Algebra and Trigonometry: Equations, Functions, Slopes, and Inequalities

A complete set of vocabulary flashcards reviewing linear equations, slope properties, systems of linear equations, rates of change, and linear inequalities.

Slope Rule for Parallel Lines

Two nonvertical lines are parallel if and only if they have the same slope; distinct vertical lines, both with undefined slopes, are also parallel.

Slope Rule for Perpendicular Lines

Two nonvertical lines are perpendicular if and only if the product of their slopes is $-1$; a horizontal line with zero slope is perpendicular to a vertical line with undefined slope.

Slope as Rate of Change

The ratio of the change in $y$ to a corresponding change in $x$, describing how fast the dependent variable changes per unit change in the independent variable.

Average Rate of Change of a Function

For distinct points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ on the graph of $f$, the average rate of change from $x_1$ to $x_2$ is given by $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$.

Secant Line and Average Rate of Change

The line passing through distinct points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ on a curve $y = f(x)$, whose slope equals the average rate of change of $f$ between $x_1$ and $x_2$.

System of Linear Equations

A collection of two or more linear equations in the form $Ax + By = C$ that represent straight lines when graphed.

Solution to a System of Linear Equations

An ordered pair $(x, y)$ that satisfies both equations in a linear system with two variables.

Consistent System

A linear system that has at least one solution.

Inconsistent System

A linear system that has no solution.

Graphical Interpretation of Linear System Solutions

The visual outcome of graphing a system of two linear equations: intersecting lines have exactly one solution, parallel lines have no solution, and coinciding lines have infinitely many solutions.

Definition of Slope

The ratio of vertical change to horizontal change through distinct points $(x_1, y_1)$ and $(x_2, y_2)$, calculated as $m = \frac{\text{change in } y}{\text{change in } x} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ where $x_2 - x_1 \neq 0$.

Rise and Run

The vertical change ($y_2 - y_1$) and horizontal change ($x_2 - x_1$) between two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, used to compute its slope.

Point-Slope Form

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

Slope-Intercept Form

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

Equation of a Horizontal Line

An equation of the form $y = b$, where $b$ is the $y$-intercept of the line, possessing a slope equal to zero.

Equation of a Vertical Line

An equation of the form $x = a$, where $a$ is the $x$-intercept of the line, possessing an undefined slope.

General Form of the Equation of a Line

An equation written in the form $Ax + By + C = 0$, where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both zero.

Linear Equation in One Variable

An equation in one variable $x$ that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers, and $a \neq 0$.

Conditional Equation

An equation that is true for at least one real number but not all real numbers.

Identity

An equation that is true for all real numbers.

Inconsistent Equation

An equation that is not true for any real number.

Solution Set of an Inequality

The set of all numbers that satisfy an inequality and make it a true statement.

Open Interval

The set $(a, b) = 1x | a < x < b2$, representing all real numbers strictly between $a$ and $b$, excluding both endpoints.

Closed Interval

The set $[a, b] = 1x | a \neq x \neq b2$, representing all real numbers between $a$ and $b$, including both endpoints.

Parentheses in Interval Notation

Symbols indicating endpoints that are not included in an interval, which are always used with $\neq \neq$ or $-\neq \neq$.

Square Brackets in Interval Notation

Symbols indicating endpoints that are included in an interval.

Linear Inequality in One Variable

An algebraic statement in $x$ with $a \neq 0$ that can be written in the form $ax + b < 0$, $ax + b \neq 0$, $ax + b > 0$, or $ax + b \neq 0$.