AEPA Mathematics (NT304) cheat sheet

Divisibility: 3 if digit sum divisible by 3; 4 if last two digits form a multiple of 4; 9 if digit sum divisible by 9; 6 if divisible by both 2 and 3.

Place value: each position is 10 times the position to its right; in decimals, each position is 1/10 of the position to its left.

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

Absolute value |a| is distance from 0 on the number line, so |a - b| gives the distance between a and b.

Percent change = (new - old)/old x 100%; percent increase then decrease of the same rate does not return to the original value.

Rational numbers can be written as a/b with integers a, b and b not 0; their decimal expansions terminate or repeat.

Irrational numbers have non-terminating, non-repeating decimals; examples include sqrt(2), pi, and e.

Scientific notation: a x 10^n where 1 <= |a| < 10; multiplying adds exponents, dividing subtracts exponents.

Proportional reasoning: if quantities are proportional, the ratios a/b are equal, so cross products ad = bc for a/b = c/d.

Slope m = (y2 - y1)/(x2 - x1); parallel lines have equal slopes, perpendicular lines have slopes whose product is -1.

Point-slope form: y - y1 = m(x - x1); slope-intercept form: y = mx + b; standard form: Ax + By = C.

Quadratic formula: x = (-b +/- sqrt(b^2 - 4ac))/(2a); the discriminant b^2 - 4ac gives two, one, or zero real roots when positive, zero, or negative.

Vertex of y = ax^2 + bx + c occurs at x = -b/(2a); the parabola opens upward if a > 0 and downward if a < 0.

Exponent rules: a^m * a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn); a^(-n) = 1/a^n; a^0 = 1 for a not 0.

Logarithm definition: log_b(x) = y means b^y = x; log(ab) = log a + log b and log(a/b) = log a - log b.

Arithmetic sequence: a_n = a_1 + (n - 1)d; geometric sequence: a_n = a_1 * r^(n-1); sum of finite geometric series = a_1(1 - r^n)/(1 - r).

Function notation: f(g(x)) is composition, evaluated inside-out; the inverse f^-1 undoes f, so f(f^-1(x)) = x.

Systems of two linear equations have one solution (different slopes), none (same slope, different intercepts), or infinitely many (same line).

Area formulas: rectangle A = lw; triangle A = (1/2)bh; circle A = pi*r^2; trapezoid A = (1/2)(b1 + b2)h.

Volume formulas: prism/cylinder V = Bh; pyramid/cone V = (1/3)Bh; sphere V = (4/3)pi*r^3.

Surface area of a sphere is 4pir^2; circumference of a circle is 2pir or pi*d.

Pythagorean theorem: a^2 + b^2 = c^2 for right triangles, where c is the hypotenuse; converse also holds.

Sum of interior angles of an n-gon is (n - 2)*180 degrees; each exterior angle of a regular n-gon measures 360/n degrees.

Similar figures have equal corresponding angles and proportional sides; the ratio of areas is the square of the scale factor, and volumes the cube.

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Congruence criteria for triangles: SSS, SAS, ASA, AAS, and HL for right triangles; SSA does not guarantee congruence.

Unit conversion uses dimensional analysis: multiply by ratios equal to 1 so unwanted units cancel.

Mean = sum/n; median is the middle value of ordered data; mode is the most frequent value; range = max - min.

Standard deviation measures typical distance from the mean; adding a constant to every value shifts the mean but leaves standard deviation unchanged.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) - P(A and B).

Independent events: 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)!).

Expected value = sum of (each outcome x its probability); a fair game has expected value 0.

Counting principle: if a task has m ways and a second has n ways, together they have m*n ways.

In a normal distribution about 68% of data lies within one standard deviation of the mean, 95% within two, and 99.7% within three.

Correlation measures strength and direction of linear association but does not establish causation.

A graph's degree sum equals twice the number of edges; a path uses edges without repeating, a circuit starts and ends at the same vertex.

Euler's formula for connected planar graphs: V - E + F = 2, counting the outer region as a face.

A tree with n vertices has exactly n - 1 edges and contains no cycles.

Set operations: |A union B| = |A| + |B| - |A intersect B|; the complement of A contains elements not in A.

The pigeonhole principle: placing more than n objects into n boxes forces at least one box to hold two or more.

A number is prime if its only positive divisors are 1 and itself; every integer greater than 1 has a unique prime factorization.

Modular arithmetic: a is congruent to b mod n when n divides a - b; remainders repeat in cycles useful for pattern problems.

Recurrence relations define terms from earlier terms, such as the Fibonacci sequence F(n) = F(n-1) + F(n-2).

Venn diagrams represent intersections and unions; the number in exactly one set equals each set's total minus the overlap.

Estimation by rounding: round each factor to a convenient place value, multiply, then adjust for whether rounding was up or down.

A ratio compares two quantities; a rate is a ratio with different units, and a unit rate has denominator 1.

Fractions: a/b + c/d requires a common denominator bd; a/b divided by c/d equals a/b times d/c.

Integer operations: subtracting a negative adds; multiplying or dividing two negatives gives a positive result.

Greatest common factor and least common multiple relate by GCF(a,b) * LCM(a,b) = a * b.

Number line ordering: negative numbers with larger absolute value are smaller; between any two rationals lies another rational.

Prime factorization is found by repeated division by primes; use it to simplify radicals such as sqrt(72) = 6*sqrt(2).

Benchmark fractions (1/4, 1/3, 1/2, 2/3, 3/4) support quick comparison of fractions and decimals.

Linear functions have constant rate of change; exponential functions have constant ratio and grow by a fixed percent each period.

Solving linear inequalities: multiplying or dividing by a negative reverses the inequality symbol.

Factoring patterns: difference of squares a^2 - b^2 = (a - b)(a + b); perfect square trinomial a^2 +/- 2ab + b^2 = (a +/- b)^2.

The zero product property: if ab = 0, then a = 0 or b = 0; use it after factoring to solve quadratics.

Direct variation y = kx; inverse variation y = k/x, where the product xy stays constant.

Domain restrictions arise from division by zero and from even roots of negative numbers.

Arithmetic series sum: S = n(a_1 + a_n)/2 for n terms with first term a_1 and last term a_n.

Translations of graphs: f(x - h) shifts right h units, f(x) + k shifts up k units, and -f(x) reflects across the x-axis.

Complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees; vertical angles are equal.

Parallel lines cut by a transversal create equal alternate interior angles and equal corresponding angles.

Circle relationships: central angle measure equals its intercepted arc; inscribed angle is half its intercepted arc.

Sector area = (central angle/360) * pir^2; arc length = (central angle/360) * 2pi*r.

Transformations: reflections, rotations, and translations preserve size and shape; dilations preserve shape but change size.

Coordinate rules: reflection over x-axis maps (x, y) to (x, -y); over y-axis to (-x, y); 180-degree rotation to (-x, -y).

Volume of a composite solid is the sum or difference of the volumes of its component solids.

Slope of a line equals tangent of the angle it makes with the positive x-axis.

Sampling methods: random sampling reduces bias; a biased sample systematically favors some outcomes over others.

Outliers can strongly affect the mean but have little effect on the median, so the median resists skew.

A box plot shows minimum, first quartile, median, third quartile, and maximum; the interquartile range is Q3 - Q1.

A histogram displays frequency of quantitative data in intervals; a bar graph compares categorical data.

Mutually exclusive events cannot occur together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).

Complement rule: P(not A) = 1 - P(A), often easier than computing P(A) directly.

A combination lock uses permutations because order matters; choosing a committee uses combinations.

Matrix addition requires equal dimensions; matrix multiplication is defined when the first matrix's columns equal the second's rows.

A relation is a function if each input has exactly one output; the vertical line test confirms this on a graph.

Rational expressions simplify by factoring numerator and denominator and canceling common factors, noting excluded values.

Radical equations: isolate the radical, square both sides, then check for extraneous solutions.

Composite function domain excludes inputs not in the inner function's domain or not in the outer function's domain after substitution.

Exponential growth/decay: y = a(1 + r)^t for growth and y = a(1 - r)^t for decay, where r is the rate per period.

The axis of symmetry of a parabola is x = -b/(2a); its y-intercept is c in y = ax^2 + bx + c.

Absolute value equations |x| = k have two solutions x = k and x = -k when k > 0, one solution when k = 0, none when k < 0.

Sequences can be defined explicitly by a formula in n or recursively using previous terms.

Rounding rules: look at the digit to the right of the target place; 5 or more rounds up, less than 5 rounds down.

Converting fractions to decimals: divide numerator by denominator; converting decimals to percents: multiply by 100.

Comparing fractions: cross-multiply (a/b vs c/d compares ad and bc) to avoid converting to decimals.

Exponent and root relationship: the nth root of a^n is |a| when n is even and a when n is odd.

Ordering real numbers: convert all to decimals or common denominators before comparing.

Properties: commutative (a + b = b + a), associative (grouping), distributive (a(b + c) = ab + ac), identity, and inverse.

A percent is a ratio out of 100; to find a percent of a number, convert the percent to a decimal and multiply.

Multiplying by powers of 10 shifts the decimal point right for positive exponents and left for negative exponents.

Perimeter is the distance around a figure; for a circle it is circumference, C = 2pir.

The sum of the exterior angles of any convex polygon is 360 degrees.

Special right triangles: 45-45-90 has sides in ratio 1:1:sqrt(2); 30-60-90 has ratio 1:sqrt(3):2.

Trigonometric ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent in a right triangle.

Law of cosines: c^2 = a^2 + b^2 - 2ab*cos(C) for any triangle.

Angle of elevation is measured upward from the horizontal; angle of depression is measured downward.

Density = mass/volume; it is a derived measure combining two different quantities.

Similar solids: if the scale factor is k, surface area ratio is k^2 and volume ratio is k^3.

Law of large numbers: as trials increase, experimental probability approaches theoretical probability.

A probability distribution lists all outcomes with their probabilities; the probabilities must sum to 1.

Two-way tables organize categorical data; joint and marginal frequencies support conditional probability calculations.

Tree diagrams multiply probabilities along branches and add across branches for combined events.

The median of an even-sized data set is the average of the two middle values.

Weighted mean = sum of (value x weight) / sum of weights; final grades often use weighted means.

Scatter plots reveal trends; a line of best fit models linear association and supports prediction.

Graph coloring assigns colors so adjacent vertices differ; the chromatic number is the minimum colors needed.

Linear equation from two points: find slope, then substitute one point into y = mx + b to find b.

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

A system solved by elimination: add or subtract equations to cancel a variable after aligning coefficients.

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

Completing the square converts y = ax^2 + bx + c into vertex form y = a(x - h)^2 + k.

The vertex form reveals the maximum or minimum value k, useful for optimization word problems.

Inverse functions swap x and y and solve for y; their graphs reflect across the line y = x.

Piecewise functions use different rules on different intervals; evaluate by choosing the rule matching the input.

The area of a regular polygon is (1/2) * apothem * perimeter.

The volume of a frustum is found by subtracting the smaller cone or pyramid from the larger one.

Congruent figures have equal corresponding angles and equal corresponding side lengths.

The midpoint of a segment is equidistant from its endpoints; the perpendicular bisector passes through it at 90 degrees.

A line tangent to a circle is perpendicular to the radius at the point of tangency.

Inscribed angles subtending the same arc are equal; an angle inscribed in a semicircle is 90 degrees.

Dilation centered at the origin by factor k maps (x, y) to (kx, ky).

Surface area of a cylinder is 2pir^2 + 2pir*h (two circular bases plus the lateral rectangle).

Addition rule for overlapping events: subtract the intersection to avoid double counting.

Multiplication rule for dependent events: P(A and B) = P(A) * P(B|A).

The number of subsets of an n-element set is 2^n.

The number of ways to arrange n distinct objects is n!; with repeated items, divide by the factorial of each repeat count.

A spanning tree connects all vertices of a graph with no cycles; a minimum spanning tree minimizes total edge weight.

Euler circuits exist when every vertex has even degree; Euler paths exist with exactly zero or two odd-degree vertices.

The mean of a data set shifts by the same constant added to every value; multiplying every value by k multiplies the mean by k.

Statistical significance relates to probability of observed results under a null hypothesis, not to the size of the effect.

A number is divisible by 5 if it ends in 0 or 5; by 2 if it ends in an even digit; by 10 if it ends in 0.

The additive inverse of a is -a; the multiplicative inverse (reciprocal) of a nonzero a is 1/a.

Absolute value equations and inequalities: |x| < k means -k < x < k; |x| > k means x < -k or x > k.

Converting repeating decimals to fractions: let x equal the decimal, multiply by a power of 10 to shift the repeat, then subtract.

Square roots of perfect squares are integers; estimate others by bounding between consecutive perfect squares.

The density of rationals: between any two distinct real numbers there is a rational number and an irrational number.

Proportional scaling: doubling all dimensions of a 3D figure multiplies volume by 8, not 2.

Integers closed under addition, subtraction, and multiplication, but not division (e.g., 1 divided by 2 is not an integer).

A function's rate of change over an interval is (f(b) - f(a))/(b - a), the slope of the secant line.

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

The domain of a rational function excludes values making the denominator zero; those values may be vertical asymptotes.

Exponential equations can be solved by expressing both sides with the same base, then setting exponents equal.

Logarithmic equations: convert to exponential form to solve; check that arguments remain positive.

The sum of the roots of ax^2 + bx + c = 0 is -b/a and the product is c/a.

Systems of linear inequalities are solved by graphing and finding the overlapping shaded region.

Linear programming maximizes or minimizes an objective function at the vertices of the feasible region.

The distance from a point to a line is the perpendicular segment length from the point to the line.

The sum of the lengths of any two sides of a triangle exceeds the length of the third side (triangle inequality).

The longest side of a triangle is opposite the largest angle; the shortest side is opposite the smallest angle.

The altitude to the hypotenuse of a right triangle creates two triangles similar to the original and to each other.

The geometric mean relates altitude and segments: altitude squared equals the product of the two hypotenuse segments.

Congruent chords in a circle are equidistant from the center; a radius perpendicular to a chord bisects it.

The area of a triangle can be computed with Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)) where s is the semiperimeter.

Rotational symmetry of a regular n-gon has order n; reflectional symmetry has n lines of symmetry.

Variance is the average of squared deviations from the mean; standard deviation is its square root.

Z-score = (value - mean)/standard deviation, expressing how many standard deviations a value lies from the mean.

A census measures every member of a population; a sample measures a subset.

Stratified sampling divides the population into groups and samples within each; cluster sampling selects whole groups.

A permutation with repetition of n objects where one type repeats r times divides n! by r! (and similarly for other repeats).

The expected number of successes in n independent trials with probability p is n*p.

A graph is bipartite if its vertices can be split into two sets with edges only between sets; it contains no odd cycles.

The fundamental counting principle extends to multiple stages by multiplying the number of choices at each stage.

Order of magnitude: powers of 10 estimate scale; 10^3 = 1000, 10^6 = 1 million, 10^-3 = 0.001.

Multiplying decimals: multiply as whole numbers, then place the decimal so the total decimal places equal the sum of the factors' decimal places.

Dividing by a fraction equals multiplying by its reciprocal; dividing by a number less than 1 yields a larger result.

Percent of increase/decrease problems: part = percent x whole, rearranged to solve for any unknown.

A composite number has a factor other than 1 and itself; 1 is neither prime nor composite.

The least common multiple of denominators gives the least common denominator for adding fractions.

Scientific notation comparison: compare the exponent first, then the coefficient, to order very large or small numbers.

Estimation reasonableness: check whether an answer's magnitude matches the problem's context before accepting it.

Arithmetic sequences have a constant difference d; geometric sequences have a constant ratio r.

The nth term of an arithmetic sequence: a_n = a_1 + (n - 1)d; the sum of the first n terms is n/2 (a_1 + a_n).

An infinite geometric series converges to a_1/(1 - r) only when |r| < 1.

Recursive formulas specify the first term and a rule relating each term to the previous one.

Modeling with functions: identify whether change is additive (linear) or multiplicative (exponential) from context.

The inverse of an exponential function is logarithmic; their graphs reflect across y = x.

A quadratic model fits data with a constant second difference; a linear model fits data with a constant first difference.

Function transformations: a*f(x) stretches vertically by factor a; f(bx) compresses horizontally by factor 1/b.

The circumference-to-diameter ratio is pi, approximately 3.14159, for every circle.

Radians and degrees convert by multiplying by pi/180 (degrees to radians) or 180/pi (radians to degrees).

Arc length in radians equals r times the central angle in radians.

The area of a sector in radians equals (1/2) r^2 times the central angle in radians.

The volume of a sphere is (4/3)pir^3; its surface area is 4pi*r^2.

The lateral surface area of a cone is pirl, where l is the slant height; total surface area adds the base pi*r^2.

Surface area of a rectangular prism is 2(lw + lh + wh).

The diagonal of a rectangular box is sqrt(l^2 + w^2 + h^2), extending the Pythagorean theorem to 3D.

The mode can be absent (no repeats) or multiple (bimodal); the mean and median always exist for numeric data.

Skewed right data: mean is typically greater than median; skewed left data: mean is typically less than median.

The range is sensitive to outliers; the interquartile range is more robust.

A probability of 0 means impossible and 1 means certain; all probabilities lie between 0 and 1.

The sample space is the set of all possible outcomes of an experiment.

With replacement, probabilities stay constant across draws; without replacement, they change after each draw.

The binomial probability of exactly k successes in n trials is C(n,k) * p^k * (1-p)^(n-k).

A graph's adjacency matrix has a 1 in row i, column j when an edge connects vertex i to vertex j.

The distributive property rewrites a(b + c) as ab + ac and underlies mental math strategies such as 6 x 47 = 6 x 40 + 6 x 7.

Compatible numbers make estimation easier: adjust dividend and divisor to numbers that divide evenly.

Front-end estimation uses the leading digits and adjusts; it is fast but less precise than rounding.

A fraction is in simplest form when the numerator and denominator share no common factor greater than 1.

Improper fractions convert to mixed numbers by dividing numerator by denominator and expressing the remainder as a fraction.

Multiplying a negative by a positive gives a negative; multiplying two negatives gives a positive.

The square of any real number is nonnegative; the square root of a negative number is not real.

A number written in expanded form shows each digit multiplied by its place value, e.g., 3 x 100 + 4 x 10 + 5.

The slope-intercept form y = mx + b identifies m as the rate of change and b as the initial value in context.

The x-intercept is found by setting y = 0; the y-intercept by setting x = 0.

A linear equation in one variable has one solution, no solution (contradiction), or infinitely many (identity).

Solving absolute value inequalities requires splitting into two cases and combining with and/or.

Polynomial multiplication uses the distributive property repeatedly; FOIL applies to binomials.

The degree of a polynomial determines its maximum number of real roots and turning points (degree minus one).

Factoring by grouping: group terms, factor each group, then factor out the common binomial.

A system of equations can be solved graphically by finding the intersection point(s) of the graphs.

The area of an irregular figure can be estimated by decomposing it into rectangles, triangles, and circles.

The perimeter of a composite figure counts only the outer boundary, not interior dividing segments.

The surface area of a composite solid excludes hidden faces where solids join.

Scale drawings: the scale factor converts drawing lengths to actual lengths; area scales by the square of the scale factor.

The volume of a cylinder is pir^2h; the volume of a cone is one-third of the cylinder with the same base and height.

The volume of a pyramid is one-third the volume of the prism with the same base and height.

The sum of the interior angles of a triangle is 180 degrees; of a quadrilateral, 360 degrees.

Congruent triangles can be used to prove segments or angles equal in geometric proofs.

The mean absolute deviation is the average distance of data values from the mean.

A residual is the difference between an observed value and the value predicted by a model (observed minus predicted).

A strong correlation has an r near 1 or -1; a weak correlation has r near 0.

The coefficient of determination r^2 gives the proportion of variance in the response explained by the model.

Simulation estimates probabilities by repeatedly generating random outcomes and recording results.

Odds in favor of an event = favorable outcomes : unfavorable outcomes; probability = favorable / total.

The complement of an event in a sample space includes all outcomes not in the event.

The number of diagonals in a convex n-gon is n(n - 3)/2.

A percent greater than 100 represents more than the whole; a percent less than 1 represents less than one hundredth.

Simple interest: I = Prt; compound interest: A = P(1 + r/n)^(nt).

Tax and tip problems: multiply the base amount by the combined percent rate expressed as a decimal.

Discount problems: final price = original price times (1 - discount rate).

A ratio table organizes equivalent ratios and supports scaling up or down.

The constant of proportionality k in y = kx is the unit rate y/x.

Fraction division models: dividing by a fraction asks how many of that fraction fit into the whole.

Checking reasonableness: a product of two numbers each between 0 and 1 is smaller than either factor.

The vertex form of a quadratic is y = a(x - h)^2 + k with vertex (h, k).

The quadratic formula gives complex roots when the discriminant is negative; roots appear as conjugate pairs.

A rational function's horizontal asymptote depends on comparing degrees of numerator and denominator.

Exponential decay models use a base between 0 and 1; growth models use a base greater than 1.

Half-life problems use A = A_0 (1/2)^(t/h), where h is the half-life.

Compound interest is exponential growth; the rule of 72 estimates doubling time as 72 divided by the percent rate.

The composition f(g(x)) is generally not equal to g(f(x)); composition is not commutative.

A relation's inverse is a function only if the original passes the horizontal line test.

Two lines are perpendicular if their slopes are negative reciprocals (product -1).

The equation of a circle centered at (h, k) with radius r is (x - h)^2 + (y - k)^2 = r^2.

The distance between parallel lines y = mx + b1 and y = mx + b2 is |b1 - b2|/sqrt(m^2 + 1).

The angle between two intersecting lines can be found from the tangent of the angle between their slopes.

A regular polygon's apothem is the perpendicular distance from center to a side; it equals r*cos(pi/n) for circumradius r.

The area of a circle sector is a fraction of the whole circle's area equal to the central angle over 360 degrees.

Similar triangles have proportional corresponding sides; the ratio of perimeters equals the scale factor.

The altitude of an equilateral triangle with side s is s*sqrt(3)/2.

The median is found by ordering data and selecting the middle value; for an even count, average the two middle values.

The five-number summary consists of minimum, Q1, median, Q3, and maximum.

A normal curve is symmetric about the mean; the mean, median, and mode coincide.

Independent events have P(A and B) = P(A) * P(B); dependence means one event changes the other's probability.

The number of ways to choose r items from n when order does not matter is C(n, r) = n!/(r!(n - r)!).

The number of ways to arrange n items in a circle is (n - 1)!.

A graph with n vertices and no edges has n connected components; each edge can reduce the component count by at most one.

The handshake lemma: the sum of degrees in a graph is even, so the number of odd-degree vertices is even.

The absolute value of a product equals the product of absolute values: |ab| = |a||b|.

The square root of a product equals the product of square roots: sqrt(ab) = sqrt(a)*sqrt(b) for nonnegative a and b.

Adding fractions with unlike denominators: find the least common denominator, rewrite each fraction, then add numerators.

A repeating decimal such as 0.333... equals 1/3; 0.999... equals 1 exactly.

Multiplying by a number between 0 and 1 decreases a positive number; multiplying by a number greater than 1 increases it.

Place value rounding for decimals: the target place determines which digit to inspect; trailing zeros may be dropped after the decimal.

The greatest common factor of two numbers is the largest number dividing both; it simplifies fractions.

A rational number can be expressed as a terminating or repeating decimal; irrational numbers cannot.

The slope of a line perpendicular to a given line is the negative reciprocal of the given slope.

Parallel lines have the same slope and different y-intercepts; coincident lines share both slope and intercept.

To find where a line intersects a parabola, substitute the linear expression into the quadratic and solve.

The number of solutions to a linear-quadratic system equals the number of intersection points.

A function is one-to-one if each output comes from exactly one input; only one-to-one functions have inverses that are functions.

The average rate of change of a quadratic over an interval is the slope of the secant line connecting the endpoints.

The maximum or minimum of a quadratic occurs at the vertex, found by completing the square or using -b/(2a).

Exponential functions never equal zero; their graphs have a horizontal asymptote at y = 0 (shifted by any vertical translation).

The area of a parallelogram is base times height, where height is perpendicular to the base.

The area of a rhombus is (1/2) * d1 * d2, the product of its diagonals divided by two.

The area of a kite is (1/2) * d1 * d2, the product of its diagonals divided by two.

The sum of the interior angles of a polygon with n sides is (n - 2) * 180 degrees.

A regular polygon has equal sides and equal angles; its central angle is 360/n degrees.

The apothem of a regular polygon connects the center to the midpoint of a side and is perpendicular to that side.

The volume of a composite solid formed by removing a shape equals the larger volume minus the removed volume.

The surface area of a sphere is 4pir^2; its volume is (4/3)pi*r^3.

The mean of a data set is affected by every value; adding a new value changes the mean unless it equals the old mean.

The median is the 50th percentile; Q1 is the 25th percentile and Q3 is the 75th percentile.

An outlier is often defined as a value more than 1.5 times the interquartile range below Q1 or above Q3.

The probability of the union of mutually exclusive events is the sum of their probabilities.

The probability of the complement of an event is 1 minus the probability of the event.

The number of permutations of n objects taken r at a time is n!/(n - r)!.

A tree diagram enumerates outcomes; multiply along branches and add across branches.

In a graph, a vertex of degree 0 is isolated; a vertex of odd degree affects the existence of Euler paths.

The order of operations ensures a single value for an expression; parentheses override the default order.

Exponents apply only to the base immediately preceding them unless parentheses group a larger expression.

The absolute value of a number is never negative; |x| = 0 only when x = 0.

A negative exponent indicates a reciprocal: a^(-n) = 1/a^n, not a negative result.

The product of two numbers with the same sign is positive; with different signs, negative.

Scientific notation expresses numbers as a coefficient between 1 and 10 times a power of 10.

The least common multiple of two numbers is the smallest number divisible by both.

Estimation by rounding: round each number to the nearest convenient place, compute, then adjust for the rounding direction.

The slope formula m = (y2 - y1)/(x2 - x1) gives the rate of change between two points.

A linear function's graph is a straight line; its equation can be written in slope-intercept or standard form.

The point-slope form y - y1 = m(x - x1) is useful when a point and slope are known.

To find the equation of a line parallel to a given line, use the same slope and the given point.

To find the equation of a line perpendicular to a given line, use the negative reciprocal slope.

A quadratic equation can be solved by factoring, completing the square, or the quadratic formula.

The discriminant b^2 - 4ac determines the number and type of roots of a quadratic.

An exponential function has the form y = a*b^x where b > 0 and b is not 1.

The perimeter of a rectangle is 2(l + w); the area is lw.

The area of a triangle is (1/2)bh; the height must be perpendicular to the chosen base.

The circumference of a circle is pi times the diameter, or 2*pi times the radius.

The area of a circle is pi times the square of the radius.

The volume of a rectangular prism is length times width times height.

The volume of a cylinder is the area of its circular base times its height.

The surface area of a cube is 6 times the square of its edge length.

The Pythagorean theorem applies only to right triangles; the hypotenuse is the side opposite the right angle.

The mean is the balance point of a data set; the sum of deviations from the mean is zero.

The median divides data into two equal halves; it is resistant to extreme values.

The mode is the value that appears most often; a data set can have no mode or several modes.

The range is the difference between the maximum and minimum values.

The interquartile range is the difference between the third and first quartiles.

Probability is a number between 0 and 1 inclusive that measures the likelihood of an event.

The sample space is the set of all possible outcomes; an event is a subset of the sample space.

The counting principle multiplies the number of choices at each stage to find total outcomes.

A prime number has exactly two positive divisors: 1 and itself.

The fundamental theorem of arithmetic states every integer greater than 1 has a unique prime factorization.

The greatest common factor of two numbers is the largest number that divides both.

The least common multiple of two numbers is the smallest number divisible by both.

A fraction is simplified by dividing numerator and denominator by their greatest common factor.

A percent is a ratio with denominator 100; 25% equals 1/4 and 0.25.

Multiplying by a power of 10 shifts the decimal point one place per power.

The absolute value of a number is its distance from zero on the number line.

A function assigns exactly one output to each input in its domain.

The domain is the set of all possible inputs; the range is the set of all possible outputs.

The vertical line test determines whether a graph represents a function.

The horizontal line test determines whether a function is one-to-one and has an inverse function.

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

An exponential function has a constant ratio between successive outputs.

The vertex of a parabola is its maximum or minimum point.

The axis of symmetry of a parabola passes through its vertex and is vertical.

Complementary angles add to 90 degrees; supplementary angles add to 180 degrees.

Vertical angles are opposite each other when two lines intersect and are equal.

The sum of the angles in a triangle is 180 degrees.

The sum of the angles in a quadrilateral is 360 degrees.

Similar figures have equal corresponding angles and proportional corresponding sides.

Congruent figures have equal corresponding angles and equal corresponding sides.

The area of a circle is pi times the square of its radius.

The volume of a sphere is four-thirds pi times the cube of its radius.

The mean is the sum of values divided by the number of values.

The median is the middle value when data are ordered from least to greatest.

The mode is the most frequent value in a data set.

The range is the difference between the largest and smallest values.

The probability of an event is the number of favorable outcomes divided by the total number of equally likely outcomes.

The complement of an event has probability 1 minus the probability of the event.

The number of permutations of n objects is n factorial.

The number of combinations of n objects taken r at a time is n factorial divided by r factorial times (n minus r) factorial.

The order of operations is parentheses, exponents, multiplication and division, addition and subtraction.

A negative times a negative is a positive.

A negative times a positive is a negative.

The absolute value of a number is its distance from zero.

A fraction is in simplest form when the numerator and denominator have no common factor greater than 1.

A percent can be converted to a decimal by dividing by 100.

A decimal can be converted to a percent by multiplying by 100.

Scientific notation writes a number as a coefficient between 1 and 10 times a power of 10.

The slope of a line is the change in y divided by the change in x.

Parallel lines have equal slopes.

Perpendicular lines have slopes whose product is negative one.

The slope-intercept form of a line is y equals mx plus b.

The point-slope form of a line is y minus y1 equals m times x minus x1.

A quadratic equation can be solved by factoring, completing the square, or the quadratic formula.

The discriminant determines the number of real roots of a quadratic equation.

An exponential function has the form y equals a times b to the x power.

The perimeter of a polygon is the sum of its side lengths.

The area of a rectangle is length times width.

The area of a triangle is one-half base times height.

The area of a circle is pi times the square of the radius.

The circumference of a circle is pi times the diameter.

The volume of a rectangular prism is length times width times height.

The volume of a cylinder is pi times the square of the radius times height.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

The mean is the average of a data set.

The median is the middle value of an ordered data set.

The mode is the most frequent value in a data set.

The range is the difference between the maximum and minimum values.

The interquartile range is the difference between the third and first quartiles.

The probability of an event is a number between 0 and 1.

The probability of the complement of an event is 1 minus the probability of the event.

The number of combinations of n objects taken r at a time is n factorial divided by r factorial times (n minus r) factorial.