ACT Math cheat sheet

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

Exponent rules: x^a · x^b = x^(a+b); (x^a)^b = x^(ab); x^(−a) = 1/x^a; x^0 = 1 for x ≠ 0.

Fractional exponents: x^(m/n) = the n-th root of x^m, so x^(1/2) = √x and x^(2/3) = cube root of x².

Radical rule: √(ab) = √a · √b, and a radical is simplified only when no perfect-square factor remains under it.

Percent change = (new − old)/old × 100%; percent increase and percent decrease use the original value as the denominator.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply coefficients and add exponents, divide coefficients and subtract exponents.

Imaginary unit: i = √(−1), i² = −1, and powers of i cycle through i, −1, −i, 1 with period 4.

Absolute value: |x| = a (a ≥ 0) gives x = a or x = −a; |x| < a means −a < x < a; |x| > a splits into two separate rays.

Proportions solve by cross-multiplication: a/b = c/d implies ad = bc.

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

Line forms: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); standard Ax + By = C with slope −A/B.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so m₁·m₂ = −1.

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

Discriminant b² − 4ac: positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Factoring patterns: a² − b² = (a − b)(a + b); x² + (p + q)x + pq = (x + p)(x + q).

Parabola y = ax² + bx + c: vertex x-value is −b/(2a); opens upward if a > 0, downward if a < 0.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality symbol.

Variation: y = kx is direct variation (constant ratio), y = k/x is inverse variation (constant product).

Function notation: f(3) means substitute 3 for every x in the rule; f(x) = 0 locates the zeros, which are the x-intercepts.

Composition: (f ∘ g)(x) = f(g(x)) applies g first, then f; composition is generally not commutative.

Inverse function: swap x and y, then solve for y; the graph of f⁻¹ is f reflected across the line y = x.

Transformations: f(x − h) shifts right h, f(x) + k shifts up k, a·f(x) stretches vertically by a, −f(x) reflects over the x-axis.

Linear f(x) = mx + b changes by a constant amount per unit; exponential f(x) = a·b^x changes by a constant ratio, growing when b > 1.

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

Even function: f(−x) = f(x), symmetric about the y-axis. Odd function: f(−x) = −f(x), symmetric about the origin.

Domain restrictions come from division by zero and even roots of negatives; the vertical line test identifies whether a relation is a function.

Angle facts: triangle interior angles sum to 180°, angles on a straight line sum to 180°, vertical angles are equal, complementary angles sum to 90°.

Similar figures have equal corresponding angles and proportional corresponding sides; congruent figures have equal sides and angles.

Pythagorean theorem: a² + b² = c² for right triangles with legs a, b and hypotenuse c.

Special right triangles: 45-45-90 sides are x, x, x√2; 30-60-90 sides are x, x√3, 2x.

Circle: area = πr², circumference = 2πr; arc length = (θ/360)(2πr) and sector area = (θ/360)(πr²).

Coordinate circle: (x − h)² + (y − k)² = r² has center (h, k) and radius r.

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

Volume: prism/cylinder V = Bh; cone/pyramid V = (1/3)Bh; sphere V = (4/3)πr³.

Right-triangle ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

Mean = sum of values ÷ number of values; median = middle value of an ordered data set; mode = most frequent value.

Standard deviation measures typical distance from the mean; a larger standard deviation means a wider spread of data.

Probability of an event = favorable outcomes ÷ total outcomes; all probabilities lie between 0 and 1.

Independent events: P(A and B) = P(A) · P(B). Mutually exclusive events: P(A or B) = P(A) + P(B).

General addition rule: P(A or B) = P(A) + P(B) − P(A and B); complement rule: P(not A) = 1 − P(A).

Counting: permutations count ordered arrangements (nPr); combinations count unordered groups (nCr = nPr / r!).

Expected value = Σ(value × probability), the long-run average result of a repeated random event.

Box plots mark Q1, median, and Q3; the interquartile range IQR = Q3 − Q1 covers the middle 50% of the data.

A line of best fit predicts from a scatterplot trend, but a strong correlation does not establish causation.