AP Precalculus Vocabulary

Absolute maximum

The highest point over the entire domain of a function; the greatest output value a function attains.

Similar definitions: global maximum

Example: "The function f(x) = −x² + 4 has an          of 4 at x = 0."

Absolute minimum

The lowest point over the entire domain of a function; the smallest output value a function attains.

Similar definitions: global minimum

Example: "The function f(x) = x² + 1 has an          of 1 at x = 0."

Absolute value

The distance of a number from zero on a number line, always expressed as a non-negative value.

Example: "The          of −7 is 7 because it is 7 units from zero on the number line."

Additive transformation

A transformation that adds a constant to the input or output of a function, resulting in a horizontal or vertical shift of the graph.

Example: "Replacing f(x) with f(x) + 3 is an          that shifts the graph up 3 units."

Amplitude

The height from the center line (midline) to the peak or trough of a periodic function; half the distance between the maximum and minimum values.

Example: "The function y = 3 sin(x) has an          of 3."

Angular speed

The rate at which an angle changes over time, typically measured in radians per unit time; ω = Δθ/Δt.

Example: "A wheel that completes one full rotation every 2 seconds has an          of π radians per second."

Arc length

The distance along a circular arc, calculated as s = rθ, where r is the radius and θ is the central angle in radians.

Example: "On a circle of radius 4, an angle of π/3 radians subtends an          of 4π/3."

Arccosine

The inverse function of cosine, written as cos⁻¹(x) or arccos(x), which returns the angle whose cosine is x.

Similar definitions: inverse cosine, cos⁻¹

Example: "The          of 0.5 is π/3, because cos(π/3) = 0.5."

Arcsine

The inverse function of sine, written as sin⁻¹(x) or arcsin(x), which returns the angle whose sine is x.

Similar definitions: inverse sine, sin⁻¹

Example: "The          of 1 is π/2, because sin(π/2) = 1."

Arctangent

The inverse function of tangent, written as tan⁻¹(x) or arctan(x), which returns the angle whose tangent is x.

Similar definitions: inverse tangent, tan⁻¹

Example: "The          of 1 is π/4, because tan(π/4) = 1."

Arithmetic sequence

A sequence of numbers in which the difference between consecutive terms is constant; each term is found by adding the same value (common difference) to the previous term.

Example: "The sequence 3, 7, 11, 15, 19 is an          with a common difference of 4."

Asymptote

A line that a curve approaches more and more closely but never reaches; can be vertical, horizontal, or oblique (slant).

Example: "The graph of y = 1/x has a vertical          at x = 0 and a horizontal one at y = 0."

Average rate of change

The ratio of the change in the output values to the change in the input values over an interval; calculated as [f(b) − f(a)] / (b − a).

Similar definitions: secant line slope

Example: "The          of f(x) = x² from x = 1 to x = 3 is (9 − 1)/(3 − 1) = 4."

Axis of symmetry

A line that divides a graph into two mirror-image halves; for a parabola y = ax² + bx + c, it is the vertical line x = −b/(2a).

Example: "The parabola y = x² − 4x + 3 has an          at x = 2."

Base of a logarithm

The number that is repeatedly multiplied in a logarithmic expression; in log_b(x), b is the base, meaning b raised to some power equals x.

Example: "In the expression log₂(8) = 3, the          is 2."

Base of an exponential function

The constant value that is raised to a variable power in an exponential function f(x) = bˣ, where b > 0 and b ≠ 1.

Example: "In the function f(x) = 3ˣ, the          is 3."

Bounded function

A function whose output values stay within a fixed range; there exist real numbers m and M such that m ≤ f(x) ≤ M for all x in the domain.

Example: "The sine function is a          because its values always lie between −1 and 1."

Cardioid

A heart-shaped polar curve defined by equations of the form r = a(1 + cos θ) or r = a(1 + sin θ).

Example: "The polar equation r = 2(1 + cos θ) produces a          that passes through the pole."

Carrying capacity

The maximum population size or value that a logistic model can sustain; the horizontal asymptote of a logistic function.

Example: "In the logistic model P(t) = 500/(1 + 9e⁻²ᵗ), the          is 500."

Change of base formula

A formula that converts a logarithm from one base to another: log_b(x) = log_a(x) / log_a(b), commonly used to evaluate logarithms with a calculator.

Example: "To evaluate log₃(7) on a calculator, use the         : log(7)/log(3)."

Closed interval

An interval that includes both of its endpoints, written with square brackets as [a, b].

Example: "The domain 0 ≤ x ≤ 5 is the          [0, 5]."

Cofunction identity

A trigonometric identity stating that a function of an angle equals the complementary function of its complement; for example, sin(θ) = cos(90° − θ).

Example: "By the         , sin(30°) = cos(60°) = 0.5."

Common difference

The constant value added to each term to produce the next term in an arithmetic sequence.

Example: "In the arithmetic sequence 5, 8, 11, 14, the          is 3."

Common logarithm

A logarithm with base 10, written as log(x) or log₁₀(x).

Example: "The          of 1000 is 3, because 10³ = 1000."

Common ratio

The constant factor multiplied by each term to produce the next term in a geometric sequence.

Example: "In the geometric sequence 2, 6, 18, 54, the          is 3."

Component form (of a vector)

A representation of a vector using horizontal and vertical components, written as ⟨a, b⟩ where a is the horizontal component and b is the vertical component.

Example: "A vector with magnitude 5 at an angle of 30° has          ⟨5cos 30°, 5sin 30°⟩ = ⟨5√3/2, 5/2⟩."

Composition of functions

The process of applying one function to the result of another; written as (f ∘ g)(x) = f(g(x)), where g is applied first and then f.

Similar definitions: composite function, function composition

Example: "If f(x) = 2x and g(x) = x + 1, then the          (f ∘ g)(3) = f(4) = 8."

Compound interest

Interest calculated on both the initial principal and the accumulated interest from previous periods; modeled by A = P(1 + r/n)ⁿᵗ or A = Peʳᵗ for continuous compounding.

Example: "An investment earning 6%          quarterly grows faster than one earning simple interest at the same rate."

Concavity

A description of how a function curves; a graph is concave up when its rate of change is increasing and concave down when its rate of change is decreasing.

Example: "The graph of y = x² has upward          because its rate of change increases as x increases."

Constant function

A function that always returns the same output value regardless of the input; its graph is a horizontal line.

Example: "The function f(x) = 5 is a          because its output is always 5."

Continuous function

A function whose graph has no breaks, holes, or jumps; it can be drawn without lifting the pen from the paper.

Example: "The polynomial f(x) = x³ − 2x is a          because its graph has no breaks or gaps."

Continuous growth/decay

A model where a quantity changes at a rate proportional to its current value at every instant, described by f(t) = aeʳᵗ where r is the continuous rate.

Example: "A population modeled by P(t) = 200e⁰·⁰³ᵗ exhibits          at a rate of 3% per year."

Cosecant

A trigonometric function that is the reciprocal of sine; csc(θ) = 1/sin(θ).

Example: "Since sin(30°) = 0.5, the          of 30° is 2."

Cosine

A trigonometric function that, in a right triangle, equals the ratio of the adjacent side to the hypotenuse; on the unit circle, it gives the x-coordinate of a point.

Example: "The          of 60° is 0.5."

Cotangent

A trigonometric function that is the reciprocal of tangent; cot(θ) = cos(θ)/sin(θ) = 1/tan(θ).

Example: "Since tan(45°) = 1, the          of 45° is also 1."

Coterminal angles

Angles in standard position that share the same terminal side; they differ by a full rotation (360° or 2π radians).

Example: "The angles 30° and 390° are          because 390° − 30° = 360°."

Cubic function

A polynomial function of degree 3, written in the general form f(x) = ax³ + bx² + cx + d, where a ≠ 0.

Example: "The function f(x) = 2x³ − x + 5 is a          because its highest-degree term is x³."

Decreasing function

A function whose output values decrease as the input values increase over a given interval; f(a) > f(b) whenever a < b on that interval.

Example: "The function f(x) = −2x is a          because as x increases, the output decreases."

Degree of a polynomial

The highest power of the variable in a polynomial expression; it determines the polynomial's end behavior and maximum number of turning points.

Example: "The polynomial 4x⁵ − 3x² + 7 has a          of 5."

Degree-radian conversion

The process of converting between degree and radian angle measures using the relationship π radians = 180°; multiply degrees by π/180 to get radians, or radians by 180/π to get degrees.

Example: "Using         , 60° = 60 × π/180 = π/3 radians."

Dependent variable

The output variable of a function whose value depends on the input (independent variable); typically represented by y.

Example: "In the equation y = 2x + 1, y is the          because its value depends on x."

Determinant

A scalar value computed from a square matrix that indicates whether the matrix is invertible; for a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc.

Example: "The          of the matrix [[3, 1], [2, 4]] is 3(4) − 1(2) = 10."

Direction of a vector

The angle a vector makes with the positive x-axis, typically measured in degrees or radians.

Example: "The          of the vector ⟨3, 3⟩ is 45° because arctan(3/3) = 45°."

Discontinuity

A point at which a function is not continuous; the graph has a break, hole, or jump at that location.

Similar definitions: point of discontinuity

Example: "The function f(x) = 1/(x − 2) has a          at x = 2 because the function is undefined there."

Domain

The set of all possible input values (x-values) for which a function is defined.

Example: "The          of f(x) = √x is all non-negative real numbers, x ≥ 0."

Dot product

An operation on two vectors that produces a scalar; for vectors ⟨a, b⟩ and ⟨c, d⟩, the dot product is ac + bd.

Example: "The          of ⟨2, 3⟩ and ⟨4, −1⟩ is 2(4) + 3(−1) = 5."

Double-angle identity

Trigonometric identities that express functions of 2θ in terms of functions of θ; for example, sin(2θ) = 2sin(θ)cos(θ).

Example: "Using the         , sin(60°) can be written as 2sin(30°)cos(30°)."

Elimination (parametric)

The process of removing the parameter from a set of parametric equations to obtain a single Cartesian equation relating x and y.

Similar definitions: eliminating the parameter

Example: "Given x = 2t and y = t²,          of the parameter t gives y = x²/4."

End behavior

The behavior of a function's output values as the input values approach positive or negative infinity.

Example: "The          of f(x) = x³ shows that f(x) → ∞ as x → ∞ and f(x) → −∞ as x → −∞."

Euler's number (e)

An irrational mathematical constant approximately equal to 2.71828, used as the base of the natural logarithm and natural exponential function.

Example: "The natural exponential function f(x) = eˣ uses          as its base, approximately 2.718."

Even function

A function that satisfies f(−x) = f(x) for all x in its domain; its graph is symmetric about the y-axis.

Example: "The function f(x) = x² is an          because f(−3) = 9 = f(3)."

Exponential decay

A decrease in quantity over time where the rate of decrease is proportional to the current value; modeled by f(t) = a · bᵗ where 0 < b < 1.

Example: "A radioactive substance losing half its mass every year follows         ."

Exponential equation

An equation in which the variable appears in the exponent, often solved using logarithms.

Example: "The          3ˣ = 81 is solved by recognizing that x = 4 since 3⁴ = 81."

Exponential function

A function of the form f(x) = a · bˣ where the base b is a positive constant not equal to 1, and the variable appears in the exponent.

Example: "The population model P(t) = 100 · 2ᵗ is an          that doubles with each unit increase in t."

Exponential growth

An increase in quantity over time where the rate of growth is proportional to the current value; modeled by f(t) = a · bᵗ where b > 1.

Example: "A bacteria colony that doubles every hour exhibits         ."

Extraneous solution

A solution obtained during the solving process that does not satisfy the original equation; commonly arises when solving logarithmic, rational, or radical equations.

Example: "When solving log(x) + log(x − 3) = 1, checking reveals that x = −2 is an          because logarithms are undefined for negative inputs."

Extrema

The maximum and minimum values of a function; can be local (relative) or global (absolute).

Example: "The          of f(x) = x³ − 3x include a local maximum at x = −1 and a local minimum at x = 1."

Factor theorem

A theorem stating that (x − c) is a factor of a polynomial f(x) if and only if f(c) = 0.

Example: "By the         , since f(2) = 0 for f(x) = x² − 4, (x − 2) is a factor of f(x)."

Factoring

The process of breaking a mathematical expression into a product of simpler expressions; used to find zeros of polynomials.

Example: "By         , the polynomial x² − 5x + 6 can be written as (x − 2)(x − 3)."

Frequency

The number of complete cycles a periodic function completes per unit of the independent variable; the reciprocal of the period.

Example: "A sine wave with a period of 0.5 seconds has a          of 2 cycles per second."

Function

A relation in which each input value is paired with exactly one output value.

Example: "The equation y = 2x + 3 defines a          because every input x produces exactly one output y."

Geometric sequence

A sequence of numbers in which each term is found by multiplying the previous term by a constant factor called the common ratio.

Example: "The sequence 3, 12, 48, 192 is a          with a common ratio of 4."

Half-angle identity

Trigonometric identities that express the sine, cosine, or tangent of half an angle in terms of the full angle; for example, sin(θ/2) = ±√[(1 − cos θ)/2].

Example: "The          can be used to find the exact value of sin(15°) from cos(30°)."

Half-life

The time required for a quantity undergoing exponential decay to decrease to half its initial value.

Example: "If a substance has a          of 5 years, then 100 grams will decay to 50 grams in 5 years."

Hole (in a graph)

A point where a rational function is undefined because a common factor in the numerator and denominator was canceled; it appears as a gap in the graph.

Similar definitions: removable discontinuity

Example: "The function f(x) = (x² − 1)/(x − 1) has a          at x = 1 because the (x − 1) factor cancels."

Horizontal asymptote

A horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Example: "The function f(x) = 2x/(x + 1) has a          at y = 2."

Horizontal compression

A transformation that multiplies the input of a function by a factor greater than 1, making the graph narrower horizontally; y = f(bx) where |b| > 1.

Example: "The graph of y = sin(2x) is a          of y = sin(x) by a factor of 2, halving its period."

Horizontal line test

A method for determining whether a function is one-to-one: if every horizontal line crosses the graph at most once, the function has an inverse that is also a function.

Example: "The function f(x) = x³ passes the         , so it has an inverse function."

Horizontal shift

A transformation that moves the graph of a function left or right; f(x − h) shifts the graph h units to the right.

Similar definitions: horizontal translation

Example: "The graph of y = (x − 3)² is a          of y = x² three units to the right."

Horizontal stretch

A transformation that multiplies the input of a function by a factor between 0 and 1, making the graph wider horizontally; y = f(bx) where 0 < |b| < 1.

Example: "The graph of y = sin(x/2) is a          of y = sin(x), doubling its period."

Identity (trigonometric)

An equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

Example: "The equation sin²(θ) + cos²(θ) = 1 is a fundamental trigonometric         ."

Identity matrix

A square matrix with 1s on the main diagonal and 0s elsewhere; multiplying any matrix by the identity matrix returns the original matrix.

Example: "The 2×2          is [[1, 0], [0, 1]]."

Increasing function

A function whose output values increase as the input values increase over a given interval; f(a) < f(b) whenever a < b on that interval.

Example: "The function f(x) = 2ˣ is an          for all real numbers."

Independent variable

The input variable of a function whose value is freely chosen; typically represented by x.

Example: "In the equation y = 3x − 7, x is the          because its value is chosen first."

Initial side

The starting position of a ray when measuring an angle in standard position; typically lies along the positive x-axis.

Example: "When measuring an angle in standard position, the          is the ray on the positive x-axis."

Intermediate value theorem

A theorem stating that if a continuous function takes values f(a) and f(b) on an interval [a, b], then it also takes every value between f(a) and f(b) at some point in that interval.

Example: "By the         , since f(1) = −2 and f(3) = 4, the continuous function f must equal zero at some x between 1 and 3."

Interval notation

A compact way to describe a set of numbers using parentheses and brackets: (a, b) for open intervals and [a, b] for closed intervals.

Example: "The domain x > 3 is written in          as (3, ∞)."

Inverse function

A function that reverses the operation of another function; if f(a) = b, then f⁻¹(b) = a. The graph of an inverse function is a reflection of the original over the line y = x.

Example: "The          of f(x) = 2x + 3 is f⁻¹(x) = (x − 3)/2."

Inverse matrix

A matrix A⁻¹ such that A × A⁻¹ = I (the identity matrix); used to solve matrix equations and systems of equations.

Example: "To solve the matrix equation AX = B, multiply both sides by the          of A: X = A⁻¹B."

Inverse trigonometric function

A function that returns the angle whose trigonometric value equals a given number; the domain of the original trigonometric function must be restricted to ensure the inverse is a function.

Example: "The          sin⁻¹(1/2) = π/6 because sin(π/6) = 1/2."

Law of cosines

A formula relating the sides and angles of any triangle: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c.

Example: "To find the third side of a triangle with sides 5 and 7 and an included angle of 60°, use the         ."

Law of sines

A formula stating that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C).

Example: "To find a missing angle in a triangle given two sides and a non-included angle, apply the         ."

Leading coefficient

The coefficient of the term with the highest degree in a polynomial; it determines the end behavior of the polynomial's graph.

Example: "In the polynomial −3x⁴ + x² − 7, the          is −3."

Lemniscate

A figure-eight shaped polar curve defined by equations of the form r² = a² cos(2θ) or r² = a² sin(2θ).

Example: "The polar equation r² = 9cos(2θ) produces a          that passes through the pole twice."

Limaçon

A polar curve of the form r = a + b cos θ or r = a + b sin θ; depending on the ratio a/b, it may have an inner loop, be a cardioid, or have a dimple.

Example: "The polar equation r = 1 + 2cos θ is a          with an inner loop because |b| > |a|."

Linear function

A function of the form f(x) = mx + b whose graph is a straight line, where m is the slope and b is the y-intercept.

Example: "The equation y = −2x + 5 is a          with slope −2 and y-intercept 5."

Linear speed

The rate at which a point moves along a circular path, calculated as v = rω, where r is the radius and ω is the angular speed.

Example: "A point on the edge of a wheel with radius 3 feet rotating at 2 radians per second has a          of 6 feet per second."

Local maximum

A point where the function's value is greater than or equal to the values at all nearby points; the highest point in a local region of the graph.

Similar definitions: relative maximum

Example: "The function f(x) = −x⁴ + 2x² has a          at x = 1."

Local minimum

A point where the function's value is less than or equal to the values at all nearby points; the lowest point in a local region of the graph.

Similar definitions: relative minimum

Example: "The function f(x) = x³ − 3x has a          at x = 1."

Logarithm

The inverse operation of exponentiation; log_b(x) = y means b^y = x. It answers the question: to what power must the base be raised to produce a given number?

Example: "The          base 2 of 8 is 3, because 2³ = 8."

Logarithmic equation

An equation that involves the logarithm of an expression containing the variable, solved by using properties of logarithms or converting to exponential form.

Example: "The          log₂(x + 3) = 5 is solved by rewriting as x + 3 = 2⁵ = 32, so x = 29."

Logarithmic function

A function of the form f(x) = log_b(x), the inverse of the exponential function; its domain is all positive real numbers and it has a vertical asymptote at x = 0.

Example: "The          f(x) = log₂(x) passes through the point (8, 3)."

Logarithmic scale

A scale in which equal distances represent equal ratios (multiplicative changes) rather than equal differences; used to linearize exponential data.

Example: "On a         , the distance from 1 to 10 is the same as the distance from 10 to 100."

Logistic function

A function that models growth with a carrying capacity, following an S-shaped curve; often written as f(t) = L / (1 + ae^(−kt)).

Example: "The spread of a disease in a limited population can be modeled with a         ."

Magnitude of a vector

The length or size of a vector, calculated using the Pythagorean theorem; for vector ⟨a, b⟩, the magnitude is √(a² + b²).

Similar definitions: norm, length of a vector

Example: "The          of the vector ⟨3, 4⟩ is √(9 + 16) = 5."

Matrix

A rectangular array of numbers arranged in rows and columns, used to represent and solve systems of equations or perform linear transformations.

Example: "A 2×2          can be used to represent a rotation transformation in the plane."

Matrix multiplication

An operation that combines two matrices by taking the dot product of rows from the first matrix with columns of the second matrix; the number of columns in the first must equal the number of rows in the second.

Example: "In         , a 2×3 matrix times a 3×2 matrix produces a 2×2 matrix."

Midline

The horizontal line halfway between the maximum and minimum values of a periodic function; for y = a·sin(bx) + d, the midline is y = d.

Example: "The function y = 3sin(x) + 2 has a          at y = 2."

Multiplicative transformation

A transformation that multiplies the input or output of a function by a constant, resulting in a stretch, compression, or reflection of the graph.

Example: "Replacing f(x) with 2f(x) is a          that vertically stretches the graph by a factor of 2."

Multiplicity

The number of times a particular factor appears in the factored form of a polynomial; it determines whether the graph crosses or touches the x-axis at that zero.

Example: "In f(x) = (x − 2)³, the zero x = 2 has a          of 3, so the graph crosses the x-axis at that point."