Algebra 1 — Free Practice Questions, Study Guides & Mock Exams

Learn equations, functions, expressions, systems, and core algebra fluency. In compliance with U.S. Public School Curriculum.

Topics covered

• True/False
• MCQ
• Word Bank
• numeric
• Parsons
• Trap Classification

Free Algebra 1 study guides

• Variables and Expressions — Build reusable algebraic models, read signed terms, protect grouping, translate phrases, and evaluate expressions with precision.
• Order of Operations — Compile complex expressions in four passes while keeping grouping, exponent scope, and fraction bars under control.
• Properties of Numbers — Justify every algebraic rewrite with number, equality, and substitution properties that preserve value or truth.
• The Distributive Property — Expand every branch, track negative signs, combine only like terms, and reverse distribution to factor expressions.
• Equations — Interpret equality, reverse an equation’s construction on both sides, translate words, and verify solutions in the original equation.
• Relations — Trace input-output pairings across ordered pairs, tables, mappings, and graphs, then extract domain and range without duplicates.
• Functions — Apply the one-output-per-input rule across pairs, tables, mappings, and graphs, then evaluate function notation by substitution.
• Interpreting Graphs of Functions — Read axes and scale, describe intervals, interpret intercepts and extrema, calculate rate of change, and use context to judge continuity.
• Writing Equations — Translate verbal, cost, and geometry relationships into equations with defined variables, preserved structure, and compatible units.
• Solving One-Step Equations — Isolate a variable with inverse operations, preserve equation balance, handle signed and fractional coefficients, and check each solution.
• Solving Multi-Step Equations — Simplify, distribute, combine like terms, remove operation layers, and verify multi-step equation solutions in the original equation.
• Solving Equations with the Variable on Each Side — Simplify and collect variable terms on one side, then distinguish one solution, no solution, and infinitely many solutions.
• Solving Equations Involving Absolute Value — Interpret absolute value as distance, isolate it, predict the solution count, solve valid branches, and check the original equation.
• Ratios and Proportions — Preserve ratio order, units, and corresponding positions while finding unit rates, solving proportions, and applying scale factors.
• Percent of Change — Measure percent change against the original baseline, label increase or decrease, and apply discount and markup multipliers.
• Literal Equations and Dimensional Analysis — Rearrange literal equations to isolate a target variable and orient conversion factors so unwanted units cancel.
• Weighted Averages — Compute weighted averages from percent weights, counts, grades, and mixture contributions.
• Graphing Linear Equations — Graph linear equations as complete solution sets using value tables, intercepts, special-line rules, and substitution checks.
• Solving Linear Equations by Graphing — Solve linear equations by graphing both sides, reading intersection inputs or x-intercepts, and interpreting solution counts and precision.
• Intercepts and Graphing — Find x- and y-intercepts with zero coordinates, graph standard-form lines efficiently, and interpret axis crossings in context.
• Rate of Change and Slope — Compute rate of change and slope from contexts, graphs, points, and tables while matching subtraction order, units, and direction.
• Direct Variation — Recognize and model direct variation using constant ratios, equations of the form y = kx, and graphs through the origin.
• Arithmetic Sequences as Linear Functions — Model arithmetic sequences recursively and explicitly, connect common difference to slope, and graph their discrete positive-integer domain.
• Transformations of Linear Functions — Compare linear functions through slope and y-intercept to describe vertical shifts, steepness changes, and reflections.
• Proportional and Nonproportional Relationships — Classify linear relationships using constant ratios, equation form, origin behavior, and fixed-versus-zero starting amounts.
• Scatter Plots and Lines of Fit — Describe bivariate associations, fit linear trends, build approximate models, distinguish interpolation from extrapolation, and avoid causal overreach.
• Modeling Systems of Inequalities — Translate real-world limits into inequalities, solve signed and compound constraints, interpret boundary inclusion, and round count solutions correctly.
• Graphing Equations in Slope-Intercept Form — Turn y = mx + b into a graph by anchoring the y-intercept, navigating rise over run, and checking plotted points by substitution.
• Writing Equations in Slope-Intercept Form — Translate a slope and intercept, graph, pair of points, table, or context into y = mx + b and recognize horizontal and vertical exceptions.
• Writing Equations in Point-Slope Form — Build a linear equation from one point and a slope, preserve coordinate signs, compare equivalent anchors, convert forms, and graph from the known point.
• Parallel and Perpendicular Lines — Classify lines by slope, generate negative reciprocals, handle horizontal and vertical cases, and write related lines through required points.
• Regression and Median-Fit Lines — Compare least-squares regression and median-fit lines, calculate residuals, interpret technology output, and judge linear fit from residual patterns.
• Inverse Linear Functions — Reverse a linear function by swapping inputs and outputs, reflecting across y = x, solving for the inverse, and checking both compositions.
• Solving Inequalities by Addition and Subtraction — Solve one-step addition and subtraction inequalities, graph complete solution sets, translate comparison phrases, and verify endpoints.
• Solving Inequalities by Multiplication and Division — Solve multiplication and division inequalities by checking the operation sign, reversing only for negative factors or divisors, then graphing the solved set.
• Solving Multi-Step Inequalities — Simplify, collect, isolate, and classify multi-step inequalities, including variables on both sides, negative final coefficients, all-real, and no-solution outcomes.
• Solving Compound Inequalities — Solve and graph compound inequalities by treating and as intersection, or as union, preserving endpoints, and reversing both symbols for negative chain operations.
• Graphing Inequalities in Two Variables — Map linear inequality solutions with accurate boundaries, solid and dashed line rules, test points, half-plane shading, and contextual constraints.
• Absolute Value Basics & Distance — Understand absolute value as nonnegative distance, evaluate inside the bars correctly, distinguish inside and outside signs, and measure distance between values.
• Absolute Value Equations — Solve absolute value equations through center-and-distance reasoning, positive/zero/negative classification, two linear branches, and original-equation checks.
• Multi-Step Absolute Equations — Unpack outside operations, isolate the absolute value, split valid cases, solve both linear branches, and check candidates in the original equation.
• Inequalities Involving Absolute Value — Solve absolute-value inequalities as distance regions, distinguish between and outside solutions, and handle endpoints and special bounds.
• Multi-Step Absolute Inequalities — Unwrap multi-step absolute-value inequalities, scan isolated bounds, choose between or outside solutions, and audit endpoints.
• Graphing Absolute Value Inequalities on a 1D Number Line — Turn absolute value distance rules into number-line graphs using centers, radii, endpoint caps, segments, and rays.
• Graphing Absolute Value Functions in Two Variables — Build absolute value V-graphs from the vertex, symmetry, opening direction, and arm slopes.
• Transformations of Absolute Value Functions — Run the h, k, and a controls to move, reshape, reflect, and reverse-engineer every absolute value graph.
• Graphing Systems of Equations — Use an intersection radar to graph, classify, and verify one, none, or infinitely many shared solutions.
• Substitution — Run a replacement relay: isolate, insert the entire equal expression, solve, back-substitute, and check both equations.
• Elimination Using Addition and Subtraction — Align two equations, eliminate a variable by adding or subtracting whole rows, back-substitute, and verify the ordered pair.
• Elimination Using Multiplication — Scale every term using strategic least common multiples, create opposite coefficients, eliminate, and verify the system solution.
• Classification of Systems — Classify systems as one solution, no solution, or infinitely many by matching graph and algebra evidence.
• Applying Systems of Linear Equations — Model real situations with two unknowns, translate two independent conditions, solve efficiently, and verify the answer with units.
• Systems of Inequalities — Graph systems of inequalities by preserving boundary inclusion, shading each half-plane, and keeping only the shared feasible region.
• Linear Programming — Build the feasible region, move the objective, compare every vertex, and turn the winning corner into a real decision.
• Multiplication Properties of Exponents — Route exponent freight by structure: add same-base factor counts, multiply stacked powers, and dispatch an outside power to every factor.
• Division Properties of Exponents — Clear exponent quotients by canceling matching factors, subtracting exponents in the correct order, rewriting negative exponents, and preserving restrictions.
• Rational Exponents — Decode rational exponents as root and power controls, convert radical forms, evaluate negative exponents, and respect real-number domain limits.
• Scientific Notation — Encode, expand, multiply, divide, and normalize numbers in scientific notation.
• Exponential Functions — Identify, graph, write, and interpret exponential functions through constant ratios.
• Growth and Decay — Turn percent changes into growth or decay factors, model repeated change, read rates from exponential bases, and avoid rounding traps.
• Geometric Sequences as Exponential Functions — Find common ratios, build and use explicit geometric rules, handle negative and fractional ratios, and graph sequences as discrete exponential points.
• Recursive Formulas — Start with an initial term, use arithmetic and geometric recursive rules, generate sequences, and compare recursive with explicit formulas.
• Adding and Subtracting Polynomials — Sort terms by exponent, identify like terms and degree, add matching coefficients, distribute subtraction, and evaluate simplified polynomials.
• Multiplying a Polynomial by a Monomial — Distribute a monomial to every polynomial term, track signs, multiply coefficients, add matching-base exponents, and use area and degree checks.
• Multiplying Polynomials — Multiply polynomials with complete distribution, area grids, aligned partial products, sign control, like-term merging, and degree forecasts.
• Special Products — Recognize and derive the square-of-a-sum, square-of-a-difference, and conjugate identities, then apply and reverse-check each pattern.
• Using the Distributive Property — Reverse distribution to factor greatest common factors, control negative GCF signs, factor four terms by grouping, and verify by expanding.
• Solving x² + bx + c = 0 — Solve monic quadratic equations by factoring, use factor pairs for b and c, apply the zero-product property, and verify both roots.
• Solving ax² + bx + c = 0 — Factor and solve non-monic quadratics using the AC method, split the middle term, group matching binomials, apply zero product, and verify both roots.
• Differences of Squares — Recognize and factor A² − B² as conjugates, take square roots of complete terms, factor GCFs first, solve resulting equations, reject near-miss patterns, and verify by expansion.
• Perfect Squares — Recognize and factor perfect-square trinomials using A² ± 2AB + B²; handle signs, non-monic terms, GCFs, equations, near misses, missing coefficients, and expansion checks.
• Graphing Quadratic Functions — Graph quadratic functions from parent, vertex, and standard forms; locate vertices and axes, read transformations, plot symmetric points and intercepts, compare widths, and state domain and range.
• Solving Quadratic Equations by Graphing — Solve quadratic equations graphically by rewriting them as f(x) = 0, reading exact or approximate x-intercepts, distinguishing two, repeated, and no-real-root cases, using intersections and useful windows, and verifying each candidate.
• Transformations of Quadratic Functions — Transform quadratic functions in vertex form by reading h, k, and a; translate, stretch or compress, reflect, identify the vertex, axis, and extrema, write equations, and correct sign and width mistakes.
• Solving Quadratic Equations by Using Square Roots — Solve quadratic equations by isolating a squared expression, taking both square roots, simplifying radicals, handling shifted squares and no-real-solution cases, interpreting graph symmetry, and verifying both roots.
• Solving Quadratic Equations by Completing the Square — Complete quadratic equations and functions by making a = 1, adding (b/2)² to preserve equality, rewriting perfect-square trinomials, using both square roots, solving exact radical cases, and revealing vertex form.
• Solving Quadratic Equations by Using the Quadratic Formula — Solve any standard-form quadratic with signed coefficient substitution, the discriminant, both formula branches, exact radical simplification, root-type prediction, method selection, and substitution checks.
• Analyzing Functions with Successive Differences and Ratios — Classify equally spaced tables as linear, quadratic, or exponential using constant first differences, second differences, and defined ratios; connect quadratic second differences to the leading coefficient.
• Systems of Linear and Quadratic Equations — Solve linear–quadratic systems as shared ordered pairs by graphing and substitution; find every x-root and y-coordinate, forecast 0/1/2 intersections with the discriminant, and verify exact solutions in both equations.
• Special Functions — Evaluate and graph piecewise-defined, absolute-value, and step functions by routing inputs through domain conditions, marking open and closed endpoints, interpreting floor values, and auditing boundaries for gaps or overlaps.
• Square Root Functions — Graph and transform square-root functions from their endpoints; use perfect-square points, radicand restrictions, transformation parameters, reflection and scale, and contextual limits to determine domain and range.
• Simplifying Radical Expressions — Simplify numerical and variable radicals by extracting perfect-power factors, applying product and quotient properties, preserving absolute values, grouping higher-root factors, and rationalizing monomial denominators.
• Operations with Radical Expressions — Add, subtract, multiply, distribute, and rationalize radical expressions by simplifying first, combining only like radicals, completing every product, and using conjugates for binomial denominators.
• Radical Equations — Solve radical equations by isolating one radical, raising both sides to the matching power, re-isolating when another radical remains, screening the real domain, and rejecting extraneous candidates by checking the original equation.
• The Pythagorean Theorem — Use the Pythagorean Theorem to identify the hypotenuse, interpret squared side lengths as areas, find missing sides, test the converse, recognize scaled triples, and derive coordinate distance.
• Distance and Midpoint Formulas — Use coordinate differences to find distance, average matching coordinates to find midpoint, protect negative signs, recognize horizontal and vertical shortcuts, and apply both formulas to coordinate geometry.
• Similar Triangles — Match corresponding parts, find and interpret a scale factor, build consistent proportions, solve missing sides, prove Angle–Angle similarity, and use similar triangles for indirect measurement and dilation.
• Trigonometric Ratios — Name opposite, adjacent, and hypotenuse from a reference angle; choose sine, cosine, or tangent; solve missing sides and angles; check degree mode; and model elevation and depression.
• Inverse Variation — Model inverse variation with a constant product, find and use the variation constant, read tables and two-branch graphs, distinguish direct from inverse variation, and solve work-rate contexts.
• Rational Functions — Analyze rational-function domains, distinguish vertical asymptotes from holes, use degree rules for end behavior, find intercepts, and transform the parent reciprocal graph.
• Simplifying Rational Expressions — Factor and simplify rational expressions, cancel only matching factors, use GCF and special patterns, and preserve every restriction from the original denominator.
• Multiplying and Dividing Rational Expressions — Multiply and divide factored rational expressions, reciprocate only the divisor, cross-cancel exact factors, and track original-denominator and divisor-zero restrictions.
• Dividing Polynomials — Divide polynomials term by term or with long division, use zero placeholders, interpret remainders, check degrees, and verify by rebuilding the dividend.
• Adding and Subtracting Rational Expressions — Factor denominators, build the least common denominator, combine complete numerators, simplify, and preserve every original restriction.
• Mixed Expressions and Complex Fractions — Simplify mixed and complex rational expressions by locating the main bar, choosing reciprocal division or LCD clearing, and preserving every nested restriction.
• Rational Equations and Functions — Solve rational equations and function intersections by recording restrictions, clearing every denominator, checking candidates in the original, and interpreting domains and units.
• Samples and Studies — Design and evaluate surveys, observational studies, and experiments by separating random selection from random assignment, detecting bias, and limiting conclusions to what the study supports.
• Statistics and Parameters — Distinguish population parameters from sample statistics, calculate and interpret center and spread, and select resistant summaries when data are skewed or contain outliers.
• Distributions of Data — Read dot plots, histograms, and box plots; describe shape, center, spread, clusters, gaps, and potential outliers; and choose a display that preserves the data honestly.
• Scatter Plots and Linear Regression — Read direction, form, strength, and unusual features in scatter plots; interpret regression equations; predict responsibly; and separate association from causation.
• Two-Way Tables and Relative Frequencies — Organize two categorical variables, complete row and column totals, and choose the correct denominator for joint, marginal, and conditional relative frequencies.
• Probability Basics — Build complete sample spaces, calculate theoretical and experimental probabilities, use complements, distinguish odds, and model probability with geometric measures.
• Permutations and Combinations — Count outcomes across stages, use factorials, distinguish ordered roles from unordered groups, and choose permutations or combinations with confidence.
• Probability of Compound Events — Translate compound-event language, map sample spaces, calculate intersections and unions, use complements, update dependent sequences, and condition on a new denominator.
• Comparing Sets of Data — Compare distributions fairly using aligned scales, compatible measures of center and spread, shape, overlap, context, and claims supported by the study design.
• Simulations — Design valid chance simulations, define one trial, assign random outcomes proportionally, repeat consistently, estimate probabilities with relative frequency, and audit the model.
• Probability Distributions — Build, graph, validate, and interpret discrete probability distributions; calculate expected value; preserve payoff signs; and recognize valid binomial settings.

Units

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11
• Chapter 12

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