Algebra 1 Foundations and Review

Comprehensive vocabulary flashcards covering fundamental algebraic concepts, equation solving, linear and absolute value graphing, systems of equations, exponent rules, and polynomial operations.

Algebra

A branch of mathematics that follows the rules and four main operations of arithmetic (addition, subtraction, multiplication, division) while introducing the element of the unknown represented by symbols.

Equation

A mathematical statement indicating that the expression on one side of the equal sign has the same value as the expression on the other side.

Solving an equation

The process of simplifying and rearranging an algebraic equation to determine the value of the unknown variable.

Variable

A symbol, usually a letter of the alphabet, used to represent an unknown value or a quantity that can vary or change within an equation.

Implied multiplication

The default operation in algebra where multiplication is assumed if no operation symbol is written between two symbols, between a number and a variable, or between adjacent sets of parentheses.

Percent of change formula

The formula used to calculate relative change: $\frac{\text{new price} - \text{old price}}{\text{old price}}$.

Quadrants

The four regions of the coordinate plane labeled I, II, III, and IV, starting in the upper right-hand corner and progressing counterclockwise.

Linear equation

A mathematical equation whose solutions form a straight line when plotted on a coordinate plane.

$x$-intercept

The point where a line or curve crosses the $x$-axis, found by substituting $y = 0$ into the equation.

$y$-intercept

The point where a line or curve crosses the $y$-axis, found by substituting $x = 0$ into the equation.

Slope formula

The formula used to calculate the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Undefined slope

The slope of a vertical line (in the form $x = c$) that occurs when the denominator of the slope formula evaluates to zero, representing division by zero.

Zero slope

The slope of a horizontal line (in the form $y = c$) where the numerator in the slope formula evaluates to zero while the denominator is non-zero ($m = 0$).

Slope-intercept form

The form of a linear equation written as $y = mx + b$, where $m$ represents the slope and $b$ represents the $y$-intercept.

Point-slope form

A form of a linear equation written as $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

Parallel lines

Lines in the same plane that never intersect because their slopes are identical.

Perpendicular lines

Lines that intersect at a 90-degree angle and have slopes that are opposite reciprocals of each other (such as $\frac{1}{2}$ and $-2$).

Absolute value function

A function in the general form $y = a|x - h| + k$ that creates a symmetric V-shaped graph with a vertex at $(h, k)$.

Domain

The complete set of all possible $x$-values that can be input into a function or observed horizontally across its graph.

Range

The complete set of all possible $y$-values produced by a function or observed vertically across its graph.

Inequality reversal rule

The rule stating that when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol must flip.

Substitution method

A method for solving a system of equations by isolating one variable in an equation and replacing that variable with its equivalent expression in the second equation.

Elimination method

A technique for solving a system of equations by multiplying one or both equations by constants so that adding or subtracting the equations cancels out one variable.

Zero exponent rule

An exponent rule stating that any non-zero base raised to the zero power is equal to one ($x^0 = 1$).

Negative exponent rule

An exponent property stating that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent ($x^{-n} = \frac{1}{x^n}$).

Scientific notation

A standard format for writing numbers using a single digit from $1$ to $9$ before the decimal point, multiplied by $10$ raised to an integer power ($a \times 10^b$).

FOIL method

A mnemonic process used to multiply two binomials by calculating the sum of the products of their First, Outside, Inside, and Last terms.

Quadratic equations

A class of algebraic equations that graph as curves and are utilized to model real-world phenomena such as the trajectory of a ball, telescope lenses, or population growth.