Praxis Core Math cheat sheet

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

Percent change = (new − old) ÷ old × 100; a 20% increase followed by a 20% decrease yields a net 4% decrease, not zero.

To convert a percent to a decimal, divide by 100; to convert a decimal to a percent, multiply by 100.

A ratio a:b compares parts; a proportion a/b = c/d is solved by cross-multiplying: ad = bc.

Absolute value |x| is the distance from x to 0 on the number line, so |−7| = 7 and |x| ≥ 0 always.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiply coefficients and add exponents when multiplying powers of ten.

Rules of exponents: x^m · x^n = x^(m+n), x^m ÷ x^n = x^(m−n), (x^m)^n = x^(mn), x^0 = 1 for x ≠ 0, x^(−n) = 1/x^n.

Unit rates convert by multiplying by fractions equal to 1, e.g., 60 mi/h × (5280 ft/1 mi) × (1 h/3600 s) = 88 ft/s.

Slope between two points = (y₂ − y₁)/(x₂ − x₁); parallel lines share the same slope, perpendicular slopes are negative reciprocals (m₁ · m₂ = −1).

Slope-intercept form y = mx + b gives slope m and y-intercept b; point-slope form is y − y₁ = m(x − x₁).

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) solves ax² + bx + c = 0; the discriminant b² − 4ac gives two, one, or no real roots when positive, zero, or negative.

FOIL or distribution expands products of binomials: (x + 3)(x − 5) = x² − 2x − 15.

When solving an inequality, multiplying or dividing both sides by a negative number reverses the inequality sign.

A function assigns exactly one output to each input; evaluate f(x) by substituting the value for x, and find f(g(x)) by substituting g(x) into f.

Linear functions have a constant rate of change; exponential functions of the form y = a·b^x change by a constant factor per unit.

Systems of two linear equations can be solved by substitution or elimination; the solution is the point where the graphs intersect.

Area formulas: rectangle A = lw, triangle A = ½bh, circle A = πr², trapezoid A = ½(b₁ + b₂)h.

Circumference of a circle C = 2πr = πd; the sum of the interior angles of any triangle is 180°.

Pythagorean theorem: in a right triangle a² + b² = c², where c is the hypotenuse; common triples include 3-4-5 and 5-12-13.

Volume formulas: rectangular prism V = lwh, cylinder V = πr²h, sphere V = (4/3)πr³, cone V = (1/3)πr²h.

Similar figures have equal corresponding angles and proportional corresponding sides; scale factor k multiplies lengths, k² multiplies areas, k³ multiplies volumes.

Sum of interior angles of an n-sided polygon = (n − 2) × 180°; each interior angle of a regular n-gon measures (n − 2)180°/n.

Complementary angles sum to 90°, supplementary angles sum to 180°, and vertical angles are equal.

Coordinate geometry: distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Mean = sum of values ÷ number of values; median is the middle value of an ordered list (average the two middle values if the count is even); mode is the most frequent value.

Range = maximum − minimum; standard deviation measures typical distance from the mean, so a larger spread means a larger standard deviation.

Probability of an event = favorable outcomes ÷ total equally likely outcomes, always between 0 and 1 inclusive.

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

Complement rule: P(not A) = 1 − P(A).

A random sample supports generalization to the population it was drawn from; a biased or self-selected sample does not.

Correlation describes the strength and direction of a linear association but does not establish causation.

On a normal distribution, about 68% of data falls within one standard deviation of the mean and about 95% within two.