Math 1030 Pre-Calculus Final Review Vocabulary

Comprehensive pre-calculus vocabulary cards covering functions, graphs, polynomials, logarithms, and trigonometry based on the Final Review guide.

Relation

A set of ordered pairs, such as ${(-1, 4), (0, 1), (1, 4), (0, 1), (2, 4)}$.

Function

A relation in which each element of the domain is paired with exactly one element in the range.

Domain

The set of all possible input values for a function, typically written in interval notation.

Range

The set of all possible output values that a function can produce.

Piecewise Function

A function defined by different formulas for different parts of its domain, such as $g(x) = \sqrt{x^2 + 4}$ if $x \le 3$ and $1 - x$ if x > 3.

Function Composition

The operation of applying one function to the results of another, denoted as $(f \circ g)(x) = f(g(x))$.

x-intercept

The point or points where the graph of a function crosses or touches the x-axis, occurring where $f(x) = 0$.

y-intercept

The point where the graph of a function crosses the y-axis, occurring where $x = 0$, denoted as $f(0)$.

One-to-one Function

A function where every output value corresponds to exactly one input value; these functions have an inverse $f^{-1}$.

Inverse Function

A function denoted as $f^{-1}(x)$ that 'undoes' the action of the original function $f(x)$, such that $f(f^{-1}(x)) = x$.

Difference Quotient

A formula used to represent the average rate of change of a function, typically written as $\frac{f(x+h) - f(x)}{h}$.

Parent Function

The simplest function in a family of functions, such as $f(x) = x^2$, $f(x) = \sqrt{x}$, or $f(x) = |x|$, from which transformations are made.

Standard Form (Quadratic)

The form of a quadratic function given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Vertex

The maximum or minimum point of a parabola, represented by the coordinate pair $(h, k)$.

Multiplicity

The number of times a specific zero occurs in a polynomial function, which affects the behavior of the graph at the x-axis.

Rational Zeros Theorem

A method to list all possible rational zeros of a polynomial by finding the factors of the constant term and the leading coefficient.

Vertical Asymptote

A vertical line that the graph of a rational function approaches but never reaches, occurring at values where the denominator is zero after simplification.

Horizontal Asymptote

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

Logarithm

The exponent to which a fixed base must be raised to produce a given number, such as $\log_3(81) = 4$ since $3^4 = 81$.

Natural Logarithm (ln)

A logarithm with base $e$, where $e \approx 2.718$, denoted as $\ln(x)$.

Exponential Equation

An equation in which variable expressions occur as exponents, such as $7^{5x} = 3^{x-2}$.

Radian

A unit of angular measure where a full circle is $2\pi$ radians, equivalent to $360^\circ$.

Coterminal Angles

Two angles in standard position that share the same terminal side, found by adding or subtracting multiples of $360^\circ$ or $2\pi$.

Arc Length

The distance along a curved arc, calculated using the formula $s = r\theta$, where $r$ is the radius and $\theta$ is in radians.

Trigonometric Functions

The six functions derived from a right triangle or unit circle: sin, cos, tan, csc, sec, and cot.

Reference Angle

The acute angle $\theta'$ formed by the terminal side of an angle and the horizontal x-axis.

Amplitude

The distance from the center line to the peak (or trough) of a periodic function like $\sin(x)$ or $\cos(x)$.

Period

The horizontal distance required for a periodic function to complete one full cycle, calculated as $\frac{2\pi}{b}$ for sine and cosine.

Double Angle Identity

A trigonometric identity used to calculate functions of twice an angle, such as $\sin(2\theta) = 2\sin\theta\cos\theta$.

Sum/Difference Identity

Identities used to find exact values of trigonometric functions for non-standard angles, such as $\sin(A + B)$ or $\cos(A - B)$.