9th Grade Honors Algebra 1 Study Guide

How to use this guide

Study one section at a time. Work each example on paper before reading the solution, then explain why each step is valid. Use your class notes and teacher’s review list to confirm which topics are included in your course.

1. Algebra foundations

Order of operations and properties

Use grouping symbols first, then exponents, multiplication/division from left to right, and addition/subtraction from left to right. Keep negative signs and parentheses carefully organized.

  • Distributive property: a(b + c) = ab + ac.

  • Commutative property: a + b = b + a and ab = ba.

  • Associative property: (a + b) + c = a + (b + c) and (ab)c = a(bc).

  • Identity property: a + 0 = a and a · 1 = a.

  • Inverse property: a + (−a) = 0 and, for a ≠ 0, a · (1/a) = 1.

Example: Simplify 4 − 2(3 − 7)². First, 3 − 7 = −4; then (−4)² = 16; finally, 4 − 2(16) = −28.

Expressions, equations, and inequalities

  • An expression has numbers, variables, and operations; it does not have an equals sign.

  • An equation states that two expressions are equal. Solve by keeping both sides balanced.

  • An inequality compares values using <, >, ≤, or ≥. Reverse the inequality sign when multiplying or dividing both sides by a negative number.

  • Like terms have the same variable part with the same exponents. Combine their coefficients: 5x − 2x = 3x, but x and x² are not like terms.

  • A solution is a value that makes the equation or inequality true. Check equation solutions by substitution.

Example: 3(2x − 5) = 4x + 7 → 6x − 15 = 4x + 7 → 2x = 22 → x = 11.

Inequality example: −2(3x − 1) > 10 → −6x + 2 > 10 → −6x > 8 → x < −4/3. The sign reverses when dividing by −6.

Ratios, rates, and percent

  • A ratio compares quantities; a unit rate gives the amount per one unit.

  • Convert a percent to a decimal by dividing by 100. To find p% of n, calculate (p/100)n.

  • For a percent change, use (new − original)/original × 100%. A negative result indicates a decrease.

2. Functions and representations

  • A relation is a set of input-output pairs. A function assigns exactly one output to each input.

  • Use the vertical line test on a graph: if a vertical line crosses the graph more than once, it is not a function.

  • Domain is the set of allowed inputs; range is the set of outputs.

  • Function notation f(x) names the output for input x; it does not mean f multiplied by x.

  • A function can be shown by a table, graph, equation, mapping, or verbal rule. Connect the same feature across representations.

Example: If f(x) = 3x − 4, then f(−2) = 3(−2) − 4 = −10.

Sequences

  • An arithmetic sequence has a constant difference; a geometric sequence has a constant ratio.

  • For an arithmetic sequence, aₙ = a₁ + (n − 1)d, where d is the common difference.

  • Check whether a sequence is arithmetic or geometric by comparing consecutive terms, not just the first and last terms.

Example: 5, 9, 13, 17, … has common difference 4, so aₙ = 5 + 4(n − 1) = 4n + 1.

3. Linear equations, functions, and models

Slope and forms of a line

  • Slope measures the rate of change: m = (y₂ − y₁)/(x₂ − x₁). A zero slope is horizontal; an undefined slope is vertical.

  • Slope-intercept form: y = mx + b, where m is slope and b is the y-intercept.

  • Point-slope form: y − y₁ = m(x − x₁), for a line through (x₁, y₁).

  • Standard form: Ax + By = C, commonly with integer coefficients.

  • Parallel nonvertical lines have the same slope. Perpendicular nonvertical lines have slopes that are negative reciprocals.

  • A linear function has a constant rate of change; its graph is a straight line.

Example: Find the line through (2, 1) and (6, 9). Slope m = (9 − 1)/(6 − 2) = 2. Using y = mx + b and (2, 1): 1 = 4 + b, so b = −3. The equation is y = 2x − 3.

Interpreting and building models

  • In y = mx + b, interpret m as the change in output for each one-unit increase in input and b as the starting value (when x = 0).

  • Identify units and what each variable represents before interpreting parameters.

  • A direct variation has the form y = kx and passes through (0, 0).

  • For a word problem, define variables, write an equation or function, solve, and check whether the answer makes sense in context.

  • A graph that looks linear is not enough: check whether the context supports a constant rate of change.

Model example: A club charges a $12 sign-up fee plus $5 per visit. If v is visits and C is total cost, then C = 5v + 12. The slope 5 is dollars per visit; the y-intercept 12 is the initial fee.

4. Systems of equations and inequalities

A solution to a system is an ordered pair that makes both equations true. Graphically, it is an intersection point.

  • Substitution: solve one equation for a variable, then substitute that expression into the other equation.

  • Elimination: add or subtract equations to cancel a variable. Multiply an equation first if needed.

  • A system may have one solution (lines intersect once), no solution (parallel distinct lines), or infinitely many solutions (same line).

  • For a system of linear inequalities, graph each boundary line, use a dashed line for < or > and a solid line for ≤ or ≥, then shade the overlap.

Example: Solve 2x + y = 7 and x − y = 2. Add the equations: 3x = 9, so x = 3. Substitute into x − y = 2: y = 1. Solution: (3, 1).

5. Exponents, scientific notation, and polynomials

Exponent rules

For nonzero a and b, and integer exponents where defined:

  • Product: aᵐ · aⁿ = aᵐ⁺ⁿ.

  • Quotient: aᵐ/aⁿ = aᵐ⁻ⁿ, a ≠ 0.

  • Power of a power: (aᵐ)ⁿ = aᵐⁿ.

  • Power of a product: (ab)ⁿ = aⁿbⁿ.

  • Zero exponent: a⁰ = 1, a ≠ 0.

  • Negative exponent: a⁻ⁿ = 1/aⁿ, a ≠ 0.

A negative exponent does not make the value negative: x⁻² = 1/x². Also, (−3)² = 9, while −3² = −9 because the exponent applies before the leading negative.

Polynomials

  • A monomial has one term; a binomial has two; a trinomial has three.

  • The degree of a one-variable term is its exponent; the degree of a polynomial is the greatest exponent with a nonzero coefficient.

  • Add or subtract polynomials by combining like terms. Multiply using the distributive property; multiply every term in one factor by every term in the other.

Example: (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.

6. Factoring and quadratic functions

Factoring

Factoring reverses multiplication. Always check for a greatest common factor (GCF) first.

  • x² + bx + c = (x + p)(x + q) when p + q = b and pq = c.

  • For ax² + bx + c, look for factors that multiply to ac and add to b, or use a method taught in class.

  • Check a factorization by multiplying it back out.

Example: Factor 2x² + 7x + 3. The factors are (2x + 1)(x + 3), since multiplying gives 2x² + 7x + 3.

Quadratic graphs and solutions

  • A quadratic function has the form y = ax² + bx + c, with a ≠ 0. Its graph is a parabola.

  • In y = a(x − h)² + k, the vertex is (h, k), the axis of symmetry is x = h, and the graph opens up if a > 0 or down if a < 0.

  • The zeros or roots are x-values where y = 0; on a graph, they are x-intercepts.

  • Solve quadratics by factoring, taking square roots, completing the square, or using the quadratic formula, as required by the problem.

  • Quadratic formula: x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac indicates two real solutions if positive, one repeated real solution if zero, and no real solutions if negative.

  • The maximum or minimum value occurs at the vertex. In an application, interpret the vertex and intercepts with units and context.

Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3. Check both values in the original equation.

7. Data, scatterplots, and modeling

  • A scatterplot displays paired numerical data. Describe the direction (positive, negative, or none), form (linear or nonlinear), strength, and any outliers.

  • A positive association means larger x-values tend to pair with larger y-values; a negative association means larger x-values tend to pair with smaller y-values.

  • Correlation does not prove causation. Other variables or chance may explain an association.

  • A line of best fit summarizes a roughly linear trend; use it for reasonable estimates, not unsupported extrapolation far beyond the data.

  • A residual is observed y − predicted y. A positive residual means the point is above the model; a negative residual means it is below.

  • When comparing models, consider the pattern, residuals, context, and whether the model is appropriate for the data.

8. Common mistakes to avoid

  • Combining unlike terms, such as 3x + 2x² = 5x³. They cannot be combined.

  • Forgetting to distribute a negative sign: −(x − 4) = −x + 4.

  • Reversing an inequality sign at the wrong time—or forgetting to reverse it when dividing or multiplying by a negative.

  • Treating f(x) as multiplication instead of function notation.

  • Confusing slope with the y-intercept or forgetting units in a model.

  • Solving one equation in a system but not checking the answer in both.

  • Assuming every quadratic has two distinct real roots; use the discriminant or graph to check.

  • Losing solutions by dividing by a variable expression that could equal zero. Check the original equation.

  • Writing a decimal answer without considering whether the context requires an exact value or a particular unit.

9. Practice questions

Try these without notes. Show steps and label units when relevant.

  1. Simplify: 5 − 2(4 − 7)².

  2. Solve: 4(x − 3) + 2 = 2x + 10.

  3. Solve and graph: −3x + 6 ≤ 15.

  4. Evaluate f(−2) if f(x) = 3x − 4.

  5. Find the slope through (−1, 2) and (3, 10).

  6. Write the line with slope −2 through (1, 5) in slope-intercept form.

  7. A taxi fare is $4 plus $2.50 per mile. Write a function for the cost C(m), and find the cost of a 6-mile ride.

  8. Solve the system: x + y = 8; x − y = 2.

  9. Simplify: (2a³b⁻²)(3a²b⁴). Write with positive exponents.

  10. Expand and simplify: (x + 4)(x − 3).

  11. Factor: x² + 7x + 12.

  12. Solve: x² − 9 = 0.

  13. Solve: x² + 2x − 1 = 0 using the quadratic formula.

  14. For y = −2(x − 3)² + 5, state the vertex, axis of symmetry, and whether the parabola opens up or down.

  15. For y = x² − 6x + 8, find the x-intercepts by factoring.

  16. A data point has observed y = 14 and predicted y = 11. Find its residual and state whether the point lies above or below the model.

  17. A sequence is 12, 8, 4, 0, … Is it arithmetic or geometric? Give an explicit rule for aₙ.

  18. A student says 2(x + 5) = 2x + 5. Identify and correct the mistake.

Answer key

  1. 5 − 2(−3)² = 5 − 18 = −13.

  2. 4x − 12 + 2 = 2x + 10 → 2x = 20 → x = 10.

  3. −3x ≤ 9 → x ≥ −3. On a number line, use a closed dot at −3 and shade right.

  4. 3(−2) − 4 = −10.

  5. (10 − 2)/(3 − (−1)) = 8/4 = 2.

  6. y − 5 = −2(x − 1), so y = −2x + 7.

  7. C(m) = 2.50m + 4; C(6) = $15.

  8. Add equations: 2x = 10, so x = 5; then y = 3. (5, 3).

  9. Multiply coefficients and add exponents: 6a⁵b².

  10. x² − 3x + 4x − 12 = x² + x − 12.

  11. (x + 3)(x + 4).

  12. (x − 3)(x + 3) = 0, so x = −3 or x = 3.

  13. x = (−2 ± √(4 + 4))/2 = (−2 ± √8)/2 = −1 ± √2.

  14. Vertex (3, 5); axis x = 3; opens down because a = −2.

  15. x² − 6x + 8 = (x − 2)(x − 4), so x-intercepts are (2, 0) and (4, 0).

  16. Residual = observed − predicted = 14 − 11 = 3; it lies above the model.

  17. Arithmetic with common difference −4. aₙ = 12 − 4(n − 1), or 16 − 4n.

  18. Distribute 2 to both terms: 2(x + 5) = 2x + 10.

Quick self-check

I can simplify expressions, solve and check equations and inequalities, interpret functions, find and explain slope, solve systems, apply exponent rules, expand and factor polynomials, solve and interpret quadratics, and describe data trends with evidence.