AP Precalculus Vocabulary

Absolute maximum

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

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; $\omega = \frac{\Delta \theta}{\Delta t}$.

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

Arc length

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

Example: "On a circle of radius 4, an angle of $\frac{\pi}{3}$ radians subtends an          of $\frac{4\pi}{3}$."

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 $\frac{\pi}{3}$, because $\cos(\frac{\pi}{3}) = 0.5$."

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 $\frac{\pi}{2}$, because $\sin(\frac{\pi}{2}) = 1$."

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 $\frac{\pi}{4}$, because $\tan(\frac{\pi}{4}) = 1$."

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 $y = \frac{1}{x}$ 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 $\frac{[f(b) - f(a)]}{(b - a)}$.

Similar definitions: secant line slope

Example: "The          of $f(x) = x^{2}$ from $x = 1$ to $x = 3$ is $\frac{(9 - 1)}{(3 - 1)} = 4$."

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 $\log_2(8) = 3$, the          is 2."

Base of an exponential function

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

Example: "In the function $f(x) = 3^x$, 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 $r = a(1 + \cos(\theta))$ or $r = a(1 + \sin(\theta))$.

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

Carrying capacity

In the logistic model $P(t) = \frac{500}{1 + 9e^{-2t}}$, the          is 500.

Change of base formula

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

Example: "To evaluate $\log_3(7)$ on a calculator, use the         : $\frac{\log(7)}{\log(3)}$."

Closed interval

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

Example: "The domain $0 \leq x \leq 5$ 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, $\sin(\theta) = \cos(90° - \theta)$.

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 $\log_{10}(x)$.

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 $\langle a, b \rangle$ where $a$ is the horizontal component and $b$ is the vertical component.

Example: "A vector with magnitude 5 at an angle of $30^\circ$ has          $\langle 5 \cos(30^\circ), 5 \sin(30^\circ) \rangle = \langle 5 \frac{\sqrt{3}}{2}, 5 \cdot \frac{1}{2} \rangle$."

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 + \frac{r}{n})^{nt} or A = Pe^{rt} 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) = ae^{rt} where r is the continuous rate.

Example: "A population modeled by P(t) = 200e^{0.03t} exhibits          at a rate of 3% per year."

Cosecant

A trigonometric function that is the reciprocal of sine; $\csc(\theta) = \frac{1}{\sin(\theta)}$.

Example: "Since $\sin(30^\circ) = 0.5$, the          of $30^\circ$ 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 $60^\circ$ is 0.5."

Cotangent

A trigonometric function that is the reciprocal of tangent; $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}$.

Example: "Since $ext{tan}(45^ ext{°}) = 1$, the          of $45^ ext{°}$ is also 1."

Coterminal angles

Angles in standard position that share the same terminal side; they differ by a full rotation ($360^\circ$ or $2\pi$ radians).

Example: "The angles $30^\circ$ and $390^\circ$ are          because $390^\circ - 30^\circ = 360^\circ$."

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 4x^5 − 3x^2 + 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

An ordered list of numbers that follow a particular pattern or rule; each number in the list is called a term.

Example: "The list $1, 4, 9, 16, 25$ is a          of perfect squares."

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 $\langle 3, 3 \rangle$ is 45° because $\arctan(\frac{3}{3}) = 45°$."

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) = \frac{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 $2\theta$ in terms of functions of $\theta$; for example, $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Example: "Using the         , \sin(60^) can be written as 2\sin(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 $f(x) = x^3$ shows that $f(x) → \infty$ as $x → \infty$ and $f(x) → -\infty$ as $x → -\infty$."

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 $f(x) = e^x$ 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 $f(x) = x^{2}$ 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 $f(t) = a \cdot b^t$ 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          $3^x = 81$ is solved by recognizing that $x = 4$ since $3^4 = 81$."

Exponential function

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

Example: "The population model $P(t) = 100 \bullet 2^t$ 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 $f(t) = a \bullet b^t$ 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 $f(x) = 2^x$ 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 $f^{-1}(x) = \frac{(x - 3)}{2}$."

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          $\sin^{-1}(1/2) = \frac{\pi}{6}$ because $\sin(\frac{\pi}{6}) = 1/2.$"

Law of cosines

A formula relating the sides and angles of any triangle: $c^2 = a^2 + b^2 - 2ab \bullet \cos(C)$, 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 $60^\circ$, 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: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

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 $-3x^4 + x^2 - 7$, the          is $-3$."

Lemniscate

A figure-eight shaped polar curve defined by equations of the form $r^{2} = a^{2} \cos(2\theta)$ or $r^{2} = a^{2} \sin(2\theta)$.

Example: "The polar equation $r^{2} = 9\cos(2\theta)$ produces a          that passes through the pole twice."

Limaçon

A polar curve of the form $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$; depending on the ratio $\frac{a}{b}$, it may have an inner loop, be a cardioid, or have a dimple.

Example: "The polar equation $r = 1 + 2\cos(\theta)$ 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 $v = r \omega$, where $r$ is the radius and $\omega$ is the angular speed.

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 $f(x) = -x^{4} + 2x^{2}$ has a          at $x = 1$."

Local minimum

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 2^3 = 8."

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          $\log_{2}(x + 3) = 5$ is solved by rewriting as $x + 3 = 2^{5} = 32$, 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) = \log_2(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 $f(t) = \frac{L}{1 + ae^{-kt}}$.

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 $\langle a, b \rangle$, the magnitude is $\sqrt{a^2 + b^2}$.

Similar definitions: norm, length of a vector

Example: "The          of the vector $\langle 3, 4 \rangle$ is $\sqrt{9 + 16} = 5$."

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.

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."