HB 1.3 Recognizing Functions and Function Families

Vocabulary terms and definitions from Module 1 of Lesson 3 regarding functions, their behaviors, and specific function families like linear, exponential, and quadratic.

Relation

The mapping between a set of input values called the domain and a set of output values called the range.

Domain

The set of all input values for a relation.

Range

The set of all output values for a relation.

Function

A relation that assigns each element of the domain to exactly one element of the range.

Function notation

An algebraic way of representing functions, where $f(x)$ is read as "f of x" and indicates that $x$ is the independent variable.

Vertical Line Test

A visual method to determine if a graph is a function; if any vertical line intersects the graph at more than one point, the relation is not a function.

Discrete graph

A graph composed of isolated points.

Continuous graph

A graph of points connected by a line or smooth curve with no breaks.

Increasing function

A function where the dependent variable increases as the independent variable increases across the entire domain.

Decreasing function

A function where the dependent variable decreases as the independent variable increases across the entire domain.

Constant function

A function where the value of the dependent variable remains the same over the entire domain.

Function family

A group of functions that share certain identifiable characteristics.

Linear functions

A function family including those of the form $f(x) = ax + b$, where $a$ and $b$ are real numbers.

Exponential functions

A function family including those of the form $f(x) = a \times b^x$, where $a$ and $b$ are real numbers, and $b > 0$ but $b \neq 1$.

Quadratic functions

A function family including those of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.

Absolute minimum

A point on a graph with a $y$-coordinate that is less than the $y$-coordinate of every other point on the graph.

Absolute maximum

A point on a graph with a $y$-coordinate that is greater than the $y$-coordinate of every other point on the graph.

x-intercept

The point where a graph crosses the $x$-axis.

y-intercept

The point where a graph crosses the $y$-axis.