewriting each fraction, combining numerators, and simplifying the result
Solve rational equations by multiplying both sides by the LCD to clear fractions, solving the resulting polynomial equation, and checking solutions against excluded values to reject extraneous roots
Analyze work-rate and proportion word problems by writing and solving rational equations, interpreting solutions in terms of the original context and checking for reasonableness
7Radicals and Exponents 7 topics
State the laws of integer exponents including product rule, quotient rule, power rule, zero exponent, and negative exponent, and apply each rule to simplify exponential expressions
Simplify square root and cube root radical expressions by factoring out perfect square or cube factors from the radicand and applying the product and quotient properties of radicals
Add and subtract radical expressions by identifying like radicals (same index and radicand), simplifying each radical first, and combining like radical terms
Multiply radical expressions using the distributive property and FOIL, then simplify the result by combining like terms and reducing radical factors
Rationalize the denominator of a fraction containing a square root by multiplying numerator and denominator by the conjugate or by the radical itself as appropriate
Convert between radical notation and rational exponent notation and apply the laws of exponents to expressions with fractional exponents, simplifying fully
Analyze scientific notation operations (multiplication, division, addition, subtraction) and evaluate when scientific notation is most appropriate for expressing very large or very small quantities
8Functions 6 topics
Describe the definition of a function as a relation where each input has exactly one output, and identify functions versus non-functions from tables, mapping diagrams, sets of ordered pairs, and graphs using the vertical line test
Evaluate functions using function notation f(x) by substituting numerical and algebraic expressions for x, interpreting f(a) as the output value when the input is a
Identify the domain and range of a function from its equation, table, or graph, expressing these sets using inequality notation, set-builder notation, and interval notation
Graph linear functions by plotting points or using slope-intercept form and identify increasing, decreasing, and constant behaviors from the graph and equation
Evaluate and simplify compositions of two functions (f composed with g)(x) by substituting the inner function into the outer function, distinguishing composition from multiplication of functions
Analyze a real-world scenario modeled by a linear function by identifying the rate of change and initial value, writing the function equation, and interpreting slope and intercepts in context
9Quadratic Functions and Equations 7 topics
Identify the standard form, vertex form, and factored form of a quadratic function and state the information each form reveals directly about the parabola's shape and key features
Solve quadratic equations by completing the square, transforming the equation into vertex form to find exact solutions including irrational and non-real roots
Apply the quadratic formula to solve any quadratic equation ax squared plus bx plus c equals zero, including equations with irrational solutions, and verify solutions by substitution
Calculate the discriminant b squared minus 4ac and use its sign to determine the number and type of solutions (two real, one real, or two non-real) before solving
Graph quadratic functions by identifying vertex, axis of symmetry, direction of opening, and intercepts, plotting additional points to sketch an accurate parabola
Analyze a projectile motion or area optimization problem by writing a quadratic model, finding the vertex for maximum or minimum value, and interpreting the result in the physical context
Compare solving methods for quadratic equations (factoring, completing the square, quadratic formula, square root property) and select the most efficient method based on the structure of each equation
10Graphing and Word Problems 6 topics
Name key features of a graph including intercepts, maximum and minimum points, intervals of increase and decrease, symmetry, and end behavior, using correct mathematical vocabulary
Graph absolute value and piecewise-defined functions by evaluating the function over each piece of the domain, plotting key points, and connecting segments appropriately
Solve number, age, mixture, distance-rate-time, and percent word problems by defining variables, writing equations, solving algebraically, and verifying the reasonableness of the answer
Interpret key features of a function's graph in context of a real-world scenario, explaining what x-intercepts, y-intercept, maximum, and minimum represent in terms of the situation
Evaluate and compare multiple representations of a function (equation, table, graph, verbal description) and translate fluently among all four representations for linear and quadratic functions
Analyze a scatter plot to determine whether a linear or quadratic model better fits the data, write a function equation for the trend, and use the model to make predictions and assess their validity
Scope
Included Topics
- Linear equations (one-variable, two-variable, standard and slope-intercept forms), inequalities (linear, compound, absolute value), systems of equations and inequalities (substitution, elimination, graphing), polynomials (operations, degree, special products), factoring (GCF, trinomials, difference of squares, sum/difference of cubes), rational expressions (simplification, operations, complex fractions), radicals and rational exponents (simplification, operations, rationalizing denominators), integer and rational exponents (laws of exponents, scientific notation), functions (definition, notation, domain/range, evaluation, composition), quadratic functions and equations (completing the square, quadratic formula, discriminant, vertex form), graphing (lines, parabolas, absolute value, piecewise), word problems and modeling
Not Covered
- Calculus concepts beyond introductory limits
- Matrices and determinants (covered in Linear Algebra)
- Complex number arithmetic beyond the discriminant context
- Statistics and probability (covered in separate domains)
- Trigonometry (covered in Trigonometry domain)