Math definitions

144 math terms from Achievable exam prep courses, each defined in plain language with links into the course textbook.

• 30-60-90 triangleA 30-60-90 triangle is a right triangle with angles of 30°, 60°, and 90°. Its sides always follow the fixed ratio x : x√3 : 2x, so knowing one side length lets you find the other two without trigonometry.
• 45-45-90 triangleA 45-45-90 triangle is a right triangle whose two acute angles both measure 45°, making it an isosceles right triangle. Its sides are always in the ratio x : x : x√2, so the hypotenuse equals a leg times √2.
• Absolute valueAbsolute value is a number's distance from zero on the number line, written with vertical bars: |x|. Because distance is never negative, absolute value is always zero or positive — |−5| and |5| both equal 5.
• Accumulation functionsAn accumulation function is a function defined by a definite integral with a variable upper limit, such as F(x) = ∫ from a to x of f(t) dt. It measures the net area accumulated under a curve from a fixed starting point up to x.
• Area and perimeterPerimeter is the total distance around the outside of a two-dimensional shape, while area is the amount of space the shape covers. Perimeter is measured in linear units and area in square units.
• Area between curvesThe area between two curves is found by integrating the difference between the upper and lower functions over the interval where they bound a region. For curves y = f(x) and y = g(x) with f above g, the area is the integral of f(x) − g(x) from a to b.
• Area of a triangleThe area of a triangle is half the product of its base and height: A = ½bh. The height must be measured perpendicular to the chosen base, and any of the three sides can serve as the base.
• Arithmetic and geometric sequencesAn arithmetic sequence adds the same number to get from one term to the next, while a geometric sequence multiplies by the same number. The repeated amount is called the common difference in an arithmetic sequence and the common ratio in a geometric sequence.
• Average value of a functionThe average value of a function on an interval [a, b] is the height of the rectangle with the same area as the region under the curve. It is computed as f_avg = (1 / (b − a)) times the definite integral of f(x) from a to b.
• BijectionA bijection is a function that pairs every element of one set with exactly one element of another set, with nothing left unmatched on either side. Because the pairing is perfect, two finite sets linked by a bijection must have the same number of elements.
• Binomial distributionThe binomial distribution gives the probability of getting exactly k successes in n independent trials when each trial has the same probability of success. It is the standard model for counting successes in a fixed number of yes-or-no trials.
• Categorical and quantitative variablesA categorical variable places each individual into a group or label, while a quantitative variable records a numerical measurement you can meaningfully do arithmetic with. The distinction determines which graphs and summary statistics are appropriate for a data set.
• Central limit theoremThe central limit theorem states that the sampling distribution of the sample mean becomes approximately normal as the sample size grows, regardless of the population's shape. It is the foundation for most inference on means.
• Chi-square testA chi-square test is a statistical hypothesis test for categorical data that compares observed counts to expected counts. Common versions test goodness of fit, independence between two variables, and homogeneity across populations.
• Combining exponentsCombining exponents means simplifying expressions with powers using the exponent rules: add exponents when multiplying like bases, subtract when dividing, and multiply when raising a power to a power.
• Completing the squareCompleting the square is an algebraic technique that rewrites a quadratic expression x² + bx as a perfect square, (x + b/2)² − (b/2)². It is used to solve quadratic equations, find vertex form, and write equations of circles.
• ConcavityConcavity describes which way a curve bends: a function is concave up where its graph curves like a cup and concave down where it curves like a frown. It is determined by the sign of the second derivative.
• Confidence intervalA confidence interval is a range of values, built from sample data, that is likely to contain an unknown population parameter such as a mean or proportion. It takes the form point estimate ± margin of error at a stated confidence level, commonly 95%.
• Conic sectionsConic sections are the four curves formed when a plane slices through a double cone: the circle, ellipse, parabola, and hyperbola. Each has a standard equation that reveals its center, vertices, and shape.
• Continuity (calculus)Continuity in calculus means a function has no breaks, holes, or jumps. A function is continuous at a point when the limit exists there, the function is defined there, and the limit equals the function's value.
• CorrelationCorrelation is a statistical measure of how strongly two variables move together. It ranges from −1 (perfect negative relationship) through 0 (no linear relationship) to +1 (perfect positive relationship).
• Correlation coefficientA correlation coefficient, written r, measures the strength and direction of the linear relationship between two variables. It always falls between −1 and +1, where values near ±1 indicate a strong linear relationship and values near 0 indicate almost none.
• Corresponding anglesCorresponding angles are pairs of angles that sit in the same relative position where a transversal crosses two lines. When the two lines are parallel, corresponding angles are congruent.
• Critical points of a functionCritical points of a function are the points in its domain where the derivative equals zero or does not exist. They are the only candidates for local maximums and minimums, making them the starting point for optimization problems.
• Cyclic quadrilateralA cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle. Its defining property is that opposite angles are supplementary — each pair sums to 180°.
• Definite integralA definite integral computes the signed area between a function and the x-axis over an interval [a, b], written ∫ from a to b of f(x) dx. It evaluates to a number, found by the Fundamental Theorem of Calculus as F(b) − F(a).
• Diophantine equationsA Diophantine equation is a polynomial equation for which only integer solutions are sought, such as finding all whole numbers x and y satisfying 3x + 5y = 47. They are named after the ancient Greek mathematician Diophantus.
• DivisibilityDivisibility means one integer divides another with no remainder — 12 is divisible by 3 because 12 ÷ 3 = 4 exactly. Divisibility rules and prime factorization make these checks fast without long division.
• Divisibility rulesDivisibility rules are shortcuts for deciding whether one whole number divides evenly into another without doing long division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.
• DivisorA divisor is the number you divide by in a division problem. More broadly, a divisor of an integer is any number that divides it evenly with no remainder — for example, 1, 2, 3, 4, 6, and 12 are the divisors of 12.
• Domain and rangeThe domain of a function is the set of all valid input (x) values, and the range is the set of all possible output (y) values. Together they describe exactly where a function lives on the coordinate plane.
• Empirical rule (68-95-99.7)The empirical rule states that in a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It links standard deviations to percentages and percentiles.
• Equation of a circleThe equation of a circle in standard form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. Every point (x, y) on the circle sits exactly r units from the center.
• Equations with variables on both sidesAn equation with variables on both sides, such as 3x + 5 = x + 13, is solved by collecting all variable terms on one side and all constants on the other. Subtracting x and then 5 from both sides gives 2x = 8, so x = 4.
• Evaluating functionsEvaluating a function means substituting a specific input for the function's variable and simplifying to find the output. For f(x) = 2x + 3, evaluating f(4) means replacing x with 4 to get f(4) = 11.
• Expected valueExpected value is the probability-weighted average of all possible outcomes of a random variable, computed as E(X) = Σ x · P(x). It represents the long-run average result you would observe if the random process were repeated many times.
• Experimental vs. observational studiesAn experimental study assigns treatments to subjects — ideally at random — to measure their effects, while an observational study records data without intervening. Only well-designed experiments can support cause-and-effect conclusions.
• Exponent rulesThe exponent rules are the properties that let you simplify expressions containing powers — including multiplying like bases by adding exponents, dividing by subtracting them, and raising a power to a power by multiplying them. They apply to any nonzero base.
• Exponential functionAn exponential function is a function of the form f(x) = a · bˣ, where the variable appears in the exponent and b is a positive constant other than 1. It models quantities that grow or decay by a constant percentage.
• Factoring quadraticsFactoring quadratics is the process of rewriting a quadratic expression like x² + 5x + 6 as a product of two binomials, such as (x + 2)(x + 3). It is one of the fastest ways to solve quadratic equations and find the roots of a parabola.
• First derivative testThe first derivative test uses the sign of a function's derivative around a critical point to classify it: if f′ changes from positive to negative, the point is a local maximum; from negative to positive, a local minimum.
• Five number summaryA five number summary describes a data set using five values: the minimum, first quartile, median, third quartile, and maximum. Together they show the center, spread, and shape of the data, and they define the box plot.
• FractionsFractions are numbers that represent parts of a whole, written as one integer over another: the numerator counts the parts, and the denominator tells how many equal parts make the whole.
• Fundamental theorem of calculusThe fundamental theorem of calculus states that differentiation and integration are inverse operations. It lets you evaluate a definite integral by finding an antiderivative and taking the difference of its values at the endpoints.
• GCF and LCMThe GCF (greatest common factor) is the largest number that divides evenly into two or more numbers, while the LCM (least common multiple) is the smallest number that is a multiple of all of them.
• Geometric distributionThe geometric distribution models the number of independent trials needed to get the first success, when each trial has the same probability of success p. Its probability formula is P(X = k) = (1 − p)^(k−1) · p, and its mean is 1/p.
• Graphs of trigonometric functionsThe graphs of sine and cosine are smooth, repeating waves that oscillate between −1 and 1 with period 2π, while tangent produces repeating curves with vertical asymptotes every π units.
• Hypothesis testingHypothesis testing is a statistical procedure for deciding whether sample data provide enough evidence to reject a stated assumption about a population. It compares a null hypothesis against an alternative using a test statistic and a p-value.
• Imaginary and complex numbersImaginary numbers are built on i, defined so that i² = −1, and a complex number combines a real and an imaginary part in the form a + bi. Together they let every polynomial equation have a solution.
• Implicit differentiationImplicit differentiation is a technique for finding dy/dx when an equation relates x and y without solving for y. You differentiate both sides with respect to x, applying the chain rule to every y term, then solve algebraically for dy/dx.
• Indefinite integralAn indefinite integral is the family of all antiderivatives of a function, written ∫f(x) dx = F(x) + C. It reverses differentiation, and the constant C accounts for the fact that every vertical shift of F has the same derivative.
• Independent eventsTwo events are independent if the occurrence of one does not change the probability of the other. When events A and B are independent, the probability that both occur equals P(A) × P(B).
• InequalitiesAn inequality is a mathematical statement that compares two expressions using <, >, ≤, or ≥ rather than an equals sign. Its solution is usually a range of values instead of a single number.
• Inflection pointsAn inflection point is a point on a curve where the concavity changes — the graph switches from curving upward to curving downward, or the reverse. It occurs where the second derivative changes sign.
• IntegerAn integer is a number with no fractional or decimal part: the positive counting numbers, their negatives, and zero (…, −3, −2, −1, 0, 1, 2, 3, …). Fractions and decimals like 1/2 or 3.7 are not integers.
• Integration using long divisionIntegration using long division is a technique for integrating a rational function whose numerator degree is greater than or equal to its denominator degree. Dividing first rewrites the integrand as a polynomial plus a simple remainder term that can be integrated directly.
• Intercepts (x- and y-intercepts)An intercept is a point where a graph crosses an axis. The x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0).
• Interior anglesInterior angles are the angles inside a polygon at each of its vertices. Their sum depends only on the number of sides: for a polygon with n sides, the interior angles add up to (n − 2) × 180°.
• Inverse trig derivativesInverse trig derivatives are the derivative formulas for the inverse trigonometric functions, such as d/dx arcsin(x) = 1/√(1 − x²) and d/dx arctan(x) = 1/(1 + x²). Notably, none of them contain trigonometric functions — they are all algebraic expressions.
• Isolating the variableIsolating the variable means rearranging an equation with inverse operations until the variable stands alone on one side. It is the core technique for solving equations, as in turning 2x + 3 = 11 into x = 4.
• Law of large numbersThe law of large numbers says that as the number of independent trials grows, the observed average converges toward the true expected value. Flip a fair coin enough times and the proportion of heads approaches 0.5.
• Least squares regressionLeast squares regression is a method for fitting a straight line to paired data by minimizing the sum of the squared vertical distances between the observed points and the line. The resulting line is the standard tool for predicting one variable from another.
• Linear approximationLinear approximation is a calculus technique that estimates a function's value near a point by using its tangent line. The formula is L(x) = f(a) + f′(a)(x − a), where a is a nearby point where the function is easy to evaluate.
• Linear equationA linear equation is an equation whose variables appear only to the first power, with no products of variables, roots, or variables in denominators. Graphed in two variables, a linear equation always produces a straight line.
• Linear functionA linear function is a function whose graph is a straight line, written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its defining feature is a constant rate of change: equal changes in x always produce equal changes in y.
• Linear inequalitiesA linear inequality compares two linear expressions using <, >, ≤, or ≥ instead of an equals sign. Solving one works like solving a linear equation, except that multiplying or dividing both sides by a negative number reverses the inequality symbol.
• Linear regressionLinear regression is a statistical method that fits a straight line through paired data to describe how one variable changes with another. The fitted line, written ŷ = a + bx, is chosen to make the sum of the squared vertical distances between the data points and the line as small as possible.
• Logarithmic functionsA logarithmic function is the inverse of an exponential function: y = log_b(x) is the exponent you must raise the base b to in order to get x. It answers the question "b to what power equals x?"
• Matrix multiplicationMatrix multiplication combines two matrices by taking dot products of rows and columns. It is defined only when the number of columns in the first matrix equals the number of rows in the second.
• Mean, median, mode, and rangeMean, median, mode, and range are the four basic ways to summarize a data set: the mean is the average, the median is the middle value, the mode is the most frequent value, and the range is the difference between the largest and smallest values.
• Measures of spreadMeasures of spread describe how far the values in a data set are from one another. Range, interquartile range, variance, and standard deviation are the common ones, and they tell you how much variability sits behind an average.
• Modular arithmeticModular arithmetic is a system of arithmetic that works with remainders: two numbers are congruent mod n if they leave the same remainder when divided by n. It is written a ≡ b (mod n) and underlies much of number theory.
• Nonlinear functionA nonlinear function is any function whose graph is not a straight line — it cannot be written in the form y = mx + b. Its rate of change varies, producing curves such as parabolas, exponential growth curves, and hyperbolas.
• Normal distributionThe normal distribution is a symmetric, bell-shaped probability distribution defined by its mean and standard deviation. Many natural measurements follow it, and it underlies z-scores, the empirical rule, and much of inferential statistics.
• Odd integersOdd integers are integers that are not divisible by 2, such as −3, −1, 1, 3, and 5. Every odd integer can be written in the form 2n + 1, where n is an integer.
• Optimization (calculus)Optimization in calculus is the process of using derivatives to find the maximum or minimum value of a quantity, such as the largest area or lowest cost, often subject to a constraint.
• Order of operationsThe order of operations is the agreed-upon sequence for evaluating math expressions: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right) — remembered by the acronym PEMDAS.
• Ordinary differential equationsAn ordinary differential equation (ODE) is an equation relating a function of a single variable to its derivatives. Solving an ODE means finding the function or family of functions that satisfies the equation.
• Outliers (statistics)Outliers are data points that fall far outside the overall pattern of a dataset. A common rule flags values more than 1.5 times the interquartile range below the first quartile or above the third quartile.
• ParabolaA parabola is the U-shaped curve that graphs a quadratic function such as y = ax² + bx + c. It is symmetric about a vertical line through its vertex, the point where the curve turns around.
• Parallel and perpendicular linesParallel lines lie in the same plane and never intersect; perpendicular lines intersect at a 90° angle. In coordinate geometry, parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
• Percent changePercent change measures how much a quantity grew or shrank relative to its starting value: (new − old) ÷ old, expressed as a percent. A positive result is an increase, a negative result is a decrease.
• PercentileA percentile is a value at or below which a given percentage of the data in a distribution falls. Scoring in the 80th percentile, for example, means you performed as well as or better than 80% of the group.
• Permutations and combinationsPermutations and combinations are counting methods for selecting items from a group. Permutations count arrangements where order matters; combinations count selections where order does not matter.
• Place valuePlace value is the value a digit represents based on its position within a number. In the base-10 system, each place is worth ten times the place to its right, so the 5 in 352 represents 50, while the 5 in 3.52 represents 5 tenths.
• Position, velocity, and accelerationIn calculus, velocity is the derivative of position and acceleration is the derivative of velocity. Working backward, integrating acceleration gives velocity, and integrating velocity gives position.
• Positive slopeA positive slope describes a line that rises from left to right on a graph — as x increases, y increases. Numerically, it means the slope value m in y = mx + b is greater than zero.
• Power ruleThe power rule is the differentiation shortcut stating that the derivative of xⁿ is n·xⁿ⁻¹: bring the exponent down as a coefficient, then reduce the exponent by one. It works for any real number exponent.
• Prime factorizationPrime factorization is the process of writing a whole number as a product of prime numbers. Every integer greater than 1 has exactly one prime factorization — for example, 60 = 2² × 3 × 5.
• Prime numbersA prime number is a whole number greater than 1 whose only positive divisors are 1 and itself, such as 2, 3, 5, 7, and 11. Numbers greater than 1 that are not prime are called composite.
• ProbabilityProbability is a number between 0 and 1 that measures how likely an event is to occur. For equally likely outcomes, it equals favorable outcomes divided by total possible outcomes — 0 means impossible, 1 means certain.
• Probability distributionA probability distribution describes every value a random variable can take and how likely each value is. The probabilities across all possible outcomes always sum to 1, whether the variable is discrete or continuous.
• Product and quotient rulesThe product and quotient rules are differentiation rules for functions built from two other functions. The product rule states (fg)′ = f′g + fg′, and the quotient rule states (f/g)′ = (f′g − fg′) / g².
• Properties of fractionsThe properties of fractions are the basic rules governing how fractions behave: a zero numerator makes the fraction equal zero, a zero denominator is undefined, and multiplying or dividing both parts by the same number produces an equivalent fraction.
• Properties of logarithmsThe properties of logarithms are the rules that let you expand or condense logarithmic expressions: the product rule, the quotient rule, the power rule, and the change of base formula. They follow directly from the definition that log_b(x) = y means b^y = x.
• ProportionsA proportion is an equation stating that two ratios are equal, such as 3/4 = 6/8. Proportions are solved by cross-multiplying and are the standard tool for scaling recipes, maps, similar figures, and unit rates.
• Pythagorean theoremThe Pythagorean theorem states that in a right triangle, the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². It only applies to right triangles.
• Quadratic polynomialA quadratic polynomial is a polynomial of degree 2, written in standard form as ax² + bx + c with a ≠ 0. Its graph is a parabola, and it has at most two real roots.
• QuadrilateralA quadrilateral is a polygon with four sides, four vertices, and interior angles that always sum to 360°. Squares, rectangles, parallelograms, rhombuses, trapezoids, and kites are all quadrilaterals.
• Radical formRadical form is a way of writing roots using the radical symbol (√) instead of fractional exponents. For example, x^(1/2) in exponent form is √x in radical form.
• Radicals and rational exponentsRadicals and rational exponents are two notations for the same idea: a fractional power. The expression x^(m/n) equals the nth root of x raised to the mth power, so square roots, cube roots, and other radicals can be rewritten as exponents.
• RatioA ratio is a comparison of two quantities showing how many times one contains the other. It can be written as 3:2, 3/2, or "3 to 2," and it underlies proportions, rates, and scale problems throughout math.
• Rational expressionsA rational expression is a fraction whose numerator and denominator are polynomials, such as (x² − 9)/(x + 3). They follow the same rules as numeric fractions, with the added restriction that the denominator cannot equal zero.
• Regression analysisRegression analysis is a statistical method that estimates the relationship between a dependent variable and one or more independent variables, producing an equation used to explain patterns and forecast future values.
• Regular polygonA regular polygon is a polygon whose sides are all the same length and whose interior angles are all equal. Equilateral triangles and squares are the simplest examples.
• Related ratesRelated rates problems use calculus to find how fast one quantity changes when you know how fast a connected quantity changes. You write an equation relating the variables, differentiate both sides with respect to time, and solve for the unknown rate.
• Reverse FOILReverse FOIL is a method for factoring a quadratic trinomial back into two binomials. It undoes the FOIL multiplication pattern by finding two numbers that multiply to the constant term and add to the middle coefficient.
• RhombusA rhombus is a quadrilateral with four sides of equal length. Its opposite angles are equal, and its diagonals bisect each other at right angles.
• Riemann sumsA Riemann sum approximates the area under a curve by dividing the interval into subintervals and adding the areas of rectangles built on them. As the rectangles get narrower, the sum approaches the exact value of the definite integral.
• Right triangle trigonometryRight triangle trigonometry uses the ratios sine, cosine, and tangent to relate the acute angles of a right triangle to the lengths of its sides. Given one acute angle and one side, these ratios let you find every remaining side and angle.
• Sampling distributionA sampling distribution is the distribution of a statistic — such as the sample mean or sample proportion — across all possible samples of a given size from a population. It describes how much a statistic varies from sample to sample.
• ScatterplotA scatterplot is a graph that displays paired data as individual points on a coordinate plane, with one variable on each axis. It shows whether two quantitative variables are related and how strong that relationship is.
• Scientific notationScientific notation writes a number as a value between 1 and 10 multiplied by a power of 10, such as 3.2 × 10⁴ for 32,000. It makes very large and very small numbers compact and easy to compare or multiply.
• Second derivative testThe second derivative test classifies a critical point of a function by checking the sign of the second derivative there. If f″(c) > 0 the point is a local minimum, if f″(c) < 0 it is a local maximum, and if f″(c) = 0 the test is inconclusive.
• Sector of a circleA sector of a circle is the pie-slice region bounded by two radii and the arc between them. Its area is (θ/360) × πr², where θ is the central angle in degrees and r is the radius.
• Separation of variablesSeparation of variables is a method for solving differential equations by moving all terms with one variable to one side and all terms with the other variable to the opposite side, then integrating both sides. It works when dy/dx can be written as a function of x times a function of y.
• Similarity (geometry)In geometry, two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are in a constant ratio called the scale factor.
• Simple and compound interestSimple interest is calculated only on the original principal, using I = Prt. Compound interest is calculated on the principal plus all previously earned interest, using A = P(1 + r/n)^(nt), so a balance grows faster than under simple interest.
• Slope fieldsA slope field is a graph of short line segments drawn at grid points, each showing the slope a differential equation assigns to that point. It lets you visualize the family of solution curves without solving the equation.
• SOHCAHTOASOHCAHTOA is a mnemonic for the three basic trigonometric ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.
• Solving quadratic equationsSolving a quadratic equation means finding the values of x that satisfy ax² + bx + c = 0. The four standard methods are factoring, taking square roots, completing the square, and applying the quadratic formula.
• Special trigonometric limitsThe special trigonometric limits are two memorized calculus facts: as x approaches 0, sin(x)/x approaches 1 and (1 − cos(x))/x approaches 0. They resolve limits that direct substitution turns into 0/0.
• Standard deviationStandard deviation measures how spread out a data set is around its mean. A small standard deviation means values cluster tightly near the average; a large one means they are widely dispersed.
• Standard form of a linear equationThe standard form of a linear equation is Ax + By = C, where A, B, and C are constants (usually integers) and A and B are not both zero. It puts both variables on the same side of the equation.
• Stars and barsStars and bars is a counting technique for distributing identical objects into distinct groups. Representing the objects as stars and the dividers between groups as bars turns the problem into a straightforward combination count.
• Systems of equationsA system of equations is a set of two or more equations sharing the same variables, solved together to find values that satisfy every equation at once. The solution to a two-variable linear system is the point where the lines intersect.
• t-distributionThe t-distribution is a bell-shaped probability distribution used in place of the normal distribution when the population standard deviation is unknown and must be estimated from a sample. It is wider in the tails than the normal curve, and its shape depends on the degrees of freedom.
• Tangent line to a curveA tangent line to a curve is the straight line that touches the curve at a single point and has the same slope as the curve at that point. In calculus, that slope is the value of the derivative at the point.
• Time series analysisTime series analysis is the study of data points recorded in time order to identify patterns and forecast future values. It decomposes a series into components such as trend, seasonal variation, cyclical movement, and random noise.
• Translating functionsTranslating a function means shifting its graph horizontally or vertically without changing its shape. Adding k outside the function, f(x) + k, shifts the graph up k units, while replacing x with x − h, f(x − h), shifts it right h units.
• Triangle inequality theoremThe triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It determines which side lengths can and cannot form a triangle.
• Trigonometric functionsTrigonometric functions relate an angle to ratios of the sides of a right triangle. The three primary functions are sine, cosine, and tangent, along with their reciprocals cosecant, secant, and cotangent.
• Two-way tablesA two-way table organizes data by two categorical variables at once, with one variable defining the rows and the other the columns. Each cell shows how many observations share that row-column combination.
• Type I and type II errorsA type I error occurs when a hypothesis test rejects a null hypothesis that is actually true — a false positive. A type II error occurs when a test fails to reject a null hypothesis that is actually false — a false negative.
• Types of numbersThe types of numbers are nested classifications of the real numbers: natural numbers, whole numbers, integers, rational numbers, and irrational numbers, with the rationals and irrationals together making up the reals.
• Types of trianglesTriangles are classified two ways: by their sides as equilateral, isosceles, or scalene, and by their angles as acute, right, or obtuse. Every triangle fits one category from each list, so a triangle can be both isosceles and right.
• u-substitutionu-substitution is an integration technique that replaces part of an integrand with a new variable u to turn a complicated integral into a simpler one. It is the reverse of the chain rule for derivatives.
• Unit circleThe unit circle is a circle of radius 1 centered at the origin of the coordinate plane. For any angle θ measured from the positive x-axis, the point where its ray meets the circle has coordinates (cos θ, sin θ).
• Unit conversionsUnit conversions change a measurement from one unit to another — such as feet to meters or hours to seconds — by multiplying by conversion factors equal to 1. The systematic method is called dimensional analysis.
• VectorsA vector is a quantity with both magnitude and direction, such as velocity, force, or displacement. Vectors are usually written as components and combined using vector addition, subtraction, and scalar multiplication.
• Venn diagramA Venn diagram is a visual representation of sets using overlapping circles, where the overlap shows elements the sets share. It is used to organize set relationships, count overlapping groups, and reason through logic problems.
• Vertex formVertex form is a way of writing a quadratic equation as y = a(x − h)² + k, where the point (h, k) is the vertex of the parabola. It makes the parabola's highest or lowest point easy to read directly from the equation.
• Volume and surface areaVolume measures the amount of three-dimensional space a solid occupies, while surface area measures the total area of all its outer faces. Each common solid — prism, cylinder, cone, sphere — has its own formulas for both.
• Z-test and t-testThe z-test and t-test are hypothesis tests that compare a sample mean to a claimed value. The z-test applies when the population standard deviation is known or the sample is large; the t-test applies when it is unknown and estimated from a small sample.