Glossary of Terms

Amplitude

For a sinusoidal function \(y = A\sin(Bx) + D\), the amplitude is \(|A|\), the distance from the midline to a peak or trough.

Example: \(y = 3\sin x\) has amplitude \(3\).

Amplitude from Data

The amplitude extracted from data as half the difference between the maximum and minimum values: \(A = (\text{max} - \text{min})/2\).

Example: Temperatures ranging from \(30^\circ\)F to \(70^\circ\)F give amplitude \(20^\circ\)F.

Angle Definition

A figure formed by two rays sharing a common endpoint (the vertex), representing rotation from an initial side to a terminal side. Measured in degrees or radians.

Example: A right angle is formed when one ray is rotated \(90^\circ\) from another.

Arc Length

The distance measured along a circular arc. For a circle of radius \(r\) and central angle \(\theta\) in radians, arc length is \(s = r\theta\).

Example: A sector of radius \(4\) with central angle \(\pi/2\) has arc length \(s = 4 \cdot \pi/2 = 2\pi\).

Arccosine

The inverse of cosine on its restricted domain \([0, \pi]\). \(\arccos x\) returns the angle in this range whose cosine is \(x\), for \(x \in [-1, 1]\).

Example: \(\arccos(1/2) = \pi/3\) because \(\cos(\pi/3) = 1/2\).

Arcsine

The inverse of sine on its restricted domain \([-\pi/2, \pi/2]\). \(\arcsin x\) returns the angle in this range whose sine is \(x\), for \(x \in [-1, 1]\).

Example: \(\arcsin(1/2) = \pi/6\) because \(\sin(\pi/6) = 1/2\).

Arctangent

The inverse of tangent on its restricted domain \((-\pi/2, \pi/2)\). \(\arctan x\) returns the angle in this range whose tangent is \(x\), for all real \(x\).

Example: \(\arctan(1) = \pi/4\) because \(\tan(\pi/4) = 1\).

Arithmetic Sequence

A sequence in which consecutive terms differ by a constant amount called the common difference.

Example: \(3, 7, 11, 15, \ldots\) has common difference \(4\).

Arithmetic Sequence Formula

The explicit rule \(a_n = a_1 + (n-1)d\), giving the \(n\)th term of an arithmetic sequence with first term \(a_1\) and common difference \(d\).

Example: If \(a_1 = 2\) and \(d = 5\), then \(a_{10} = 2 + 9 \cdot 5 = 47\).

Asymptotes of Tangent

Vertical lines where \(\cos x = 0\), i.e., \(x = \pi/2 + k\pi\) for integer \(k\). The tangent function grows without bound near these lines.

Average Rate of Change

The ratio of the change in output to the change in input over an interval. Geometrically, this equals the slope of the secant line connecting two points on a curve.

Example: For \(f(x) = x^2\), the average rate of change from \(x = 1\) to \(x = 3\) is \((9-1)/(3-1) = 4\).

Axis of Symmetry

A line about which a graph is a mirror image of itself. For the parabola \(y = ax^2 + bx + c\), the axis of symmetry is \(x = -\tfrac{b}{2a}\).

Example: For \(y = x^2 - 6x + 5\), the axis of symmetry is \(x = 3\).

Base of Exponential

The positive constant \(b\) in \(f(x) = a \cdot b^x\). The base determines whether the function grows (\(b > 1\)) or decays (\(0 < b < 1\)).

Binomial

A polynomial with exactly two terms connected by addition or subtraction.

Example: \(x^2 - 9\) and \(3x + 4\) are binomials.

Bounded Functions

A function is bounded on an interval if there exist constants \(m\) and \(M\) with \(m \leq f(x) \leq M\) for all \(x\) in the interval. Otherwise it is unbounded.

Example: \(\sin x\) is bounded on \(\mathbb{R}\) since \(-1 \leq \sin x \leq 1\).

Calculator Strategies

Techniques for efficient and accurate use of a graphing calculator on the AP Pre-Calculus exam, including graphing to locate solutions, using numeric solve for roots, and checking modes (degree vs radian).

Cardioids

A heart-shaped polar curve with equation \(r = a(1 + \cos\theta)\) or \(r = a(1 + \sin\theta)\). The name comes from the Greek word for "heart."

Example: \(r = 1 + \cos\theta\) produces a cardioid passing through the origin when \(\theta = \pi\).

Change in Tandem

The simultaneous, coordinated variation of two related quantities, such as how output changes when input changes. Understanding this covariation is a foundational idea for rates of change.

Example: As radius \(r\) increases, the area \(A = \pi r^2\) of a circle changes in tandem with it.

Change of Base Formula

The identity \(\log_b x = \dfrac{\log_a x}{\log_a b}\), converting a logarithm from base \(b\) to any other positive base \(a \neq 1\). Useful for calculator evaluation.

Circles in Polar Form

Circles whose equations simplify nicely in polar coordinates: \(r = a\) (centered at origin), \(r = 2a\cos\theta\) (centered on \(x\)-axis), \(r = 2a\sin\theta\) (centered on \(y\)-axis).

Example: \(r = 4\cos\theta\) is a circle of radius \(2\) centered at \((2, 0)\).

Cofunction Identities

The identities relating trig functions of complementary angles: \(\sin(\pi/2 - \theta) = \cos\theta\), \(\tan(\pi/2 - \theta) = \cot\theta\), \(\sec(\pi/2 - \theta) = \csc\theta\).

Example: \(\cos(60^\circ) = \sin(30^\circ) = 1/2\).

Combined Transformations

A sequence of two or more elementary transformations applied to a parent function, producing a graph described by a compact formula such as \(y = a f(b(x - h)) + k\).

Example: \(y = -2(x+1)^2 + 3\) combines a horizontal shift, vertical stretch, reflection, and vertical shift of \(y = x^2\).

Common Difference

The constant value \(d\) added to each term of an arithmetic sequence to produce the next term: \(a_{n+1} - a_n = d\).

Example: In \(5, 8, 11, 14, \ldots\) the common difference is \(d = 3\).

Common Logarithm

The logarithm to base \(10\), denoted \(\log x\) (or \(\log_{10} x\)). Historically used for numerical computation and in scientific scales.

Example: \(\log 1000 = 3\).

Common Ratio

The constant value \(r\) that multiplies each term of a geometric sequence to give the next: \(a_{n+1}/a_n = r\).

Example: In \(80, 40, 20, 10, \ldots\) the common ratio is \(r = 1/2\).

Competing Models

Two or more candidate models for the same data, compared using goodness-of-fit measures, residual patterns, simplicity, and predictive performance to choose the most appropriate.

Example: Choosing between a quadratic and exponential model for a growth dataset.

Completing the Square

An algebraic procedure that rewrites \(ax^2 + bx + c\) in vertex form \(a(x - h)^2 + k\) by creating a perfect square trinomial. It reveals the vertex and enables solving.

Example: \(x^2 + 6x + 5 = (x + 3)^2 - 4\).

Complex Conjugate Zeros

For a polynomial with real coefficients, non-real zeros occur in conjugate pairs: if \(a + bi\) is a zero, so is \(a - bi\). This keeps the product of factors real.

Example: \(x^2 + 4\) has zeros \(2i\) and \(-2i\).

Complex Numbers

Numbers of the form \(a + bi\) where \(a, b \in \mathbb{R}\) and \(i^2 = -1\). They extend the real numbers and make every polynomial equation solvable.

Example: \(3 + 4i\) is a complex number with real part \(3\) and imaginary part \(4\).

Complex Zeros

Values of \(x\) that make a polynomial equal zero but lie outside the real numbers, arising as \(a \pm bi\) with \(b \neq 0\). For polynomials with real coefficients, complex zeros occur in conjugate pairs.

Example: \(x^2 + 1 = 0\) has complex zeros \(x = \pm i\).

Component Form

The representation of a vector by its ordered components: \(\vec{v} = \langle a, b \rangle\) in two dimensions or \(\langle a, b, c \rangle\) in three. Components equal terminal-minus-initial coordinates.

Example: From \((1, 2)\) to \((4, 6)\): \(\vec{v} = \langle 3, 4 \rangle\).

Composing Trig Inverses

Simplifying expressions like \(\sin(\arccos x)\) using right-triangle diagrams or identities: \(\sin(\arccos x) = \sqrt{1 - x^2}\) for \(x \in [-1, 1]\).

Example: \(\tan(\arcsin x) = \dfrac{x}{\sqrt{1 - x^2}}\).

Composition of Functions

A function formed by applying one function to the result of another: \((f \circ g)(x) = f(g(x))\), read "\(f\) of \(g\) of \(x\)".

Example: If \(f(x) = x + 1\) and \(g(x) = x^2\), then \((f \circ g)(x) = x^2 + 1\).

Compound Interest

A model for investment growth: \(A = P\left(1 + \tfrac{r}{n}\right)^{nt}\), where \(P\) is principal, \(r\) the annual rate, \(n\) compounding periods per year, and \(t\) time in years.

Concavity from Rates

Concavity can be identified by observing how average rates of change behave on adjacent intervals: increasing rates indicate concave-up behavior, decreasing rates indicate concave-down.

Example: \(f(x) = x^2\) has increasing average rates of change, signaling concave-up behavior.

Constant Functions

A function of the form \(f(x) = c\) for some constant \(c\). Its graph is a horizontal line, and its range contains the single value \(c\).

Example: \(f(x) = 7\) gives the same output \(7\) for every input.

Continuous Growth

Exponential growth modeled with continuous compounding: \(A(t) = A_0 e^{kt}\), where \(k\) is the continuous growth rate. It arises as the limit of ever-more-frequent compounding.

Example: \(A(t) = 1000 e^{0.04 t}\) models a deposit growing at \(4\%\) continuously.

Coordinate Plane

A two-dimensional plane formed by two perpendicular number lines, the \(x\)-axis and \(y\)-axis, intersecting at the origin. Every point is specified by an ordered pair \((x, y)\).

Correlation

A numerical measure of the strength and direction of a linear relationship between two variables, typically the correlation coefficient \(r \in [-1, 1]\). Values near \(\pm 1\) indicate strong linear association.

Example: Height and shoe size in adults typically show a strong positive correlation.

Cosecant Function

The reciprocal of sine: \(\csc\theta = 1/\sin\theta\), defined wherever \(\sin\theta \neq 0\). Period \(2\pi\), range \((-\infty, -1] \cup [1, \infty)\).

Example: \(\csc(\pi/2) = 1\).

Cosecant Graph

The graph of \(y = \csc x\): U-shaped branches with period \(2\pi\), opening up where \(\sin x > 0\) and down where \(\sin x < 0\), with vertical asymptotes at zeros of \(\sin x\).

Cosine Function

The function \(\cos\theta\) defined as the \(x\)-coordinate of the point on the unit circle at angle \(\theta\). Periodic with period \(2\pi\) and range \([-1, 1]\).

Example: \(\cos(0) = 1\).

Cosine Graph

The graph of \(y = \cos x\): a smooth wave oscillating between \(-1\) and \(1\), starting at \((0, 1)\), with period \(2\pi\). It is the sine graph shifted left by \(\pi/2\).

Cotangent Function

The reciprocal of tangent: \(\cot\theta = \cos\theta/\sin\theta\), defined wherever \(\sin\theta \neq 0\). Period \(\pi\), range \(\mathbb{R}\).

Example: \(\cot(\pi/4) = 1\).

Cotangent Graph

The graph of \(y = \cot x\): a decreasing curve on each interval of length \(\pi\), with vertical asymptotes at \(x = k\pi\). Period is \(\pi\).

Coterminal Angles

Two angles in standard position that share the same terminal side, differing by a whole number of full rotations (\(360^\circ\) or \(2\pi\)).

Example: \(30^\circ\), \(390^\circ\), and \(-330^\circ\) are all coterminal.

Data Analysis for Models

The examination of data through plots, summary statistics, and pattern detection to inform the choice and parameters of a mathematical model.

Example: Plotting population versus year on semi-log axes reveals whether exponential growth is plausible.

Daylight Hour Models

Sinusoidal models for hours of daylight versus day of year, with period \(365\), amplitude depending on latitude, and a phase shift placing maxima near the summer solstice.

Example: At \(45^\circ\) latitude, daylight ranges from roughly \(9\) to \(15\) hours over the year.

Decay Factor

The multiplier per unit of input in an exponential decay model: if \(f(x) = a \cdot b^x\) with \(0 < b < 1\), then \(b\) is the decay factor and \(1 - b\) is the decay rate.

Example: A quantity losing \(20\%\) per year has decay factor \(0.80\).

Decomposing Functions

Writing a given function \(h\) as a composition \(f \circ g\) of simpler functions. Useful for understanding structure and for applying composition-based techniques.

Example: \(h(x) = (2x + 1)^3\) decomposes as \(f(u) = u^3\) and \(g(x) = 2x + 1\).

Decreasing Functions

A function \(f\) is decreasing on an interval if \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\) for all \(x_1, x_2\) in the interval. The graph falls from left to right.

Example: \(f(x) = -x + 4\) is decreasing on \(\mathbb{R}\).

Decreasing Rate of Change

A situation in which successive average rates of change decrease as one moves along the domain, indicating a concave-down shape.

Example: An object slowing to a stop shows decreasing rates of change in position.

Degree Measure

An angular unit in which one full revolution equals \(360\) degrees (\(360^\circ\)). Each degree is subdivided into \(60\) minutes and each minute into \(60\) seconds.

Example: A right angle measures \(90^\circ\).

Degree of a Polynomial

The highest power of \(x\) that appears with a non-zero coefficient in a polynomial. The degree governs the shape, end behavior, and maximum number of roots.

Example: \(P(x) = x^4 - 3x + 7\) has degree \(4\).

Degree-Radian Conversion

The relation \(180^\circ = \pi\) radians, so multiply degrees by \(\pi/180\) to convert to radians and multiply radians by \(180/\pi\) to convert to degrees.

Example: \(60^\circ = 60 \cdot \pi/180 = \pi/3\) radians.

Dependent Variable

The variable whose value is determined by the value of another variable through a functional relationship. Usually plotted on the vertical axis.

Example: In \(d = 5t\), distance \(d\) depends on time \(t\), so \(d\) is the dependent variable.

Determinant

A scalar associated with a square matrix that measures scaling factor and invertibility. For \(\begin{bmatrix} a & b \\ c & d\end{bmatrix}\), the determinant is \(ad - bc\); a matrix is invertible iff its determinant is nonzero.

Example: \(\det\begin{bmatrix} 2 & 3 \\ 1 & 4\end{bmatrix} = 8 - 3 = 5\).

Direction of a Vector

The angle a vector makes with a reference axis, typically the positive \(x\)-axis. For \(\vec{v} = \langle a, b\rangle\), direction is \(\theta = \arctan(b/a)\), adjusted for quadrant.

Example: \(\langle 1, 1 \rangle\) has direction \(\pi/4\).

Direction of Motion

The direction in which a particle is moving along a parametric curve at a given time, indicated by the velocity vector \((x'(t), y'(t))\).

Example: At \(t = 1\) for \(x = t, y = t^2\), the direction is \((1, 2)\), pointing up and to the right.

Discriminant

The quantity \(\Delta = b^2 - 4ac\) associated with the quadratic \(ax^2 + bx + c = 0\). Its sign determines the nature and number of real roots.

Domain

The set of all permitted input values of a function. For \(f(x)\), it is every \(x\) for which \(f(x)\) is defined.

Example: The domain of \(f(x) = \sqrt{x}\) is \(\{x \in \mathbb{R} : x \geq 0\}\).

Domain of Compositions

The set of \(x\) in the domain of \(g\) such that \(g(x)\) lies in the domain of \(f\). It may be smaller than the domain of \(g\) alone.

Example: For \(f(x) = \sqrt{x}\) and \(g(x) = x - 4\), the domain of \(f \circ g\) is \(x \geq 4\).

Domain Restrictions

Explicit limits placed on the domain of a function to exclude values that make the formula undefined, such as zero denominators or negatives under even roots.

Example: The domain of \(f(x) = \tfrac{1}{x - 3}\) is restricted to \(x \neq 3\).

Dot Product

An operation producing a scalar from two vectors: \(\langle a, b\rangle \cdot \langle c, d\rangle = ac + bd\). Geometrically, \(\vec{u} \cdot \vec{v} = |\vec u||\vec v|\cos\theta\).

Example: \(\langle 1, 2\rangle \cdot \langle 3, 4\rangle = 3 + 8 = 11\).

Double Angle Formulas

Identities for trig functions of \(2\theta\): \(\sin(2\theta) = 2\sin\theta\cos\theta\), \(\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1\).

Example: \(\sin(60^\circ) = 2\sin(30^\circ)\cos(30^\circ) = 2 \cdot \tfrac{1}{2} \cdot \tfrac{\sqrt 3}{2} = \tfrac{\sqrt 3}{2}\).

Doubling Time

The time required for an exponentially growing quantity to double in size. For the model \(P(t) = P_0 e^{kt}\), doubling time is \(t = \ln(2)/k\).

Example: If a population grows at \(5\%\) per year, doubling time is \(\ln(2)/0.05 \approx 13.9\) years.

Eliminating the Parameter

Converting parametric equations to a rectangular relation between \(x\) and \(y\) by solving one equation for \(t\) and substituting, or by using identities.

Example: From \(x = \cos t, y = \sin t\), eliminating \(t\) via \(\cos^2 t + \sin^2 t = 1\) gives \(x^2 + y^2 = 1\).

End Behavior of Rationals

The limits of a rational function as \(x \to \pm \infty\), determined by comparing the degrees of numerator and denominator: ratio of leading coefficients, zero, or unbounded.

Example: \(\tfrac{3x^2 + 1}{x^2 - 4}\) has horizontal asymptote \(y = 3\) as \(x \to \pm\infty\).

Even Functions

A function satisfying \(f(-x) = f(x)\) for all \(x\) in its domain. Its graph is symmetric about the \(y\)-axis.

Example: \(f(x) = x^2\) and \(f(x) = \cos x\) are even.

Even-Odd Identities

The identities \(\cos(-\theta) = \cos\theta\) (even), \(\sin(-\theta) = -\sin\theta\) (odd), and \(\tan(-\theta) = -\tan\theta\) (odd), reflecting symmetry of the graphs.

Example: \(\sin(-\pi/6) = -\sin(\pi/6) = -1/2\).

Exact Trig Values

Values of trigonometric functions at special angles expressed symbolically (using \(\sqrt{}\), fractions, and \(\pi\)) rather than as decimal approximations.

Example: Exact: \(\cos(\pi/6) = \sqrt{3}/2\); approximate: \(0.866\).

Explicit Formulas

A rule giving \(a_n\) directly as a function of \(n\), without reference to previous terms. Contrasts with a recursive definition.

Example: \(a_n = 2n + 1\) produces \(3, 5, 7, 9, \ldots\).

Exponential Decay

Change modeled by \(f(x) = a \cdot b^x\) with \(0 < b < 1\), where the quantity shrinks by the same factor over equal intervals, approaching zero.

Example: A drug concentration halving every \(6\) hours.

Exponential Equations

Equations in which the unknown appears in an exponent, such as \(2^x = 16\) or \(e^{2x} = 5\). They are typically solved using logarithms.

Example: \(3^x = 81\) gives \(x = 4\).

Exponential Function Def

A function of the form \(f(x) = a \cdot b^x\) with \(a \neq 0\), \(b > 0\), \(b \neq 1\). The variable appears in the exponent, giving characteristic rapid growth or decay.

Example: \(f(x) = 3 \cdot 2^x\).

Exponential Graphs

Sketches of \(y = a \cdot b^x\) identifying the \(y\)-intercept \(a\), the horizontal asymptote \(y = 0\), and whether the function grows (\(b > 1\)) or decays (\(0 < b < 1\)).

Exponential Growth

Change modeled by \(f(x) = a \cdot b^x\) with \(b > 1\), in which the quantity multiplies by the same factor over equal intervals. It accelerates as \(x\) increases.

Example: A bacterial population doubling each hour follows \(P(t) = P_0 \cdot 2^t\).

Exponential Modeling

Using exponential functions to represent real-world phenomena characterized by constant percentage change, such as population growth, radioactive decay, and investments.

Example: A radioactive sample with \(10\%\) annual decay: \(A(t) = A_0 (0.90)^t\).

Exponential Regression

A statistical procedure that fits a function of the form \(y = a b^x\) to data, typically after linearizing via logarithms or using nonlinear least squares.

Example: Fitting bacterial counts versus time to \(y = 100 \cdot 1.5^t\).

Exponential Transformations

Shifts, stretches, and reflections applied to \(y = b^x\), producing \(y = a \cdot b^{k(x - h)} + c\). The horizontal asymptote moves with the vertical shift \(c\).

Example: \(y = 2^{x-3} + 4\) shifts \(y = 2^x\) three right and four up, with asymptote \(y = 4\).

Exponential-Log Conversion

The equivalence \(b^y = x \iff \log_b x = y\), used to switch between exponential and logarithmic equations when solving.

Example: Converting \(e^x = 7\) gives \(x = \ln 7\).

Extrapolation

Using a fitted model to estimate values of the dependent variable outside the range of the observed data. Extrapolation is less reliable than interpolation.

Example: Predicting population in year \(2050\) from data collected through \(2025\).

Factor Theorem

For a polynomial \(P(x)\), \((x - a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). It links factorization and roots directly.

Example: Since \(P(2) = 0\) for \(P(x) = x^2 - 4\), \((x - 2)\) is a factor.

Factored Form

A quadratic written as \(f(x) = a(x - p)(x - q)\), where \(p\) and \(q\) are the roots. This form reveals the \(x\)-intercepts directly.

Example: \(f(x) = 2(x - 1)(x + 3)\) has roots \(1\) and \(-3\).

Factoring Polynomials

The process of writing a polynomial as a product of lower-degree polynomial factors. Techniques include common factoring, grouping, special patterns, and the factor theorem.

Example: \(x^3 - x = x(x - 1)(x + 1)\).

Finding Inverses Algebraically

The algebraic process: write \(y = f(x)\), swap \(x\) and \(y\), then solve the new equation for \(y\) to obtain \(y = f^{-1}(x)\).

Example: For \(y = 3x - 1\), swap to get \(x = 3y - 1\), then \(y = \tfrac{x + 1}{3}\).

Free Response Strategy

An approach to AP free-response questions emphasizing careful reading, setting up with definitions and notation, showing all relevant steps, labeling answers with units, and writing interpretations in context.

Frequency

The number of complete cycles of a periodic function per unit of input, equal to the reciprocal of the period: \(f = 1/T\).

Example: A wave with period \(\pi\) has frequency \(1/\pi\) cycles per unit.

Function Behavior

A systematic description of a function in terms of domain, range, intercepts, symmetry, intervals of increase/decrease, turning points, asymptotes, and end behavior.

Function Definition

A rule that assigns to each element \(x\) in a set called the domain exactly one element \(y\) in a set called the codomain. Written \(f: x \mapsto f(x)\) or \(y = f(x)\).

Example: \(f(x) = x^2\) assigns to each real number its square, so \(f(3) = 9\).

Function Evaluation

The process of substituting a specific input value into a function's rule and computing the resulting output. Produces a single numeric value.

Example: For \(f(x) = 2x^2 - 1\), evaluating at \(x = 3\) gives \(f(3) = 17\).

Function Model Selection

The process of choosing which family of functions (linear, quadratic, exponential, logarithmic, trigonometric, etc.) best fits a situation's behavior before fitting specific parameters.

Example: Constant percent growth signals an exponential model rather than a linear one.

Function Notation

The symbolic convention \(f(x)\) used to represent the output of a function \(f\) at input \(x\). It makes the function's name and argument explicit.

Example: If \(f(x) = x^2 + 1\), then \(f(4) = 17\) uses function notation to evaluate the function at \(x = 4\).

Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one complex root. A polynomial of degree \(n\) has exactly \(n\) roots in \(\mathbb{C}\), counted with multiplicity.

Example: \(x^3 + 1 = 0\) has exactly three complex roots.

General Solutions

The complete solution set of a trig equation, including all angles differing from a primary solution by integer multiples of the function's period.

Example: \(\sin x = 1/2\) has general solution \(x = \pi/6 + 2k\pi\) or \(x = 5\pi/6 + 2k\pi\), \(k \in \mathbb{Z}\).

Geometric Sequence

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed nonzero constant called the common ratio.

Example: \(2, 6, 18, 54, \ldots\) has common ratio \(3\).

Geometric Sequence Formula

The explicit rule \(a_n = a_1 \cdot r^{n-1}\), giving the \(n\)th term of a geometric sequence with first term \(a_1\) and common ratio \(r\).

Example: If \(a_1 = 3\) and \(r = 2\), then \(a_5 = 3 \cdot 2^4 = 48\).

Global Maximum

The largest value of a function over its entire domain. Also called absolute maximum. May occur at a critical point or an endpoint.

Example: \(f(x) = -x^2\) has a global maximum of \(0\) at \(x = 0\).

Global Minimum

The smallest value of a function over its entire domain. Also called absolute minimum. May occur at a critical point or an endpoint.

Example: \(f(x) = x^2\) has a global minimum of \(0\) at \(x = 0\).

Goodness of Fit

A measure of how well a model describes data, commonly reported as the coefficient of determination \(R^2\), with values near \(1\) indicating strong fit.

Example: An \(R^2 = 0.98\) means the model explains \(98\%\) of the variation in the data.

Graphing Functions

The process of creating a visual representation of a function's input-output relationship by plotting the set of points \((x, f(x))\). Key features include intercepts, asymptotes, extrema, and end behavior.

Example: Graphing \(f(x) = x^2\) produces a parabola opening upward with vertex at the origin.

Graphing Trig Transformations

The process of sketching \(y = A \cdot \text{trig}(B(x - C)) + D\) by applying amplitude scaling, period change from \(B\), phase shift \(C\), and vertical shift \(D\) to the parent trig graph.

Example: \(y = 2\sin(\pi(x - 1)) + 3\) has amplitude \(2\), period \(2\), shift right \(1\), up \(3\).

Growth Factor

The multiplier per unit of input in an exponential growth model: if \(f(x) = a \cdot b^x\) with \(b > 1\), then \(b\) is the growth factor and \(b - 1\) is the growth rate as a decimal.

Example: A population growing by \(5\%\) per year has growth factor \(1.05\).

Half-Life

The time required for an exponentially decaying quantity to reduce to half its initial value. For \(A(t) = A_0 e^{-kt}\), half-life is \(t_{1/2} = \ln(2)/k\).

Example: Carbon-14 has a half-life of approximately \(5{,}730\) years.

Holes in Graphs

A removable discontinuity at \(x = a\) where the factor \((x - a)\) cancels in both numerator and denominator. The graph has a missing point rather than an asymptote.

Example: \(f(x) = \tfrac{(x - 1)(x + 2)}{x - 1}\) has a hole at \(x = 1\).

Horizontal Asymptotes

A horizontal line \(y = L\) that the graph of \(f\) approaches as \(x \to \pm\infty\), meaning \(f(x) \to L\) for large \(|x|\).

Example: \(f(x) = \tfrac{1}{x}\) has horizontal asymptote \(y = 0\).

Horizontal Compression

The transformation \(y = f(bx)\) with \(b > 1\): every \(x\)-coordinate is divided by \(b\), squeezing the graph horizontally toward the \(y\)-axis.

Example: \(y = \sin(2x)\) is a horizontal compression of \(\sin x\) by factor \(\tfrac{1}{2}\).

Horizontal Line Test

A graphical test: \(f\) is one-to-one (and therefore has an inverse) if and only if no horizontal line meets its graph at more than one point.

Horizontal Stretch

The transformation \(y = f\left(\tfrac{x}{b}\right)\) with \(b > 1\): every \(x\)-coordinate is multiplied by \(b\), widening the graph horizontally.

Example: \(y = \sin\left(\tfrac{x}{2}\right)\) is a horizontal stretch of \(\sin x\) by factor \(2\).

Horizontal Translation

The transformation \(y = f(x - h)\), which shifts the graph of \(f\) by \(h\) units to the right (or \(|h|\) units left if \(h < 0\)).

Example: \(y = (x - 3)^2\) is \(y = x^2\) shifted \(3\) right.

Identity Matrix

The square matrix \(I_n\) with \(1\)s on the main diagonal and \(0\)s elsewhere. It acts as the multiplicative identity: \(AI = IA = A\) when dimensions allow.

Example: \(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}\).

Imaginary Unit

The number \(i\), defined by \(i^2 = -1\). It is the fundamental building block of complex numbers and allows square roots of negatives to be expressed.

Example: \(\sqrt{-9} = 3i\).

Increasing Functions

A function \(f\) is increasing on an interval if \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\) for all \(x_1, x_2\) in the interval. The graph rises from left to right.

Example: \(f(x) = 2x + 1\) is increasing on \(\mathbb{R}\).

Increasing Rate of Change

A situation in which successive average rates of change are themselves increasing as one moves along the domain, indicating a concave-up shape.

Example: Compound interest growth produces increasing rates of change over time.

Independent Variable

The variable whose value is chosen freely and which determines the value of the dependent variable. Usually plotted on the horizontal axis.

Example: In \(d = 5t\), time \(t\) is the independent variable.

Input and Output

The input is a value fed into a function (from the domain); the output is the resulting value produced by the function (in the range). A function pairs each input with exactly one output.

Example: For \(f(x) = x + 5\), input \(3\) produces output \(8\).

Intercepts

The points where the graph of a function crosses a coordinate axis. \(x\)-intercepts occur where \(f(x) = 0\); the \(y\)-intercept is the value \(f(0)\).

Example: The graph of \(y = 2x - 6\) has \(x\)-intercept \((3,0)\) and \(y\)-intercept \((0,-6)\).

Intermediate Value Theorem

A theorem stating that a continuous function on \([a, b]\) takes every value between \(f(a)\) and \(f(b)\). Useful for proving existence of roots without finding them explicitly.

Example: Since \(f(x) = x^2 - 2\) is continuous with \(f(1) = -1\) and \(f(2) = 2\), there exists \(c \in (1, 2)\) where \(f(c) = 0\).

Interpolation

Using a fitted model or known values to estimate the dependent variable at an input lying within the observed data range. Usually more reliable than extrapolation.

Interval Notation

A compact way of describing a set of real numbers using brackets: \([a, b]\) includes endpoints, \((a, b)\) excludes them, and \(\infty\) is always open.

Example: \([0, 5)\) means all real \(x\) with \(0 \leq x < 5\).

Inverse Function Definition

For a one-to-one function \(f\), the function \(f^{-1}\) satisfying \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f\), and \(f(f^{-1}(y)) = y\) for all \(y\) in the range.

Example: If \(f(x) = 2x + 3\), then \(f^{-1}(x) = \tfrac{x - 3}{2}\).

Inverse Function Graphs

The graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y = x\), because \((a, b)\) on \(f\) corresponds to \((b, a)\) on \(f^{-1}\).

Example: The graphs of \(y = e^x\) and \(y = \ln x\) are mirror images across \(y = x\).

Inverse of a Matrix

For a square matrix \(A\) with nonzero determinant, the matrix \(A^{-1}\) satisfying \(A A^{-1} = A^{-1} A = I\). For \(2\times 2\): \(A^{-1} = \tfrac{1}{\det A}\begin{bmatrix} d & -b \\ -c & a\end{bmatrix}\).

Example: \(A = \begin{bmatrix} 2 & 1 \\ 1 & 1\end{bmatrix}\) has \(A^{-1} = \begin{bmatrix} 1 & -1 \\ -1 & 2\end{bmatrix}\).

Inverse Trig Domains

The input sets of inverse trig functions: \(\arcsin\) and \(\arccos\) require \(x \in [-1, 1]\); \(\arctan\) accepts all real \(x\); inverse reciprocal functions have corresponding restrictions.

Inverse Trig Graphs

Graphs obtained by reflecting restricted trig graphs across \(y = x\). They are increasing (for arcsin, arctan) or decreasing (for arccos) within their ranges.

Example: The graph of \(y = \arctan x\) has horizontal asymptotes \(y = \pm \pi/2\).

Inverse Trig Ranges

The output sets (restricted to make the inverse single-valued): \(\arcsin: [-\pi/2, \pi/2]\); \(\arccos: [0, \pi]\); \(\arctan: (-\pi/2, \pi/2)\).

Leading Coefficient

The coefficient \(a_n\) of the highest-degree term in a polynomial. Its sign and magnitude control end behavior and vertical scaling.

Example: The leading coefficient of \(P(x) = -2x^3 + x\) is \(-2\).

Leading Term Test

A rule for determining polynomial end behavior from the leading term \(a_n x^n\): the sign of \(a_n\) and the parity of \(n\) dictate the left and right tail directions.

Example: For \(P(x) = 2x^4 - \dots\), both ends rise because \(a_n > 0\) and \(n\) is even.

Limaçons

A polar curve with equation \(r = a + b\cos(\theta)\) or \(r = a + b\sin(\theta)\). The shape varies from dimpled to cardioid to looped depending on the ratio \(a/b\).

Example: \(r = 2 + \cos(\theta)\) is a limaçon with an inner loop when \(|b| > |a|\).

Linear Function Definition

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

Example: \(f(x) = 3x - 2\) is linear with slope \(3\).

Linear Rate of Change

A rate of change that is constant across the entire domain, characteristic of linear functions. The slope equals the rate of change at every point.

Example: A car traveling at a constant \(60\) mph has a linear rate of change of position.

Linear Regression

A statistical procedure that fits a line \(y = mx + b\) to bivariate data by minimizing the sum of squared residuals, producing the best-fitting linear model.

Example: Given data pairs of study time and test score, linear regression might yield \(y = 5.2x + 60\).

Linear Transformations

Functions from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) that preserve addition and scalar multiplication. Every linear transformation can be represented by a matrix acting on coordinate vectors.

Example: Rotation, reflection, and scaling in the plane are linear transformations.

Linearizing Exp Data

Transforming data that follow \(y = a b^x\) by taking the logarithm of \(y\), producing \(\log y = \log a + x \log b\), which is linear in \(x\). Enables linear-regression fitting of the exponent.

Example: If \(\log y\) vs \(x\) lies on a line of slope \(0.30\), then \(b \approx 10^{0.30} \approx 2\).

Local Maximum

A point where the function value is larger than all nearby values. A local max occurs where \(f'\) changes from positive to negative.

Example: \(f(x) = -x^2\) has a local (and global) maximum at \(x = 0\).

Local Minimum

A point where the function value is smaller than all nearby values. A local min occurs where \(f'\) changes from negative to positive.

Example: \(f(x) = x^2\) has a local (and global) minimum at \(x = 0\).

Log-Log Plots

A graph with logarithmic scales on both axes. Power functions \(y = a x^k\) appear as straight lines with slope \(k\) on a log-log plot.

Example: Plotting \(y = 5 x^2\) on log-log axes yields a line of slope \(2\).

Logarithm Definition

The inverse of an exponential: \(\log_b x = y\) means \(b^y = x\), defined for \(x > 0\), \(b > 0\), \(b \neq 1\). It answers the question "to what power must \(b\) be raised to give \(x\)?"

Example: \(\log_2 8 = 3\) because \(2^3 = 8\).

Logarithmic Equations

Equations in which the unknown appears inside a logarithm, solved by applying log properties, converting to exponential form, and checking domain validity.

Example: \(\log_2(x + 1) = 3\) gives \(x + 1 = 8\), so \(x = 7\).

Logarithmic Form

The equation \(\log_b x = y\), which is equivalent to the exponential form \(b^y = x\). Rewriting in logarithmic form lets us isolate an unknown exponent.

Example: \(2^5 = 32\) in logarithmic form is \(\log_2 32 = 5\).

Logarithmic Graphs

Sketches of \(y = \log_b x\) identifying the \(x\)-intercept \((1, 0)\), the vertical asymptote \(x = 0\), and the domain \(x > 0\). Shape is increasing for \(b > 1\), decreasing for \(0 < b < 1\).

Logarithmic Modeling

Using logarithmic functions to represent phenomena that grow quickly at first and then level off, such as reaction time, perception scales, and rate of information accumulation.

Example: Perceived sound loudness in decibels uses a logarithmic model of sound intensity.

Logarithmic Scale

A scale where equal distances represent equal ratios rather than equal differences. Used in decibels, pH, and the Richter scale to handle quantities spanning many orders of magnitude.

Logarithmic Transformations

Shifts, stretches, and reflections applied to \(y = \log_b x\), producing \(y = a \log_b(k(x - h)) + c\). The vertical asymptote moves with the horizontal shift \(h\).

Example: \(y = \log_2(x - 3) + 1\) shifts the parent right \(3\) and up \(1\).

Long Division of Polynomials

An algorithm mirroring long division of integers, producing a quotient and remainder when one polynomial is divided by another: \(P(x) = D(x)Q(x) + R(x)\) with \(\deg R < \deg D\).

Example: \((x^3 - 1) \div (x - 1) = x^2 + x + 1\) with remainder \(0\).

Magnitude of a Vector

The length of a vector, calculated for \(\vec{v} = \langle a, b \rangle\) as \(|\vec{v}| = \sqrt{a^2 + b^2}\). Extended to three dimensions by including \(c\).

Example: \(|\langle 3, 4 \rangle| = 5\).

Matrix Addition

The operation combining two matrices of the same dimensions by adding corresponding entries. \((A + B)_{ij} = A_{ij} + B_{ij}\).

Example: \(\begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix} + \begin{bmatrix} 0 & 5 \\ 6 & 1\end{bmatrix} = \begin{bmatrix} 1 & 7 \\ 9 & 5\end{bmatrix}\).

Matrix Applications

Uses of matrices in solving linear systems, performing geometric transformations, encoding graphs and networks, representing population transitions, and storing tabular data.

Example: A population transition matrix predicts next-year distribution from this year's.

Matrix Definition

A rectangular array of numbers arranged in rows and columns, typically enclosed in brackets. Matrices encode linear transformations, systems of equations, and organized data.

Example: \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) is a \(2 \times 2\) matrix.

Matrix Dimensions

The size of a matrix, given as rows \(\times\) columns. A matrix with \(m\) rows and \(n\) columns is called an \(m \times n\) matrix.

Example: A \(3 \times 2\) matrix has three rows and two columns.

Matrix Multiplication

The operation producing a matrix \(AB\) whose \((i, j)\) entry is the dot product of row \(i\) of \(A\) with column \(j\) of \(B\). Requires the number of columns of \(A\) to equal the number of rows of \(B\).

Example: \(\begin{bmatrix} 1 & 2\end{bmatrix} \begin{bmatrix} 3 \\ 4\end{bmatrix} = [11]\).

Matrix Subtraction

The operation combining two matrices of the same dimensions by subtracting corresponding entries. \((A - B)_{ij} = A_{ij} - B_{ij}\).

Example: \(\begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 4 & 4\end{bmatrix}\).

Midline

The horizontal line halfway between the maximum and minimum of a sinusoidal function. For \(y = A\sin(Bx) + D\), the midline is \(y = D\).

Example: \(y = 2\sin x + 5\) has midline \(y = 5\).

Model Assumptions

The conditions or simplifications built into a mathematical model (e.g., constant rate, closed system, ideal conditions) that must hold for the model's conclusions to be trustworthy.

Example: A simple exponential growth model assumes unlimited resources.

Model Validation

The process of assessing whether a model adequately describes data and predicts new cases, using residual analysis, error metrics, and out-of-sample testing.

Example: Comparing model predictions to held-out data points checks validation.

Monomial

A polynomial with exactly one term, of the form \(a x^n\) where \(a\) is a constant and \(n\) a non-negative integer.

Example: \(7x^3\) and \(-5\) are monomials.

Multiple Angle Equations

Equations involving trig functions of \(kx\) (e.g., \(\sin(2x) = \tfrac{1}{2}\)), whose solutions are found by solving for \(kx\) and then dividing to recover \(x\), preserving all solutions in the requested interval.

Example: \(\sin(2x) = 1/2\) on \([0, 2\pi)\) yields \(x = \pi/12, 5\pi/12, 13\pi/12, 17\pi/12\).

Multiple Choice Strategy

An approach to AP multiple-choice questions that includes reading all options before choosing, eliminating clearly wrong answers, plugging in choices when stuck, and pacing to avoid getting stuck on one item.

Multiple Polar Represent

A single point in the plane has infinitely many polar representations, obtained by adding multiples of \(2\pi\) to \(\theta\) or by allowing \(r\) to be negative (and adjusting \(\theta\) by \(\pi\)).

Example: \((2, \pi/4)\), \((2, \pi/4 + 2\pi)\), and \((-2, 5\pi/4)\) describe the same point.

Multiplicity of Zeros

The number of times a factor \((x - a)\) appears in the factorization of a polynomial. A zero of even multiplicity touches the \(x\)-axis; odd multiplicity crosses it.

Example: In \((x - 2)^3(x + 1)\), root \(2\) has multiplicity \(3\).

Natural Base e

The irrational constant \(e \approx 2.71828\ldots\), defined as \(\lim_{n \to \infty} \left(1 + \tfrac{1}{n}\right)^n\). It is the natural base for exponentials and logarithms.

Natural Logarithm

The logarithm to base \(e\), denoted \(\ln x\). It is the inverse of the natural exponential \(e^x\).

Example: \(\ln e^5 = 5\).

Nonlinear Rate of Change

A rate of change that varies across the domain, characteristic of functions that are not linear. Different intervals yield different average rates.

Example: The function \(f(x) = x^2\) has average rate of change \(3\) on \([1,2]\) but \(5\) on \([2,3]\).

Number Line

A straight line on which every point corresponds to exactly one real number, providing a geometric model of \(\mathbb{R}\). Positive numbers lie to the right of \(0\), negatives to the left.

Example: The solution to \(|x| < 2\) is the interval \((-2, 2)\) shown on a number line.

Odd Functions

A function satisfying \(f(-x) = -f(x)\) for all \(x\) in its domain. Its graph has rotational symmetry of order two about the origin.

Example: \(f(x) = x^3\) and \(f(x) = \sin x\) are odd.

One-to-One Functions

A function \(f\) such that different inputs always give different outputs: \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\). Only one-to-one functions have inverses.

Example: \(f(x) = 2x + 1\) is one-to-one; \(f(x) = x^2\) on \(\mathbb{R}\) is not.

Order of Operations

The agreed-upon sequence for evaluating arithmetic expressions: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right). Often remembered as PEMDAS.

Example: \(2 + 3 \cdot 4 = 2 + 12 = 14\), not \(20\).

Ordered Pairs

A pair of elements \((a, b)\) in which the order matters, used to represent a point in the coordinate plane or an element of a relation. \((a, b) \neq (b, a)\) unless \(a = b\).

Example: \((3, -2)\) locates a point three units right and two units below the origin.

Parabola

The U-shaped graph of a quadratic function. It opens upward if \(a > 0\) and downward if \(a < 0\), and is symmetric about its axis of symmetry.

Parallel Lines

Two distinct lines in the plane that never intersect. Non-vertical parallel lines have equal slopes: \(m_1 = m_2\).

Example: \(y = 2x + 1\) and \(y = 2x - 4\) are parallel.

Parameter

A quantity in a formula that is held fixed in a particular case but can be varied to generate a family of related functions or curves. In parametric equations, the parameter (often \(t\)) traces the curve.

Example: In \(x = \cos t, y = \sin t\), the parameter \(t\) traces out the unit circle.

Parametric Equations

Equations expressing coordinates as functions of a parameter: \(x = f(t)\), \(y = g(t)\). Together they define a parametric curve.

Example: \(x = t, y = t^2\) parametrizes the parabola \(y = x^2\).

Parametric Graphs

Curves produced by plotting the points \((f(t), g(t))\) as \(t\) varies through an interval. They may include loops, cusps, and self-intersections that functions of \(x\) cannot describe.

Example: \(x = t^2 - 1, y = t^3 - t\) produces a curve with a loop.

Parametric Modeling

Using parametric equations to represent real-world motion and relationships in which both coordinates change with a common parameter, such as time.

Example: Modeling the path of a satellite with \(x(t) = R\cos(\omega t)\), \(y(t) = R\sin(\omega t)\).

Parametric Motion

The motion of a point along a parametric curve as the parameter (often time) increases. Direction and speed are encoded in the parametric equations.

Example: \(x = \cos t, y = \sin t\) describes counterclockwise motion around the unit circle.

Parent Functions

The simplest forms of a family of functions (e.g., \(y = x\), \(y = x^2\), \(y = |x|\), \(y = \sqrt{x}\), \(y = \tfrac{1}{x}\), \(y = b^x\), \(y = \log x\), \(y = \sin x\)) from which others are obtained by transformation.

Example: The parent of \(y = 3(x-1)^2 + 5\) is \(y = x^2\).

Percent Rate of Change

The relative change of a quantity expressed as a percentage per unit time. For exponential models, this rate is constant and relates directly to the growth or decay factor.

Example: A growth factor of \(1.03\) corresponds to a \(3\%\) percent rate of change per unit.

Period

The smallest positive value \(T\) for which \(f(x + T) = f(x)\). For \(y = \sin(Bx)\), \(T = 2\pi/|B|\).

Example: \(y = \sin(2x)\) has period \(\pi\).

Period from Data

The period extracted from data by measuring the distance (in the input variable) between repeating features such as consecutive maxima or zeros.

Example: If successive high tides are \(12.4\) hours apart, the period is \(12.4\) hours.

Period of Tangent

The tangent function has period \(\pi\) (not \(2\pi\)), because \(\tan(x + \pi) = \tan x\). For \(y = \tan(Bx)\), the period is \(\pi/|B|\).

Example: \(y = \tan(2x)\) has period \(\pi/2\).

Periodic Data Analysis

Examining data that repeat at regular intervals to extract period, amplitude, midline, and phase, typically prior to fitting a sinusoidal model.

Example: Monthly rainfall over several years reveals an annual period.

Perpendicular Lines

Two lines that meet at a right angle. Non-vertical perpendicular lines satisfy \(m_1 \cdot m_2 = -1\).

Example: \(y = 3x\) and \(y = -\tfrac{1}{3}x + 2\) are perpendicular.

Petal Count of Roses

For \(r = a\sin(n\theta)\) or \(r = a\cos(n\theta)\): \(n\) petals if \(n\) is odd, \(2n\) petals if \(n\) is even, due to how the curve traces out as \(\theta\) sweeps through \([0, 2\pi)\).

Example: \(r = \sin(4\theta)\) has \(8\) petals.

Phase Shift

The horizontal translation of a sinusoidal function. For \(y = A\sin(B(x - C)) + D\), the phase shift is \(C\) units (right if positive, left if negative).

Example: \(y = \sin(x - \pi/4)\) has phase shift \(\pi/4\) to the right.

Piecewise Functions

A function defined by different formulas on different parts of its domain, typically using a brace notation listing each formula with its interval.

Example: \(f(x) = \begin{cases} x + 1, & x < 0 \\ x^2, & x \geq 0 \end{cases}\)

Point-Slope Form

The equation of a line through point \((x_1, y_1)\) with slope \(m\), written \(y - y_1 = m(x - x_1)\). Convenient when a point and slope are given directly.

Example: A line through \((2, 5)\) with slope \(3\): \(y - 5 = 3(x - 2)\).

Polar Coordinate System

A coordinate system using distance from the origin (\(r\)) and angle from the polar axis (\(\theta\)) to locate points in the plane. Well-suited for curves with radial symmetry.

Example: The point \((1, 1)\) in Cartesian is \((\sqrt 2, \pi/4)\) in polar coordinates.

Polar Function Definition

A function \(r = f(\theta)\) that expresses radial distance as a function of angle. Its graph is the set of polar points \((f(\theta), \theta)\).

Example: \(r = 2\cos\theta\) defines a polar function whose graph is a circle.

Polar Graph Intersections

Points where two polar curves \(r_1 = f(\theta)\) and \(r_2 = g(\theta)\) coincide, found by solving \(f(\theta) = g(\theta)\) and also checking the pole separately, since a point may have multiple polar representations.

Example: \(r = 1\) and \(r = 2\cos\theta\) intersect at \(\theta = \pm \pi/3\).

Polar Graphs

Curves produced by plotting \(r = f(\theta)\) as \(\theta\) sweeps through an interval. Common examples include circles, cardioids, limaçons, roses, and spirals.

Polar Point Plotting

The procedure of locating a polar point \((r, \theta)\): rotate from the polar axis by \(\theta\), then move distance \(r\) along that ray (backward if \(r < 0\)).

Example: \((3, \pi/3)\) is plotted \(60^\circ\) counterclockwise from the polar axis, \(3\) units out.

Polar Rate of Change

The rate at which \(r\) changes with \(\theta\), \(dr/d\theta\). It controls how rapidly the curve moves toward or away from the pole as \(\theta\) increases.

Example: For \(r = \theta\), \(dr/d\theta = 1\), producing a spiral that moves outward at constant radial speed.

Polar to Rectangular

The conversion \(x = r\cos\theta\), \(y = r\sin\theta\), turning polar coordinates into Cartesian.

Example: \((4, \pi/2) \to (0, 4)\).

Polynomial Addition

The operation of combining two polynomials by adding coefficients of like terms (terms with the same variable and exponent).

Example: \((2x^2 + 3x - 1) + (x^2 - 4x + 5) = 3x^2 - x + 4\).

Polynomial Definition

A function of the form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where the coefficients \(a_i\) are real constants and \(n\) is a non-negative integer.

Example: \(P(x) = 2x^3 - 5x + 1\).

Polynomial End Behavior

The behavior of \(P(x)\) as \(x \to +\infty\) and \(x \to -\infty\). For polynomials, it is determined by the leading term \(a_n x^n\).

Example: For \(f(x) = -x^3\), \(f(x) \to -\infty\) as \(x \to \infty\) and \(f(x) \to \infty\) as \(x \to -\infty\).

Polynomial Graphs

The graph of a polynomial function is smooth and continuous, with end behavior governed by the leading term and interior shape controlled by zeros and their multiplicities.

Example: \(P(x) = (x-1)(x+2)^2\) has a crossing at \(x=1\) and a touch at \(x=-2\).

Polynomial Multiplication

The operation of combining two polynomials by applying the distributive property so that every term of one is multiplied by every term of the other, then collecting like terms.

Example: \((x + 2)(x - 3) = x^2 - x - 6\).

Position Functions

Functions \(x(t)\) and \(y(t)\) giving the location of a moving object at time \(t\). Together they form a vector-valued position \((x(t), y(t))\).

Example: \(x(t) = 2t, y(t) = -16t^2 + 80t\) models a thrown ball.

Positive and Negative Regions

Intervals of the domain where the function's output is positive (graph above the \(x\)-axis) or negative (graph below). Boundaries occur at \(x\)-intercepts and vertical asymptotes.

Example: \(f(x) = x^2 - 4\) is positive on \((-\infty,-2)\cup(2,\infty)\) and negative on \((-2,2)\).

Power Regression

A statistical procedure that fits a function of the form \(y = a x^k\) to data, typically after linearizing by taking logarithms of both variables.

Example: Kepler's third law \(T^2 \propto a^3\) is revealed by power regression on planetary data.

Power Rule for Logs

The identity \(\log_b(x^n) = n \log_b x\), valid for \(x > 0\). A logarithm of a power equals the exponent times the logarithm.

Example: \(\log_5(25^3) = 3 \log_5 25 = 3 \cdot 2 = 6\).

Product Rule for Logs

The identity \(\log_b(xy) = \log_b x + \log_b y\), valid for \(x, y > 0\). A logarithm of a product equals the sum of the logarithms.

Example: \(\log_2(8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5\).

Projectile Motion Model

A parametric model of an object under gravity: \(x(t) = v_0\cos(\theta) t\), \(y(t) = v_0\sin(\theta) t - \tfrac{1}{2}g t^2\), given initial speed \(v_0\) and launch angle \(\theta\).

Example: A ball thrown at \(20\) m/s at \(45^\circ\) follows a parabolic path.

Properties of Logarithms

A set of identities including \(\log_b(xy) = \log_b x + \log_b y\), \(\log_b(x/y) = \log_b x - \log_b y\), and \(\log_b(x^n) = n \log_b x\), valid for \(x, y > 0\).

Pythagorean Identity

The identity \(\sin^2\theta + \cos^2\theta = 1\), true for all real \(\theta\). It follows directly from the unit-circle definition.

Example: Dividing by \(\cos^2\theta\) gives \(\tan^2\theta + 1 = \sec^2\theta\).

Quadrantal Angles

Angles whose terminal sides lie on a coordinate axis: \(0, \pi/2, \pi, 3\pi/2\) and their coterminal equivalents. Their unit-circle coordinates are \((\pm 1, 0)\) or \((0, \pm 1)\).

Example: \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\).

Quadratic Formula

The formula giving the roots of \(ax^2 + bx + c = 0\): \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It solves any quadratic equation, real or complex.

Quadratic Function Definition

A polynomial function of degree two: \(f(x) = ax^2 + bx + c\) with \(a \neq 0\). Its graph is a parabola.

Example: \(f(x) = 2x^2 - 3x + 1\) is quadratic.

Quotient Identities

The identities \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\) and \(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\), expressing tangent and cotangent in terms of sine and cosine.

Quotient Rule for Logs

The identity \(\log_b(x/y) = \log_b x - \log_b y\), valid for \(x, y > 0\). A logarithm of a quotient equals the difference of the logarithms.

Example: \(\log_{10}(100/10) = 2 - 1 = 1\).

Radian Measure

An angle measure where one radian equals the angle subtended by an arc equal in length to the radius. One full revolution is \(2\pi\) radians.

Example: \(180^\circ = \pi\) radians, so \(90^\circ = \pi/2\) radians.

Range

The set of all output values a function actually produces as its input varies over the domain. The range is a subset of the codomain.

Example: The range of \(f(x) = x^2\) with domain \(\mathbb{R}\) is \(\{y \in \mathbb{R} : y \geq 0\}\).

Rate of Change Units

The units attached to a rate of change, expressed as output-units per input-unit. They carry the physical or contextual meaning of the rate.

Example: If distance is in miles and time in hours, then rate of change has units of miles per hour.

Rational Expressions

An algebraic expression that is a ratio of two polynomials, such as \(\dfrac{x^2 - 1}{x + 3}\). Operations follow rules analogous to arithmetic with fractions.

Example: \(\dfrac{2x}{x^2 + 1}\) is a rational expression.

Rational Function Definition

A function of the form \(f(x) = \tfrac{P(x)}{Q(x)}\), where \(P\) and \(Q\) are polynomials and \(Q(x) \neq 0\). Its domain excludes the zeros of \(Q\).

Example: \(f(x) = \tfrac{x + 1}{x^2 - 4}\) is rational with domain \(x \neq \pm 2\).

Rational Function Domain

The set of all real numbers except those that make the denominator zero. Excluded values correspond to vertical asymptotes or holes.

Example: The domain of \(f(x) = \tfrac{1}{x^2 - 4}\) is \(x \neq \pm 2\).

Rational Function Graphs

Sketches of \(f(x) = \tfrac{P(x)}{Q(x)}\) built by identifying intercepts, vertical and horizontal (or slant) asymptotes, holes, and behavior on each interval between discontinuities.

Rational Function Zeros

The \(x\)-values for which a rational function \(f(x) = \tfrac{P(x)}{Q(x)}\) equals zero. They are the zeros of \(P(x)\) that are not also zeros of \(Q(x)\).

Example: \(f(x) = \tfrac{x-2}{x+1}\) has a zero at \(x = 2\).

Rational Inequalities

Inequalities of the form \(\tfrac{P(x)}{Q(x)} > 0\) (or \(\geq, <, \leq\)), typically solved with a sign chart over the zeros of \(P\) and \(Q\), respecting excluded domain values.

Example: \(\tfrac{x-1}{x+2} > 0\) has solution \(x < -2\) or \(x > 1\).

Real Number System

The set \(\mathbb{R}\) consisting of all rational and irrational numbers, corresponding to every point on the real number line. It includes natural numbers, integers, rational numbers, and irrationals such as \(\pi\) and \(\sqrt{2}\).

Example: \(-3\), \(0\), \(\tfrac{1}{2}\), \(\sqrt{2}\), and \(\pi\) are all real numbers.

Real Zeros

Zeros of a polynomial that are real numbers, visible as \(x\)-intercepts on the graph. A polynomial of degree \(n\) has at most \(n\) real zeros.

Example: \(P(x) = x^3 - x\) has real zeros \(-1, 0, 1\).

Reciprocal Identities

The identities \(\csc\theta = 1/\sin\theta\), \(\sec\theta = 1/\cos\theta\), \(\cot\theta = 1/\tan\theta\), relating each trig function to its reciprocal.

Reciprocal Trig Functions

The functions secant, cosecant, and cotangent, defined as reciprocals of cosine, sine, and tangent respectively: \(\sec\theta = 1/\cos\theta\), \(\csc\theta = 1/\sin\theta\), \(\cot\theta = 1/\tan\theta\).

Rectangular to Polar

The conversion \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan(y/x)\) (adjusted for quadrant), turning Cartesian coordinates into polar.

Example: \((1, \sqrt 3) \to (r, \theta) = (2, \pi/3)\).

Recursive Formulas

A rule that defines each term of a sequence using one or more preceding terms, along with initial values. Useful when a pattern is easier to express step-by-step.

Example: \(a_1 = 1\), \(a_n = a_{n-1} + 3\) defines \(1, 4, 7, 10, \ldots\).

Reference Angle

The positive acute angle between the terminal side of an angle in standard position and the nearest part of the \(x\)-axis. Used to relate any angle's trig values to those of an acute angle.

Example: The reference angle of \(210^\circ\) is \(30^\circ\).

Reflection Matrix

A matrix that reflects vectors across a line through the origin. Common examples: \(\begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}\) reflects across the \(x\)-axis, \(\begin{bmatrix} -1 & 0 \\ 0 & 1\end{bmatrix}\) across the \(y\)-axis.

Example: Reflecting \(\langle 3, 4\rangle\) across the \(x\)-axis gives \(\langle 3, -4\rangle\).

Reflection Over X-Axis

The transformation \(y = -f(x)\), which reflects the graph across the \(x\)-axis by negating every \(y\)-coordinate.

Example: \(y = -x^2\) is \(y = x^2\) reflected in the \(x\)-axis.

Reflection Over Y-Axis

The transformation \(y = f(-x)\), which reflects the graph across the \(y\)-axis by negating every \(x\)-coordinate.

Example: \(y = \sqrt{-x}\) is \(y = \sqrt{x}\) reflected in the \(y\)-axis.

Regression Models

Models produced by fitting parameters of a chosen function family (linear, quadratic, exponential, power, sinusoidal) to data by minimizing an error measure such as sum of squared residuals.

Example: Linear regression fits \(y = mx + b\) to bivariate data.

Remainder Theorem

When a polynomial \(P(x)\) is divided by \((x - a)\), the remainder equals \(P(a)\). It provides a quick evaluation method.

Example: Dividing \(P(x) = x^3 + 1\) by \((x - 2)\) gives remainder \(P(2) = 9\).

Residual Analysis

Inspecting the residuals (observed minus predicted values) for patterns; random scatter suggests a good model fit, while systematic patterns reveal missing structure.

Example: A U-shaped residual plot from a linear fit suggests a quadratic model is better.

Residuals

The signed differences between observed data values and the values predicted by a model: \(\text{residual} = y_{\text{obs}} - y_{\text{pred}}\). They measure how well the model fits each data point.

Example: If a model predicts \(y = 12\) and the observation is \(14\), the residual is \(+2\).

Restricting Domains

Limiting the domain of a non-one-to-one function to an interval on which it becomes one-to-one so that an inverse can be defined.

Example: Restricting \(f(x) = x^2\) to \(x \geq 0\) allows \(f^{-1}(x) = \sqrt{x}\).

Right Triangle Trig

The definitions of trigonometric ratios for acute angles in a right triangle: \(\sin = \text{opp}/\text{hyp}\), \(\cos = \text{adj}/\text{hyp}\), \(\tan = \text{opp}/\text{adj}\).

Example: In a 3-4-5 right triangle with the angle opposite side 3: \(\sin = 3/5\), \(\cos = 4/5\).

Rose Curves

A polar curve with equation \(r = a\cos(n\theta)\) or \(r = a\sin(n\theta)\) that forms petal shapes. Has \(n\) petals if \(n\) is odd, \(2n\) petals if \(n\) is even.

Example: \(r = \cos(3\theta)\) is a three-petaled rose.

Rotation Matrix

The \(2\times 2\) matrix \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}\), which rotates vectors in the plane counterclockwise by angle \(\theta\) about the origin.

Example: Rotating \(\langle 1, 0\rangle\) by \(\pi/2\) yields \(\langle 0, 1\rangle\).

Scalar Mult of Matrices

The operation of multiplying every entry of a matrix by a real number (scalar): \((kA)_{ij} = k \cdot A_{ij}\).

Example: \(3 \begin{bmatrix} 1 & -2 \\ 0 & 4\end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 0 & 12\end{bmatrix}\).

Scalar Multiplication

The operation of multiplying a vector by a real number (scalar), scaling its magnitude and (if negative) reversing direction: \(k\langle a, b\rangle = \langle ka, kb\rangle\).

Example: \(3 \langle 2, -1\rangle = \langle 6, -3\rangle\).

Secant Function

The reciprocal of cosine: \(\sec\theta = 1/\cos\theta\), defined wherever \(\cos\theta \neq 0\). Period \(2\pi\), range \((-\infty, -1] \cup [1, \infty)\).

Example: \(\sec(0) = 1\).

Secant Graph

The graph of \(y = \sec x\): U-shaped branches with period \(2\pi\), opening up where \(\cos x > 0\) and down where \(\cos x < 0\), with vertical asymptotes at zeros of \(\cos x\).

Secant Line

A line passing through two points on a curve. The slope of a secant line equals the average rate of change over that interval.

Example: The secant line through \((1, 1)\) and \((3, 9)\) on \(y = x^2\) has slope \(4\).

Sector Area

The area of a pie-slice region of a circle with radius \(r\) and central angle \(\theta\) in radians: \(A = \tfrac{1}{2} r^2 \theta\).

Example: A sector with \(r = 6\) and \(\theta = \pi/3\) has area \(A = \tfrac{1}{2}(36)(\pi/3) = 6\pi\).

Semi-Log Plot

A graph with a logarithmic scale on one axis (usually \(y\)) and a linear scale on the other. Exponential functions appear as straight lines on a semi-log plot.

Example: Plotting \(y = 3 \cdot 2^x\) on a semi-log plot yields a straight line.

Sequence Convergence

A sequence converges if its terms approach a finite limit \(L\) as \(n \to \infty\). Written \(\lim_{n\to\infty} a_n = L\).

Example: The sequence \(a_n = 1/n\) converges to \(0\).

Sequence Definition

An ordered list of numbers following a pattern, written as \(\{a_n\}\). Each number is called a term, and the domain is typically the natural numbers.

Example: The sequence \(a_n = 1/n\) is \(1, 1/2, 1/3, 1/4, \ldots\)

Sequence Divergence

A sequence diverges if it does not approach a finite limit. Its terms may grow without bound, oscillate, or behave chaotically.

Example: \(a_n = (-1)^n\) diverges because it oscillates between \(-1\) and \(1\).

Set-Builder Notation

A notation for defining a set by stating a property its members satisfy, written \(\{x : P(x)\}\) or \(\{x \mid P(x)\}\).

Example: \(\{x \in \mathbb{R} : x > 2\}\) is the set of all real numbers greater than \(2\).

Showing Work on Free Resp

The practice of presenting enough written reasoning on free-response questions that a reader can follow the method: definitions, setup equations, key algebra, and final answers with units and context.

Signs by Quadrant

The signs of \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) vary by quadrant. In Q1 all positive; Q2 only sine positive; Q3 only tangent positive; Q4 only cosine positive (remembered as "All Students Take Calculus").

Example: In Q3, \(\sin\theta < 0\) and \(\cos\theta < 0\), so \(\tan\theta > 0\).

Simplifying Rational Expr

The process of reducing a rational expression to lowest terms by factoring numerator and denominator and canceling common factors, noting any resulting domain restrictions.

Example: \(\dfrac{x^2 - 1}{x - 1} = x + 1\) for \(x \neq 1\).

Simplifying Trig Expr

Reducing a trigonometric expression to a shorter equivalent form using identities, factoring, and combining fractions.

Example: \(\dfrac{\sin^2\theta}{1 - \cos\theta} = 1 + \cos\theta\) by using \(\sin^2\theta = 1 - \cos^2\theta\).

Sine Function

The function \(\sin\theta\) defined as the \(y\)-coordinate of the point on the unit circle at angle \(\theta\). Periodic with period \(2\pi\) and range \([-1, 1]\).

Example: \(\sin(\pi/2) = 1\).

Sine Graph

The graph of \(y = \sin x\): a smooth wave oscillating between \(-1\) and \(1\), passing through the origin, with period \(2\pi\).

Sinusoidal Functions

Functions of the form \(y = A\sin(B(x - C)) + D\) or \(y = A\cos(B(x - C)) + D\), produced by transforming sine or cosine. They model periodic behavior.

Example: \(y = 2\sin(3x - \pi) + 1\) is a sinusoidal function.

Sinusoidal Modeling

Using sinusoidal functions \(y = A\sin(B(x - C)) + D\) to model real-world periodic phenomena such as tides, daylight hours, and seasonal temperature.

Example: Hours of daylight per day is modeled by a sinusoid with period \(365\).

Sinusoidal Regression

A statistical procedure that fits \(y = A\sin(B(x - C)) + D\) to data using nonlinear least squares, extracting amplitude, period, phase, and midline from observations.

Example: Fitting monthly temperature data to a sinusoid yields an annual cycle.

Slant Asymptotes

A slanted line \(y = mx + c\) (with \(m \neq 0\)) that the graph of a rational function approaches as \(x \to \pm\infty\). It occurs when \(\deg P = \deg Q + 1\).

Example: \(f(x) = \tfrac{x^2 + 1}{x}\) has slant asymptote \(y = x\).

Slope

The measure of steepness of a line, equal to the ratio of vertical change to horizontal change between any two points on the line.

Example: A line rising \(2\) units for every \(1\) unit to the right has slope \(m = 2\).

Slope Formula

The formula \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\), computing slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) with \(x_1 \neq x_2\).

Example: Between \((1, 2)\) and \((4, 11)\): \(m = \tfrac{11 - 2}{4 - 1} = 3\).

Slope-Intercept Form

The equation of a straight line written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. It makes both features immediately visible.

Example: \(y = -\tfrac{1}{2}x + 3\) has slope \(-\tfrac{1}{2}\) and \(y\)-intercept \(3\).

SOH-CAH-TOA

A mnemonic for right-triangle trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Solutions in an Interval

The specific solutions to a trig equation that lie within a given interval (often \([0, 2\pi)\) or \([0^\circ, 360^\circ)\)), selected from the general solution.

Example: \(\tan x = 1\) on \([0, 2\pi)\) gives \(x = \pi/4\) and \(x = 5\pi/4\).

Solving Trig Equations

Finding angles that satisfy a trigonometric equation, using identities, inverse trig functions, and knowledge of periodicity to list all solutions within the requested interval.

Example: \(2\sin x - 1 = 0\) on \([0, 2\pi)\) gives \(x = \pi/6\) or \(5\pi/6\).

Special Angles

Angles whose trig values are exactly expressible, typically \(0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\), and their multiples. Their exact coordinates form the backbone of the unit circle.

Example: \(\sin(\pi/6) = 1/2\) and \(\cos(\pi/6) = \sqrt{3}/2\).

Standard Form of a Line

The equation of a straight line written as \(Ax + By = C\), where \(A\), \(B\), \(C\) are constants and \(A\), \(B\) not both zero. Useful for systems and for lines with undefined slope.

Example: \(2x - 3y = 6\) is in standard form.

Standard Form of Quadratic

The quadratic expressed as \(f(x) = ax^2 + bx + c\). This form makes the \(y\)-intercept (\(c\)) and the coefficients for the discriminant and quadratic formula explicit.

Example: \(f(x) = x^2 + 4x + 3\).

Standard Position

An angle positioned with its vertex at the origin and its initial side along the positive \(x\)-axis. Counterclockwise rotation is positive, clockwise is negative.

Example: A \(135^\circ\) angle in standard position has its terminal side in the second quadrant.

Sum and Difference Formulas

Identities expressing trig functions of \(A \pm B\) in terms of functions of \(A\) and \(B\). For example, \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\).

Example: \(\cos(75^\circ) = \cos(45^\circ + 30^\circ) = \tfrac{\sqrt 6 - \sqrt 2}{4}\).

Symmetry

A property describing graph invariance under reflection or rotation. Common forms include \(y\)-axis symmetry (even functions), origin symmetry (odd functions), and symmetry about a vertical line (parabolas).

Example: The parabola \(y = (x-3)^2\) is symmetric about the line \(x = 3\).

Symmetry of Polar Graphs

Tests for polar symmetry: replace \(\theta\) with \(-\theta\) (symmetry about polar axis), replace \(\theta\) with \(\pi - \theta\) (symmetry about \(\pi/2\) axis), replace \(r\) with \(-r\) (symmetry about pole).

Example: \(r = 1 + \cos\theta\) is symmetric about the polar axis.

Synthetic Division

A shorthand algorithm for dividing a polynomial by a linear factor \((x - a)\), using only the coefficients. Faster than long division when applicable.

Example: Synthetic division of \(x^3 - 2x + 5\) by \((x - 1)\) quickly yields the quotient and remainder.

Tangent Function

The function \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\), defined wherever \(\cos\theta \neq 0\). Periodic with period \(\pi\) and range \(\mathbb{R}\).

Example: \(\tan(\pi/4) = 1\).

Tangent Graph

The graph of \(y = \tan x\): a repeating curve with period \(\pi\), passing through the origin, with vertical asymptotes at \(x = \pi/2 + k\pi\).

Temperature Models

Sinusoidal models of temperature versus day of year, capturing seasonal variation with period \(365\), amplitude from half the annual range, and midline from the annual mean.

Example: \(T(d) = 20\sin\left(\tfrac{2\pi}{365}(d - 80)\right) + 50\).

Terminal Side

The ray of an angle in standard position after the rotation, distinguished from the initial side. It determines the quadrant and reference angle.

Example: The terminal side of a \(210^\circ\) angle lies in the third quadrant.

Tidal Models

Sinusoidal models of tide height versus time, with period roughly \(12.4\) hours, amplitude given by the tidal range, and midline given by mean sea level.

Example: \(h(t) = 3\sin\left(\tfrac{2\pi}{12.4} t\right) + 5\) models a tide with range \(6\) ft around mean level \(5\) ft.

Time Management on Exams

The planning and pacing of work on a timed exam, allocating minutes per question based on point values, reserving time for review, and skipping and returning rather than stalling.

Transformation Order

The conventional order of applying composite transformations: horizontal stretches/reflections, then horizontal translations, then vertical stretches/reflections, then vertical translations. Order matters because the result generally depends on it.

Trig Function Evaluation

The process of computing the value of a trigonometric function at a given angle, using unit-circle coordinates, special-angle values, reference angles, or technology.

Example: \(\sin(5\pi/6) = \sin(\pi/6) = 1/2\) via reference angle.

Trig Inequalities

Inequalities involving trig functions (e.g., \(\sin x > 1/2\)), solved by finding boundary solutions and testing intervals on the unit circle or graph.

Example: \(\cos x < 0\) on \([0, 2\pi)\) holds for \(x \in (\pi/2, 3\pi/2)\).

Trinomial

A polynomial with exactly three terms connected by addition or subtraction.

Example: \(x^2 + 5x + 6\) is a trinomial.

Turning Points

Points on a graph where the function changes from increasing to decreasing or vice versa. They are local maxima or local minima.

Example: The parabola \(y = x^2 - 4\) has one turning point at \((0, -4)\).

Unit Circle Coordinates

For an angle \(\theta\) in standard position, the point where its terminal side meets the unit circle has coordinates \((\cos\theta, \sin\theta)\).

Example: At \(\theta = \pi/4\), the unit-circle point is \((\sqrt 2/2, \sqrt 2/2)\).

Unit Circle Definition

The circle of radius \(1\) centered at the origin: \(x^2 + y^2 = 1\). It provides coordinates \((\cos\theta, \sin\theta)\) for each angle \(\theta\) in standard position.

Unit Vectors

Vectors of magnitude \(1\), used to specify direction. The standard unit vectors are \(\mathbf{i} = \langle 1, 0\rangle\) and \(\mathbf{j} = \langle 0, 1\rangle\). Any vector divided by its magnitude is a unit vector in the same direction.

Example: A unit vector in the direction of \(\langle 3, 4\rangle\) is \(\langle 3/5, 4/5\rangle\).

Variables and Expressions

A variable is a symbol representing a number that may change within a problem; an expression is a combination of numbers, variables, and operations that represents a value without an equality sign.

Example: In \(3x^2 - 5x + 2\), \(x\) is a variable and the whole string is an expression.

Vector Addition

The operation combining two vectors by adding their corresponding components: \(\langle a, b\rangle + \langle c, d\rangle = \langle a + c, b + d\rangle\). Geometrically, the tip-to-tail rule produces the sum.

Example: \(\langle 2, 3\rangle + \langle 4, -1\rangle = \langle 6, 2\rangle\).

Vector Applications

Uses of vectors in modeling physical quantities such as force, velocity, displacement, and navigation, where both size and direction are essential.

Example: Computing ground speed and heading of an airplane in a crosswind.

Vector Definition

A quantity with both magnitude and direction, often represented as an arrow or as an ordered list of components.

Example: A wind blowing east at \(10\) mph is a vector with magnitude \(10\) and direction east.

Vector Notation

Symbolic conventions for vectors, including boldface (\(\mathbf{v}\)), an arrow overhead (\(\vec{v}\)), or component form \(\langle a, b \rangle\). Components are enclosed in angle brackets to distinguish from points.

Example: \(\vec{v} = \langle 3, -4 \rangle\).

Vector Subtraction

The operation combining two vectors by subtracting corresponding components: \(\langle a, b\rangle - \langle c, d\rangle = \langle a - c, b - d\rangle\). Equivalent to adding the negative.

Example: \(\langle 5, 2\rangle - \langle 3, 6\rangle = \langle 2, -4\rangle\).

Velocity from Parametrics

The instantaneous velocity vector \((x'(t), y'(t))\) derived from parametric position functions. Its magnitude is speed; its direction is the direction of motion.

Example: If \(x(t) = t^2, y(t) = t^3\), velocity is \((2t, 3t^2)\).

Verifying Identities

The process of demonstrating that a trigonometric equation holds for all values in its domain by transforming one side into the other using known identities and algebraic manipulation.

Example: Verify \(\tan\theta + \cot\theta = \sec\theta\csc\theta\) by combining fractions and using the Pythagorean identity.

Vertex

The turning point of a parabola, where the axis of symmetry meets the curve. For \(f(x) = ax^2 + bx + c\), it lies at \(x = -\tfrac{b}{2a}\).

Example: \(y = x^2 - 4x + 1\) has vertex \((2, -3)\).

Vertex Form

The quadratic written as \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. This form reveals the turning point and axis of symmetry directly.

Example: \(f(x) = (x - 2)^2 - 5\) has vertex \((2, -5)\).

Vertical Asymptotes

A vertical line \(x = a\) that the graph of \(f\) approaches but never meets, typically where the function's output grows without bound as \(x \to a\).

Example: \(f(x) = \tfrac{1}{x - 2}\) has a vertical asymptote at \(x = 2\).

Vertical Compression

The transformation \(y = a f(x)\) with \(0 < a < 1\): every \(y\)-coordinate is multiplied by \(a\), shrinking the graph vertically toward the \(x\)-axis.

Example: \(y = \tfrac{1}{2}x^2\) is a vertical compression of \(y = x^2\) by factor \(\tfrac{1}{2}\).

Vertical Line Test

A graphical test: a curve in the Cartesian plane represents a function of \(x\) if and only if no vertical line intersects the curve at more than one point.

Example: A circle fails the vertical line test, so \(x^2 + y^2 = 1\) is not a function of \(x\).

Vertical Shift of Trig

The constant \(D\) in \(y = A\sin(B(x - C)) + D\), which moves the midline (and whole graph) up or down by \(D\) units.

Example: \(y = \cos x + 3\) shifts the cosine graph up by \(3\).

Vertical Stretch

The transformation \(y = a f(x)\) with \(a > 0\): every \(y\)-coordinate is multiplied by \(a\). Values \(a > 1\) stretch; \(0 < a < 1\) compress.

Example: \(y = 3x^2\) is a vertical stretch of \(y = x^2\) by factor \(3\).

Vertical Translation

The transformation \(y = f(x) + k\), which shifts the graph of \(f\) by \(k\) units upward (or \(|k|\) units downward if \(k < 0\)).

Example: \(y = x^2 + 4\) is \(y = x^2\) shifted \(4\) up.

Writing Transformed Eqns

The skill of producing an algebraic rule for a transformed function given a description (shifts, stretches, reflections) or a graph, typically in the form \(y = a f(b(x - h)) + k\).

Example: A graph of \(y = x^2\) shifted \(2\) right and \(5\) up gives \(y = (x - 2)^2 + 5\).

X-Intercept

The \(x\)-coordinate of a point where a graph crosses the \(x\)-axis; a value of \(x\) for which \(f(x) = 0\). Also called a root or zero of the function.

Example: The \(x\)-intercepts of \(y = x^2 - 4\) are \(x = \pm 2\).

Y-Intercept

The \(y\)-coordinate of the point where a graph crosses the \(y\)-axis; the value of \(f(0)\). For a line \(y = mx + c\), the \(y\)-intercept is \(c\).

Example: The \(y\)-intercept of \(y = 2x + 5\) is \(5\).

Zeros of Polynomials

Values of \(x\) for which \(P(x) = 0\). They correspond to \(x\)-intercepts (if real) and to factors \((x - a)\) via the factor theorem.

Example: The zeros of \(P(x) = x^2 - 9\) are \(x = \pm 3\).