Algebra 2 Definitions Flashcards

Algebra 2 Definitions

• The opposite or additive inverse of any number b is represented as -b.
• The reciprocal, also called the multiplicative inverse of a nonzero number b, is 1/b.
• A numerical expression includes numbers, operations, and grouping symbols.
• A power indicates repeated multiplication of the same factor.
• An exponent indicates the number of times the base is used in a power.
• The base of a power is the factor that is multiplied in repeated multiplication.
• A variable is a letter used to represent numbers.
• An algebraic expression consists of numbers, variables, operations, and grouping and is often referred to as a variable expression.
• Terms of an expression are the individual parts that are added together within that expression.
• A variable term contains a variable part.
• A constant term contains only a number and lacks a variable part.
• A coefficient is the number in front of a variable in a term.
• Like terms have the same variable parts, including constant terms.
• Equivalent expressions yield the same result for all variable values.
• An identity equates two equivalent expressions.
• An equation states that two expressions are equal.
• A linear equation in one variable is expressed in the form ax + b = 0.
• The solution of an equation in one variable produces a true statement when a number replaces the variable.
• Equivalent equations have the same solution set.
• A formula expresses the relationship between quantities, often using variables.
• To solve for a variable, rewrite an equation to isolate the variable on one side.
• A verbal model is a narrative representation of a real-life mathematical problem.
• A linear inequality in one variable may be expressed as ax + b < 0, ≤ 0, > 0, ≥ 0.
• A solution of an inequality in one variable produces a true statement when substituted into the inequality.
• The graph of an inequality in one variable spans all points on a number line that satisfy the inequality.
• A compound inequality combines two inequalities linked by "and" or "or".
• Equivalent inequalities demonstrate the same solution set.
• Absolute value |x| measures the distance of x from zero on a number line.
• An extraneous solution appears valid but does not satisfy the original equation.
• A relation maps input values to output values.
• The domain is the set of input values for a relation.
• The range is the set of output values for a relation.
• A function ensures each input has one unique output.
• An equation in two variables contains two variables.
• The solution of an equation in two variables is an ordered pair (x, y) making the equation true.
• The graph of an equation in two variables is the set of all points (x, y) that satisfy the equation.
• An independent variable denotes the input in an equation, while the dependent variable denotes the output.
• A linear function takes the form y = mx + b where m and b are constants.
• Function notation uses symbols like f(x) to denote the dependent variable of a function.
• The slope is determined by the ratio of vertical change (rise) to horizontal change (run) between points on a line.
• Parallel lines do not intersect within the same plane.
• Perpendicular lines intersect at right angles within the same plane.
• Rate of change compares how much one quantity changes relative to another.
• A parent function is the simplest form within a function family.
• The y-intercept is where a graph intersects the y-axis.
• Slope-intercept form of a linear equation is y = mx + b, establishing slope m and y-intercept b.
• The x-intercept is where a graph intersects the x-axis.
• The standard form of a linear equation is Ax + By = C where A and B are not both zero.
• Point-slope form of a line is y - y₁ = m(x - x₁), defined around a specific point (x₁, y₁).
• Direct variation indicates y = ax when two variables, x and y, are directly proportional, with a as a nonzero constant.
• The constant of variation a is essential in variations (direct, inverse, or joint).
• A scatter plot graphs pairs of data points (x, y) to evaluate variable relationships.
• Positive correlation occurs if y increases as x increases, while negative correlation means y decreases as x increases.
• The correlation coefficient r (between -1 and 1) measures how well a line fits data pairs.
• The best fitting line closely aligns with all data points in a scatter plot.
• An absolute value function incorporates an absolute value expression.
• The vertex of an absolute value graph indicates the peak or trough of the function.
• A transformation modifies a graph's size, shape, or position.
• A translation shifts a graph without altering size or shape.
• A reflection flips a graph or figure across a line.
• A linear inequality in two variables states Ax + By < C, ≤ C, > C, or ≥ C.
• The solution of a linear inequality in two variables is an ordered pair satisfying each inequality.
• The graph of a linear inequality depicts all viable solutions on a coordinate plane.
• Half-planes emerge as regions created by a linear inequality's boundary line.
• A system of two linear equations consists of equations in the forms Ax + By = C and Dx + Ey = F.
• The solution of a system of linear equations is an ordered pair satisfying all equations.
• A consistent system has at least one solution, while an inconsistent system lacks a solution.
• An independent system has exactly one solution, while a dependent system contains infinitely many solutions.
• The substitution method solves a system by substituting one variable into another equation.
• The elimination method eliminates variables by manipulating equations then combining them.
• A system of linear inequalities includes two or more inequalities.
• The solution of a system of linear inequalities consists of ordered pairs meeting all conditions.
• The graph of a system of linear inequalities represents all solutions graphically.
• A linear equation in three variables is expressed as ax + by + cz = d, with a, b, c not all zero.
• A system of three linear equations features three linear equations in three variables.
• The solution of a system of linear equations in three variables is an ordered triple (x, y, z) that satisfies all equations.
• An ordered triple is a set of three numbers (x, y, z) representing a point in a three-dimensional space.
• A matrix is a rectangular array of numbers organized in rows and columns.
• The dimensions of a matrix are defined by the number of rows and columns, noted as m × n.
• Each number in a matrix is termed an element.
• Equal matrices have matching dimensions and corresponding elements.
• A scalar refers to a real number.
• Scalar multiplication involves multiplying each matrix element by a scalar.
• The determinant is a real number tied to square matrices, denoted as det A or |A|, calculated as ad - cb for specific 2x2 matrices.
• Cramer's rule uses determinants to solve systems of linear equations under certain conditions.
• The coefficient matrix for the system ax + by = e, cx + dy = f is structured as |a b| |c d|.
• An identity matrix has ones along its main diagonal and zeros elsewhere.
• An inverse matrix exists if the product of two n × n matrices returns the identity matrix.
• The matrix of variables in a linear system includes variables expressed as a matrix.
• The matrix of constants includes constants from the linear equation systems presented as a matrix.
• A quadratic function takes the form y = ax² + bx + c, where a cannot equal zero.
• The standard form of a quadratic equation is also y = ax² + bx + c with a not equal to zero.
• A parabola is the set of points equidistant from a focus and a directrix, visible in the graph of a quadratic function.