Engineering Calculus I - Relations and Functions

These vocabulary flashcards cover foundational concepts from Engineering Calculus I, including relations, functions, linear and quadratic equations, intercepts, and symmetry.

Relation

A set of ordered pairs from a set $X$ to set $Y$ such that to each object in $X$, there corresponds at least one object in $Y$.

Domain of a Relation

The set that contains the first components (also called independent variables) of all the ordered pairs in a relation $R$, denoted by $\text{dom}R$.

Range of a Relation

The set that contains the second components (also called dependent variables) of all the ordered pairs in a relation $R$, denoted by $\text{ran}R$.

Function

A rule from a set $X$ to a set $Y$ that assigns to each object in $X$ exactly one object in set $Y$.

Abscissa

The technical term for the common $x$-component or first coordinate in an ordered pair $(x, y)$.

Vertical Line Test

A graphical test to determine if a relation is a function; a vertical line must intersect the graph at exactly one point only.

Ordinate

The technical term for the common $y$-coordinate or second component of an ordered pair $(x, y)$.

Even Function

A function where $f(-x) = f(x)$ for any $x$ in the domain, characterized by symmetry across the $y$-axis.

Odd Function

A function where $f(-x) = -f(x)$ for any $x$ in the domain, characterized by $180$-degree symmetry about the origin.

x-intercept

The abscissa of the point where a graph crosses the $x$-axis, obtained by setting $y = 0$ in the function's equation.

y-intercept

The ordinate of the point where a graph crosses the $y$-axis, obtained by setting $x = 0$ in the function's equation.

Polynomial Function

A function of degree $n$ expressed in the form $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a nonnegative integer.

Leading Coefficient

The coefficient $a_n$ of the highest power term in a polynomial function.

Leading Term

The term $a_n x^n$ associated with the highest power in a polynomial function.

Constant Function

A polynomial function of the form $f(x) = a_0$, whose graph is a horizontal line passing through the point $(0, a_0)$.

Linear Function

A polynomial function of the form $f(x) = mx + b$, whose graph is a straight line.

Quadratic Function

A polynomial function of the form $f(x) = ax^2 + bx + c$, whose graph is a parabola.

Slope (m)

A measure of steepness in a linear function where $m > 0$ indicates an increasing line and $m < 0$ indicates a decreasing line.

General Form of the Equation of a Line

The equation expressed as $Ax + By + C = 0$, where $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero.

Slope-Intercept Form

The equation of a line with slope $m$ and $y$-intercept $b$, written as $y = mx + b$.

Point-Slope Form

The equation of a line passing through point $(x_1, y_1)$ with slope $m$, written as $y - y_1 = m(x - x_1)$.

Intercept Form of the Equation of a Line

The equation of a line with $x$-intercept $a$ and $y$-intercept $b$, written as $\frac{x}{a} + \frac{y}{b} = 1$ where $a, b \neq 0$.

Vertex of a Quadratic Function

The highest or lowest point of a parabola, located at the coordinates $(\frac{-b}{2a}, \frac{4ac - b^2}{4a})$.