Praxis Algebra I cheat sheet

Closure: integers are closed under addition, subtraction, and multiplication but not division (e.g., 3 ÷ 2 is not an integer).

Order of operations: grouping symbols, then exponents/roots, then multiplication/division left to right, then addition/subtraction left to right.

Absolute value |a| is the distance from a to 0 on the number line; |a| ≥ 0 always, and |a − b| is the distance between a and b.

Properties: a + b = b + a and ab = ba (commutative); (a + b) + c = a + (b + c) and (ab)c = a(bc) (associative); a(b + c) = ab + ac (distributive).

Additive inverse of a is −a; multiplicative inverse (reciprocal) of nonzero a is 1/a; a + (−a) = 0 and a · (1/a) = 1.

Subtracting b equals adding −b: a − b = a + (−b). Dividing by b equals multiplying by 1/b: a ÷ b = a · (1/b) for b ≠ 0.

Rational numbers can be written p/q with integers p, q and q ≠ 0; their decimal expansions terminate or repeat. Irrational numbers (√2, π) do neither.

Rules for signs: product or quotient of two numbers with the same sign is positive; with opposite signs it is negative.

Multiplication by a negative reverses an inequality: if a < b, then −a > −b.

Distribute then combine like terms: 3(2x − 5) + 4x = 6x − 15 + 4x = 10x − 15.

Like terms have identical variable parts; 5x² and 5x are not like terms because the exponents differ.

Product rule: xᵐ · xⁿ = xᵐ⁺ⁿ; quotient rule: xᵐ / xⁿ = xᵐ⁻ⁿ; power rule: (xᵐ)ⁿ = xᵐⁿ; negative exponent: x⁻ⁿ = 1/xⁿ for x ≠ 0.

Zero exponent: x⁰ = 1 for any x ≠ 0; 0⁰ is undefined.

Multiply binomials with FOIL or distribution: (x + 3)(x − 5) = x² − 2x − 15.

Perfect-square patterns: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² (the middle term is essential).

Difference of squares: a² − b² = (a + b)(a − b).

Factor by grouping: for ax + ay + bx + by, group as a(x + y) + b(x + y) = (a + b)(x + y).

Degree of a polynomial is the largest exponent; the leading coefficient multiplies the highest-power term.

Solve linear equations by undoing operations in reverse order: first add or subtract, then multiply or divide both sides.

Clearing fractions: multiply every term by the least common denominator, then solve the resulting integer equation.

If solving leads to a false statement such as 3 = 7, there is no solution; if it leads to a true statement such as 0 = 0, every real number is a solution.

Isolate a variable in a formula by treating other variables as constants: from A = P + Prt, r = (A − P)/(Pt).

Absolute value equation |x| = c with c > 0 gives two cases, x = c and x = −c; with c < 0 it has no solution.

Multiplying or dividing an inequality by a negative number reverses the inequality symbol.

Compound inequality a < x < b is an intersection; 'x < a or x > b' is a union and is written with two separate inequalities.

Slope-intercept form y = mx + b: m is the slope and b is the y-intercept, the value of y when x = 0.

Point-slope form y − y₁ = m(x − x₁) builds an equation from one point and the slope.

Slope formula: m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have slope 0, vertical lines have undefined slope.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (product = −1).

Standard form Ax + By = C has intercepts at (C/A, 0) and (0, C/B) when A and B are nonzero.

A relation is a function if each input has exactly one output; the vertical line test fails a graph that is not a function.

Function notation: f(3) means substitute x = 3; the domain is the set of allowed inputs and the range the set of outputs.

Solving a system by substitution: solve one equation for a variable, substitute into the other, then back-substitute.

Solving a system by elimination: multiply equations to create opposite coefficients for one variable, add, then solve.

A linear system has one solution if the slopes differ, none if slopes are equal and intercepts differ, and infinitely many if the equations are equivalent.

Rate of change of a linear function equals its slope and is constant over any interval.

Product, quotient, and power rules apply to all real exponents: xᵐ · xⁿ = xᵐ⁺ⁿ, xᵐ/xⁿ = xᵐ⁻ⁿ, (xᵐ)ⁿ = xᵐⁿ, (xy)ⁿ = xⁿyⁿ.

Rational exponents: x^(1/n) = ⁿ√x and x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ, defined for x ≥ 0 when n is even.

Simplify radicals by removing perfect-square factors: √72 = √(36 · 2) = 6√2.

Combine radicals only when the radicands match: 3√5 + 2√5 = 5√5, but √2 + √3 cannot be combined.

Rationalize a denominator by multiplying by the conjugate or the radical itself: 1/√3 = √3/3.

Exponential function form f(x) = a · bˣ: a is the initial value and b the growth factor; b > 1 is growth, 0 < b < 1 is decay.

Compound interest: A = P(1 + r/n)^(nt), where r is the annual rate, n the compounding periods per year, and t the years.

Doubling time for growth factor b is log base 2 of 2 divided by log base 2 of b; equivalently solve bᵗ = 2.

Graph of f(x) = a · bˣ has y-intercept (0, a), is always positive, and has the x-axis as a horizontal asymptote.

Standard form y = ax² + bx + c: the parabola opens upward if a > 0 and downward if a < 0.

Axis of symmetry: x = −b/(2a); the vertex is at that x-value with y = f(−b/(2a)).

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) solves ax² + bx + c = 0 for a ≠ 0.

Discriminant b² − 4ac: positive means two real solutions, zero means one repeated real solution, negative means no real solutions.

Factoring method: find two numbers whose product is ac and whose sum is b, split the middle term, then factor by grouping.

Zero product property: if (x − r)(x − s) = 0, then x = r or x = s.

Vertex form y = a(x − h)² + k has vertex (h, k); completing the square converts standard form to vertex form.

The x-intercepts of a quadratic are the real solutions of ax² + bx + c = 0; the y-intercept is (0, c).

Sum of roots = −b/a and product of roots = c/a for ax² + bx + c = 0.

Function notation: f(g(2)) means evaluate g at 2 first, then apply f to that result.

Domain restrictions arise from division by zero (excluded) and even roots of negatives (excluded); polynomial domains are all real numbers.

Average rate of change over [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line.

Arithmetic sequence: aₙ = a₁ + (n − 1)d, where d is the common difference; the graph is linear.

Geometric sequence: aₙ = a₁ · rⁿ⁻¹, where r is the common ratio; the graph is exponential.

Recursive rules define a term from previous terms, e.g., a₁ = 3, aₙ = aₙ₋₁ + 5.

Translations: f(x) + k shifts the graph up k units; f(x − h) shifts it right h units; −f(x) reflects across the x-axis.

A model's slope or growth factor gives the constant rate of change per unit of the independent variable.

Mean = sum of values divided by the number of values; median is the middle value of ordered data or the mean of the two middle values.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1 and measures the spread of the middle half of the data.

Outliers are often flagged when a value is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves it unchanged, while multiplying by a constant scales it.

A line of best fit summarizes a linear association; the slope estimates the predicted change in y per one-unit increase in x.

Correlation coefficient r ranges from −1 to 1; r near 0 indicates little linear association, and correlation does not imply causation.

Two-way tables: joint relative frequency divides a cell count by the grand total; conditional relative frequency divides by a row or column total.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A) = 1 − P(A).

For independent events, P(A and B) = P(A) · P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).