Linear Functions, Parallelism, and Perpendicularity

Vocabulary flashcards reviewing core algebraic concepts including linear functions, domain, range, slope formulas, line equations, parallel lines, and perpendicular lines based on the lecture transcript.

Linear Function

A function that produces a straight line when graphed and can be written in the form $y = mx + b$.

Domain

All possible $x$-values for a function; for a linear function, the domain is any real number.

Range

All possible $y$-values for a function; for a linear function, the range is any real number.

Slope ($m$)

The measure of the steepness of a line, defined as $\frac{\text{rise}}{\text{run}}$ and calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

$y$-intercept

The value of $y$ where the line crosses the $y$-axis, occurring when $x$ is $0$.

$x$-intercept

The value of $x$ where the line crosses the $x$-axis, occurring when $y$ is $0$.

Linear Relationship

A relationship between two quantities that change at a constant rate, such as total distance growing evenly over time.

Slope-Intercept Form

The equation of a straight line expressed as $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Point-Slope Form

The equation of a straight line expressed as $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a specific point on the line.

Parallel Lines

Two or more straight lines on the same flat surface that stay the same distance apart and never cross or touch.

Properties of Parallel Lines

Characteristics of lines that have no intersection, maintain equal distance from one another, and possess equal slopes ($m_1 = m_2$).

Perpendicular Lines

Two lines that intersect at a single point to form right angles ($90^\text{o}$), creating four equal angles.

Properties of Perpendicular Lines

Characteristics of lines that intersect at right angles at a single point and have slopes that are negative reciprocals ($m_1 \times m_2 = -1$).

Negative Reciprocal

A fraction whose numerator and denominator are swapped and sign is inverted (e.g., $\frac{2}{5}$ becomes $-\frac{5}{2}$), such that the product of the original slope and its negative reciprocal equals $-1$.