Linear Functions and Arithmetic Sequences Vocabulary

Vocabulary flashcards covering linear functions, arithmetic sequences, transformations, forms of equations, and line relationships from the lecture transcript.

Domain of an Arithmetic Sequence

The set of inputs for an arithmetic sequence, which consists of natural numbers only for term numbers.

Domain of a Linear Function

The set of inputs for a linear function, which includes all real numbers.

Common Difference

The constant rate of change $d$ in an arithmetic sequence, which corresponds to the slope $m$ of a linear function.

Explicit Form of an Arithmetic Sequence

The arithmetic sequence equation written as $a_n = a_1 + d(n - 1)$.

Point-Slope Form

A linear equation form written as $y - y_1 = m(x - x_1)$, which can be derived from the explicit form of an arithmetic sequence.

Slope-Intercept Form

A linear equation form expressed as $y = mx + b$.

General Form of a Linear Function

The standard linear equation form expressed as $f(x) = mx + b$.

Factored Form of a Linear Function

A linear function form written as $f(x) = a(x - r)$, where $a$ represents the slope and $r$ represents the root.

Root

Also known as the zeros of a function, representing the $x$-intercept(s).

Parent Linear Function

The simplest form of a linear function, given by $f(x) = x$ or $y = x$.

$k$ Value

The constant added or subtracted in $f(x) = a(x - h) + k$ that causes a vertical shift ($+k$ shifts up, $-k$ shifts down) without changing the slope.

$a$ Value

The coefficient in $f(x) = a(x - h) + k$ that affects the slope, dilating the graph or reflecting it if negative.

Vertical Stretch

A dilation transformation that occurs when $|a| > 1$.

Vertical Compression

A dilation transformation that occurs when $|a| < 1$.

Perpendicular Lines

Two lines that intersect to form $90^\text{o}$ angles and have slopes that are opposite signs and reciprocals.

Parallel Lines

Lines that have the same slope.

Finding Intercepts

An algebraic method where one variable is replaced with $0$ to solve for the other intercept.