Georgia Unified Math cheat sheet

Rational exponents: a^(m/n) = (a^(1/n))^m = (ⁿ√a)^m; the denominator n is the root, the numerator m is the power, and a ≥ 0 when n is even.

Negative exponent: a^(−n) = 1/a^n and (a/b)^(−n) = (b/a)^n; a negative exponent never makes the result negative.

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n b^n.

Simplify radicals by extracting perfect powers: √(a²b) = a√b for a ≥ 0; combine like radicals only when the index and radicand match.

Rationalize a denominator by multiplying by the conjugate: 1/(√a+√b) = (√a−√b)/(a−b).

Complex numbers: i² = −1, i³ = −i, i⁴ = 1; divide by multiplying numerator and denominator by the conjugate of the denominator.

Scientific notation: (a×10^m)(b×10^n) = (ab)×10^(m+n); adjust the coefficient so that 1 ≤ |a| < 10.

Unit conversion is dimensional analysis: multiply by ratios equal to 1 so unwanted units cancel, e.g. km/h → m/s uses 1000 m/1 km and 1 h/3600 s.

Vectors in component form: magnitude ‖v‖ = √(x²+y²); direction angle θ = arctan(y/x) adjusted for the quadrant.

Quadratic formula: x = (−b ± √(b²−4ac))/(2a); the discriminant b²−4ac gives two real roots if positive, one if zero, and two complex roots if negative.

Vertex of y = ax²+bx+c occurs at x = −b/(2a); the vertex form y = a(x−h)²+k shows vertex (h, k) directly.

Factor by grouping: for ax²+bx+c, find two numbers whose product is ac and whose sum is b, then split the middle term.

Zero product property: if (x−p)(x−q) = 0, then x = p or x = q; always set one side to zero before factoring.

Systems of two linear equations: solve by substitution or elimination; parallel lines (equal slopes, different intercepts) give no solution, identical lines give infinitely many.

Exponential growth/decay: A = P(1+r)^t for growth, A = P(1−r)^t for decay, and A = Pe^(rt) for continuous change.

Logarithms invert exponentials: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(x^k) = k log_b x, and log_b b = 1.

Change of base: log_b x = ln x / ln b; use it to evaluate any logarithm with a calculator.

Solve radical equations by isolating the radical, squaring both sides, then checking for extraneous roots introduced by squaring.

A relation is a function when every input in the domain maps to exactly one output; the vertical line test detects violations on a graph.

Function notation: f(g(x)) means substitute g(x) into f; composition is not generally commutative, so f(g(x)) ≠ g(f(x)) in general.

Inverse functions undo each other: f⁻¹(f(x)) = x; find f⁻¹ by swapping x and y, then solving for y. The graph of f⁻¹ reflects f over y = x.

Domain restrictions: exclude x-values that make a denominator zero or make an even-index radicand negative.

Transformations of y = f(x): f(x−h) shifts right h, f(x)+k shifts up k, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis, and af(x) stretches vertically by |a|.

Even functions satisfy f(−x) = f(x) (symmetric about the y-axis); odd functions satisfy f(−x) = −f(x) (symmetric about the origin).

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

Exponential functions y = ab^x grow when b > 1 and decay when 0 < b < 1; linear functions change by a constant difference while exponentials change by a constant ratio.

A sequence is arithmetic when consecutive terms differ by constant d (aₙ = a₁ + (n−1)d) and geometric when they differ by constant ratio r (aₙ = a₁r^(n−1)).

Distance formula: d = √((x₂−x₁)² + (y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2); slope = (y₂−y₁)/(x₂−x₁).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals).

Triangle angle sum is 180°; the exterior angle equals the sum of the two remote interior angles.

Similar figures have proportional corresponding sides and equal corresponding angles; areas scale by the square of the scale factor and volumes by its cube.

Pythagorean theorem a² + b² = c² for right triangles; converse: if a² + b² = c², the triangle is right.

Circle facts: circumference C = 2πr, area A = πr², arc length = (θ/360)·2πr, sector area = (θ/360)·πr².

Volume formulas: prism/cylinder V = Bh, pyramid/cone V = (1/3)Bh, sphere V = (4/3)πr³, sphere surface area SA = 4πr².

Inscribed angle theorem: an inscribed angle equals half its intercepted arc, and an angle inscribed in a semicircle is 90°.

Coordinate geometry of conics: circle (x−h)²+(y−k)² = r²; parabola y = a(x−h)²+k opens up when a > 0.

Mean = sum/n; median is the middle value of an ordered data set; mode is the most frequent value; the mean is pulled toward outliers while the median resists them.

Standard deviation measures spread about the mean: compute deviations, square them, average (dividing by n for a population or n−1 for a sample), then take the square root.

In a normal distribution about 68% of data lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

A z-score z = (x − μ)/σ states how many standard deviations a value lies from the mean.

Probability of an event P(A) = favorable outcomes / total outcomes; P(Aᶜ) = 1 − P(A); P(A or B) = P(A) + P(B) − P(A and B).

Independent events satisfy P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count ordered arrangements: nPr = n!/(n−r)!; combinations count unordered selections: nCr = n!/(r!(n−r)!).

Correlation r ranges from −1 to 1; r near ±1 indicates a strong linear association, but correlation does not establish causation.

Line of best fit predicts y from x; the residual is actual y minus predicted y, and a good fit has residuals scattered randomly about zero.