MATHEMATICS FOR ARCHITECTURES UNIT III: Functions and Their Graphs - Vocabulary

Vocabulary flashcards covering key concepts from the lecture notes on functions, graphs, coordinates, and related topics.

Function

A relation from a set A to a set B that assigns to each element in A exactly one element in B (domain A; inputs, range B; outputs).

Domain

The set of all inputs (x-values) for which the function is defined.

Range

The set of all outputs (y-values) produced by the function.

Implied Domain

The domain assumed from the algebraic expression when no domain is specified, typically all real numbers for which the expression is defined.

Input

An element of the domain; a value fed into the function.

Output

The value in the range produced by the function for a given input.

Ordered pair

A pair (x, y) representing an input-output relationship of a function; x is the input, y is the output.

Function Notation

The notation f(x) representing the value of the function f at x; f is the function name.

Rectangular (Cartesian) coordinates

A coordinate system using two perpendicular axes (x and y) with an origin to locate points in the plane.

Origin

The point (0, 0), where the x-axis and y-axis intersect.

Quadrant

One of four regions of the plane formed by the axes: I (x>0,y>0), II (x

Abscissa

The x-coordinate of a point, denoted x.

Ordinate

The y-coordinate of a point, denoted y.

Distance formula

d(P1, P2) = √[(x2 − x1)² + (y2 − y1)²].

Midpoint

The point M halfway between P1(x1,y1) and P2(x2,y2): M = ((x1+x2)/2, (y1+y2)/2).

Slope

The rate of change of a function: m = (y2 − y1)/(x2 − x1).

Slope-intercept form

The equation f(x) = mx + b, where m is the slope and b is the y-intercept.

y-intercept

The point where the graph crosses the y-axis; (0, b) in y = mx + b.

x-intercept

The point(s) where the graph crosses the x-axis; when y = 0.

Linear function

A function whose graph is a straight line, typically written as f(x) = mx + b.

Vertex

The highest or lowest point of a parabola; for f(x) = a(x−h)² + k, the vertex is (h, k).

Axis of symmetry

The vertical line x = h through which a parabola is symmetric.

Parabola

Graph of a quadratic function; generally a U-shaped curve.

Vertex form

Quadratic form f(x) = a(x − h)² + k, where (h, k) is the vertex.

General form (quadratic)

f(x) = ax² + bx + c, with a ≠ 0.

Standard form (quadratic)

Often refers to f(x) = a(x − h)² + k, a form emphasizing the vertex (h, k).

Leading coefficient

The coefficient a of the highest-degree term in a polynomial; determines end behavior and opening/width of a parabola.

Polynomial function

A function written as f(x) = an x^n + … + a1 x + a0 with n a nonnegative integer and an ≠ 0.

Monomial / Power function

A polynomial with a single term, f(x) = x^n (n > 0); a power function.

Rational function

A function that is the quotient of two polynomials: f(x) = N(x)/D(x), with D(x) not the zero polynomial.

Domain of a rational function

All real x except the zeros of the denominator D(x) (where D(x) = 0).

Vertical asymptote

A vertical line x = a where f(x) → ±∞ as x approaches a from either side.

Horizontal asymptote

A horizontal line y = b where f(x) → b as x → ±∞.

Absolute value function

f(x) = |x|, defined as x if x > 0 and −x if x < 0.

Piecewise-defined function

A function defined by two or more equations over a specified domain.

Domain

The set of all inputs for which the function is defined.

Range

The set of all outputs produced by the function.