Glossary of Terms
This glossary provides definitions for all 200 concepts covered in the Algebra I course. Each definition follows ISO 11179 metadata registry standards: precise, concise, distinct, non-circular, and free of business rules.
Absolute Value
The distance of a number from zero on the number line, always expressed as a non-negative value.
Example: The absolute value of -5 is 5, written as |-5| = 5, because -5 is 5 units away from zero.
Absolute Value Equations
Equations containing an expression within absolute value symbols that must be solved by considering both positive and negative cases.
Example: Solving |x - 3| = 7 yields two solutions: x = 10 and x = -4.
Absolute Value Function
A function that outputs the non-negative distance from zero for any input value, creating a V-shaped graph.
Example: The function f(x) = |x| produces f(-3) = 3 and f(3) = 3.
Absolute Value Inequalities
Inequalities containing absolute value expressions that require analyzing multiple cases to find solution sets.
Example: |x| < 5 means x is between -5 and 5: -5 < x < 5.
Adding Polynomials
The process of combining polynomial expressions by summing coefficients of like terms.
Example: (3x² + 2x - 1) + (x² - 5x + 4) = 4x² - 3x + 3.
And Inequalities
Compound inequalities where both conditions must be satisfied simultaneously, representing the intersection of solution sets.
Example: x > 2 AND x < 8 describes values between 2 and 8, written as 2 < x < 8.
Applications of Linear Equations
Real-world problems modeled and solved using linear equations in one variable.
Example: If a taxi charges $3 plus $2 per mile, finding the cost for a 7-mile trip: C = 3 + 2(7) = $17.
Applications of Quadratics
Real-world scenarios modeled using quadratic functions, often involving area, projectile motion, or optimization.
Example: Finding the maximum height of a ball thrown upward with height h(t) = -16t² + 32t + 6.
Applications of Systems
Problems requiring two or more equations to represent relationships between multiple unknown quantities.
Example: Determining the price of apples and oranges when two purchases with different quantities yield known total costs.
Arithmetic Sequence
An ordered list of numbers where each term after the first is found by adding a constant difference to the previous term.
Example: The sequence 3, 7, 11, 15, 19 has a common difference of 4.
Associative Property
A property stating that the grouping of numbers in addition or multiplication does not affect the result.
Example: (2 + 3) + 4 = 2 + (3 + 4), both equal 9.
Axis of Symmetry
A vertical line that divides a parabola into two mirror-image halves, passing through the vertex.
Example: For f(x) = x² - 4x + 3, the axis of symmetry is the line x = 2.
Base
The number or expression that is raised to a power in an exponential expression.
Example: In 5³, the base is 5; in x⁴, the base is x.
Binomial
A polynomial expression consisting of exactly two terms connected by addition or subtraction.
Example: 3x + 7, a² - 4, and 2x² + 5x are all binomials.
Boundary Line
The line that separates a coordinate plane into regions when graphing a linear inequality, which may be solid or dashed.
Example: For y > 2x + 1, the boundary line y = 2x + 1 is dashed because points on the line are not included.
Checking Solutions
The process of verifying whether a proposed value satisfies an equation or inequality by substitution.
Example: To check if x = 3 solves 2x + 1 = 7, substitute: 2(3) + 1 = 7, which is true.
Coefficient
A numerical factor that multiplies a variable or variables in an algebraic term.
Example: In 5x², the coefficient is 5; in -3xy, the coefficient is -3.
Combining Like Terms
The process of simplifying expressions by adding or subtracting terms with identical variable parts.
Example: 4x + 7x - 2x simplifies to 9x.
Commutative Property
A property stating that the order of numbers in addition or multiplication does not affect the result.
Example: 3 + 5 = 5 + 3, and 4 × 7 = 7 × 4.
Comparing Linear and Exponential
Analyzing differences between linear functions that change at constant rates and exponential functions that change by constant factors.
Example: Linear growth adds the same amount each period (2, 4, 6, 8), while exponential growth multiplies by the same factor (2, 4, 8, 16).
Completing the Square
A method for rewriting quadratic expressions as perfect square trinomials by adding a strategic constant term.
Example: Converting x² + 6x to (x + 3)² - 9 by adding and subtracting 9.
Complex Numbers
Numbers that include both a real part and an imaginary part, written in the form a + bi.
Example: 3 + 4i, where 3 is the real part and 4i is the imaginary part.
Compound Inequalities
Two or more inequalities joined by "and" or "or" that describe combined conditions.
Example: x < -2 OR x > 5 describes values outside the interval [-2, 5].
Compound Interest
Interest calculated on both the initial principal and accumulated interest from previous periods, growing exponentially over time.
Example: $1000 invested at 5% annual compound interest grows to $1000(1.05)ᵗ after t years.
Conditional Equation
An equation that is true only for specific values of the variable, having a finite solution set.
Example: 2x + 3 = 11 is conditional because it's true only when x = 4.
Consistent System
A system of equations that has at least one solution, where the equations represent lines that intersect.
Example: y = 2x + 1 and y = -x + 4 form a consistent system with solution (1, 3).
Constant
A fixed value that does not change, represented by a number or letter that maintains the same value throughout a problem.
Example: In the expression 3x + 7, the number 7 is a constant.
Continuous Function
A function whose graph can be drawn without lifting the pencil, having no breaks, jumps, or gaps.
Example: f(x) = x² is continuous, while a step function with jumps is not.
Contradiction
An equation that is never true, having no solution regardless of the variable's value.
Example: x + 5 = x + 3 is a contradiction because no value of x makes it true.
Coordinate Plane
A two-dimensional surface formed by two perpendicular number lines (axes) intersecting at their zero points.
Example: The point (3, -2) is located 3 units right and 2 units down from the origin.
Coordinate System
A framework using numerical coordinates to uniquely identify the position of points on a line, plane, or space.
Example: The Cartesian coordinate system uses ordered pairs (x, y) to locate points in a plane.
Correlation
A statistical relationship between two variables that indicates how they tend to change together.
Example: Height and weight typically show positive correlation—as one increases, the other tends to increase.
Cube Root
The value that, when multiplied by itself three times, produces the original number.
Example: The cube root of 27 is 3 because 3 × 3 × 3 = 27, written as ∛27 = 3.
Data Set
A collection of related numerical or categorical values gathered for analysis or study.
Example: Test scores for a class: {85, 92, 78, 95, 88, 90} form a data set.
Decay Factor
The constant multiplier less than 1 that determines the rate of exponential decrease in a decay function.
Example: In f(t) = 100(0.8)ᵗ, the decay factor is 0.8, meaning the quantity decreases by 20% each period.
Degree of Polynomial
The highest exponent of the variable in a polynomial expression.
Example: The polynomial 5x⁴ - 3x² + 7 has degree 4.
Dependent System
A system of equations with infinitely many solutions, where the equations represent the same line.
Example: y = 2x + 1 and 2y = 4x + 2 are dependent because they describe identical lines.
Dependent Variable
The output variable in a function or relationship whose value depends on the input variable.
Example: In C = 5n + 10, cost C is the dependent variable because it depends on n.
Difference of Cubes
A polynomial pattern of the form a³ - b³ that factors as (a - b)(a² + ab + b²).
Example: x³ - 8 = (x - 2)(x² + 2x + 4).
Difference of Squares
A polynomial pattern of the form a² - b² that factors as (a + b)(a - b).
Example: x² - 9 = (x + 3)(x - 3).
Discriminant
The expression b² - 4ac from the quadratic formula that determines the nature and number of solutions.
Example: For x² - 5x + 6 = 0, the discriminant is (-5)² - 4(1)(6) = 1, indicating two distinct real roots.
Distance Rate Time Problems
Word problems involving the relationship distance = rate × time, often requiring equation setup and solving.
Example: If a car travels at 60 mph for 2.5 hours, it covers 60 × 2.5 = 150 miles.
Distributive Property
A property stating that multiplication distributes over addition: a(b + c) = ab + ac.
Example: 3(x + 4) = 3x + 12.
Domain
The complete set of possible input values for which a function or relation is defined.
Example: For f(x) = √x, the domain is x ≥ 0 because square roots of negative numbers are not real.
Elimination Method
A technique for solving systems of equations by adding or subtracting equations to eliminate one variable.
Example: Adding x + y = 5 and x - y = 1 eliminates y, giving 2x = 6, so x = 3.
Equation
A mathematical statement asserting that two expressions are equal, connected by an equals sign.
Example: 2x + 5 = 13 states that the expression 2x + 5 equals 13.
Equations with Decimals
Equations containing decimal coefficients or constants that may be solved directly or by clearing decimals.
Example: 0.5x + 1.2 = 3.7 can be solved by subtracting 1.2 and dividing by 0.5, yielding x = 5.
Equations with Fractions
Equations containing fractional coefficients that may be solved by multiplying through by a common denominator.
Example: (1/2)x + 3 = 7 can be solved by multiplying all terms by 2, giving x + 6 = 14, so x = 8.
Evaluating Expressions
The process of finding the numerical value of an expression by substituting specific values for variables and simplifying.
Example: Evaluating 3x² - 2x + 1 when x = 4 gives 3(16) - 8 + 1 = 41.
Evaluating Functions
The process of determining a function's output value by substituting a specific input value.
Example: For f(x) = x² + 3, evaluating f(5) yields 5² + 3 = 28.
Expanding Expressions
The process of removing parentheses by applying the distributive property to write expressions in standard form.
Example: Expanding 3(2x - 5) yields 6x - 15.
Exponent
A number or variable that indicates how many times a base is multiplied by itself.
Example: In 2⁵, the exponent 5 indicates 2 is multiplied by itself 5 times: 2 × 2 × 2 × 2 × 2 = 32.
Exponential Decay
A decreasing pattern where quantities reduce by a constant percentage each time period, modeled by exponential functions with decay factors.
Example: Radioactive material with half-life of 10 years decreases according to A(t) = A₀(0.5)^(t/10).
Exponential Function
A function in which the variable appears as an exponent, typically in the form f(x) = a·bˣ where b > 0 and b ≠ 1.
Example: f(x) = 2ˣ doubles with each unit increase in x: f(0) = 1, f(1) = 2, f(2) = 4.
Exponential Growth
An increasing pattern where quantities multiply by a constant factor each time period, modeled by exponential functions with growth factors greater than 1.
Example: A population doubling each year follows P(t) = P₀(2)ᵗ.
Exponential Models
Mathematical representations using exponential functions to describe real-world phenomena involving multiplicative growth or decay.
Example: Bank account growth with compound interest: A(t) = P(1 + r)ᵗ.
Expression
A mathematical phrase combining numbers, variables, and operations without an equals or inequality sign.
Example: 3x² - 5x + 7 is an expression, not an equation.
Factoring
The process of rewriting an expression as a product of simpler factors.
Example: Factoring x² + 5x + 6 yields (x + 2)(x + 3).
Factoring by GCF
The process of factoring an expression by identifying and extracting the greatest common factor from all terms.
Example: 6x³ + 9x² = 3x²(2x + 3).
Factoring by Grouping
A technique for factoring polynomials with four or more terms by grouping pairs and factoring common factors from each group.
Example: x³ + 2x² + 3x + 6 = x²(x + 2) + 3(x + 2) = (x² + 3)(x + 2).
Factoring Completely
The process of factoring an expression until no further factorization is possible using integer coefficients.
Example: 2x³ - 8x = 2x(x² - 4) = 2x(x + 2)(x - 2).
Factoring Difference of Squares
The technique of factoring expressions in the form a² - b² into (a + b)(a - b).
Example: 4x² - 25 = (2x)² - 5² = (2x + 5)(2x - 5).
Factoring Perfect Squares
The process of recognizing and factoring perfect square trinomials into squared binomial form.
Example: x² + 6x + 9 = (x + 3)².
Factoring Quadratic Form
The technique of factoring expressions that resemble quadratics but involve higher or different powers.
Example: x⁴ - 5x² + 4 factors as (x² - 1)(x² - 4) using u = x² substitution.
Factoring Trinomials
The process of rewriting trinomial expressions as products of two binomials.
Example: x² + 7x + 12 = (x + 3)(x + 4).
FOIL Method
A technique for multiplying two binomials by systematically multiplying First, Outer, Inner, and Last terms.
Example: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.
Formula Manipulation
The process of rearranging formulas to solve for a different variable by applying algebraic operations.
Example: Solving A = πr² for r yields r = √(A/π).
Function
A relation where each input value corresponds to exactly one output value.
Example: f(x) = 2x + 1 is a function because each x-value produces exactly one y-value.
Function Notation
The notation f(x) used to represent the output value of a function f for input value x.
Example: If f(x) = x² - 3, then f(4) = 16 - 3 = 13.
Function Transformation
Changes to a parent function's graph through translations, reflections, stretches, or compressions.
Example: g(x) = (x - 2)² + 3 is f(x) = x² shifted right 2 units and up 3 units.
Geometric Sequence
An ordered list of numbers where each term after the first is found by multiplying the previous term by a constant ratio.
Example: The sequence 2, 6, 18, 54, 162 has a common ratio of 3.
Graph of a Line
The visual representation of all ordered pairs that satisfy a linear equation, forming a straight line.
Example: The graph of y = 2x + 1 passes through (0, 1) and (1, 3).
Graphing Exponentials
The process of plotting exponential functions, which show curves that increase or decrease rapidly without bound.
Example: The graph of y = 2ˣ passes through (0, 1) and rises steeply to the right.
Graphing Inequalities
The process of representing inequality solutions on a number line, using open or closed circles and shading.
Example: x > 3 is shown with an open circle at 3 and shading to the right.
Graphing Method
A technique for solving systems of equations by plotting both equations and identifying intersection points.
Example: Graphing y = x + 2 and y = -x + 6 reveals they intersect at (2, 4).
Graphing Quadratics
The process of plotting quadratic functions, which produce U-shaped curves called parabolas.
Example: The graph of y = x² - 4x + 3 is a parabola opening upward with vertex at (2, -1).
Graphing Systems
The visual method of finding solutions to systems of inequalities by graphing boundary lines and identifying overlapping shaded regions.
Example: Graphing y > x and y < -x + 4 shows the solution region between these lines.
Greatest Common Factor
The largest integer or expression that divides evenly into all terms of a polynomial.
Example: The GCF of 12x³, 18x², and 6x is 6x.
Growth Factor
The constant multiplier greater than 1 that determines the rate of exponential increase in a growth function.
Example: In f(t) = 50(1.2)ᵗ, the growth factor is 1.2, meaning the quantity increases by 20% each period.
Horizontal Line
A line with zero slope that runs parallel to the x-axis, having constant y-coordinate for all points.
Example: The line y = 5 is horizontal, passing through all points with y-coordinate 5.
Identity Equation
An equation that is true for all values of the variable, having infinitely many solutions.
Example: 2(x + 3) = 2x + 6 is an identity because it simplifies to a true statement for any x.
Identity Property
The property stating that adding zero or multiplying by one leaves a number unchanged.
Example: 7 + 0 = 7 and 9 × 1 = 9 demonstrate additive and multiplicative identity properties.
Imaginary Unit
The number i defined as √(-1), serving as the basis for imaginary and complex numbers.
Example: i² = -1, i³ = -i, and i⁴ = 1, creating a repeating pattern.
Inconsistent System
A system of equations with no solution, where the equations represent parallel lines that never intersect.
Example: y = 2x + 1 and y = 2x + 5 are inconsistent because they have the same slope but different y-intercepts.
Independent System
A system of equations with exactly one solution, where the equations represent lines that intersect at a single point.
Example: y = x + 1 and y = -x + 5 form an independent system with unique solution (2, 3).
Independent Variable
The input variable in a function or relationship whose value is chosen freely and determines other values.
Example: In d = 50t, time t is the independent variable because we choose the time to find distance.
Inequality
A mathematical statement comparing two expressions using symbols such as <, >, ≤, or ≥.
Example: x + 5 > 12 states that x + 5 is greater than 12.
Inequality Symbols
Mathematical notation used to express relationships of greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤).
Example: x ≥ 5 means x is 5 or any value greater than 5.
Initial Value
The starting amount in an exponential function when the input variable equals zero.
Example: In P(t) = 100(1.05)ᵗ, the initial value is 100.
Input
The independent variable value entered into a function to produce an output.
Example: For f(x) = 3x + 2, when the input is x = 4, the output is f(4) = 14.
Integer Exponents
Whole number exponents, including positive, negative, and zero values.
Example: 2³ = 8, 2⁻² = 1/4, and 2⁰ = 1 demonstrate integer exponents.
Integers
The set of whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
Example: -5, 0, and 42 are integers, but 3.5 and 2/3 are not.
Inverse Property
The property stating that every number has an additive inverse (opposite) that sums to zero and a multiplicative inverse (reciprocal) that multiplies to one.
Example: 7 + (-7) = 0 and 5 × (1/5) = 1 demonstrate inverse properties.
Leading Coefficient
The coefficient of the term with the highest degree in a polynomial written in standard form.
Example: In 3x⁴ - 2x² + 5x - 1, the leading coefficient is 3.
Like Terms
Terms in an expression that have identical variable parts with matching exponents.
Example: 5x² and -3x² are like terms, but 5x² and 5x are not.
Line of Best Fit
A straight line drawn through data points on a scatterplot that best represents the overall trend.
Example: For points roughly following an upward trend, the line of best fit might be y = 2x + 1.
Linear Combination
An expression formed by adding or subtracting multiples of equations or expressions, used in the elimination method.
Example: Adding 2(x + y = 5) and -1(2x + y = 8) creates a linear combination -y = 2.
Linear Equation in One Variable
An equation that can be written in the form ax + b = c where a ≠ 0, having at most one solution.
Example: 3x - 7 = 14 is a linear equation with solution x = 7.
Linear Function
A function whose graph is a straight line, expressible in the form f(x) = mx + b.
Example: f(x) = 2x + 3 is a linear function with slope 2 and y-intercept 3.
Linear Inequality
An inequality that can be written in a form similar to linear equations using inequality symbols instead of equals.
Example: 2x + 5 < 17 is a linear inequality with solution x < 6.
Linear Regression
A statistical method for finding the line of best fit that minimizes the distance between data points and the line.
Example: Using linear regression on height and weight data might yield the equation W = 1.5H + 10.
Literal Equations
Equations containing multiple variables where the goal is to solve for one variable in terms of the others.
Example: Solving P = 2l + 2w for l yields l = (P - 2w)/2.
Maximum Value
The largest y-coordinate on the graph of a function, occurring at the vertex of a downward-opening parabola.
Example: For f(x) = -x² + 4x + 1, the maximum value is 5 at x = 2.
Minimum Value
The smallest y-coordinate on the graph of a function, occurring at the vertex of an upward-opening parabola.
Example: For f(x) = x² - 6x + 5, the minimum value is -4 at x = 3.
Monomial
A polynomial expression consisting of a single term, which may be a number, variable, or product of numbers and variables.
Example: 7, x, and -3x²y³ are all monomials.
Multi-Step Equations
Equations requiring multiple algebraic operations to isolate the variable and find the solution.
Example: Solving 3(x - 2) + 5 = 14 requires distributing, combining terms, and inverse operations.
Multiplying Polynomials
The process of finding the product of polynomial expressions by distributing each term of one polynomial to every term of the other.
Example: (x + 2)(x² - 3x + 1) = x³ - 3x² + x + 2x² - 6x + 2 = x³ - x² - 5x + 2.
Nature of Roots
The classification of quadratic equation solutions as two distinct real roots, one repeated real root, or two complex conjugate roots based on the discriminant.
Example: If b² - 4ac > 0, the equation has two distinct real roots.
Negative Correlation
A relationship between variables where increases in one variable correspond to decreases in the other.
Example: As temperature decreases, heating costs increase, showing negative correlation.
Negative Exponents
Exponents less than zero that represent reciprocals of positive exponents.
Example: x⁻³ = 1/x³ and 2⁻² = 1/4.
Negative Slope
A slope value less than zero, indicating a line that falls from left to right.
Example: The line y = -2x + 5 has negative slope -2.
No Correlation
The absence of any predictable relationship between two variables in a data set.
Example: Shoe size and test scores typically show no correlation.
Number
A mathematical object used to count, measure, label, or calculate quantities.
Example: 5, -3, 2.7, π, and √2 are all numbers.
Number Line
A visual representation of numbers as points on a straight line, with positive numbers to the right and negative numbers to the left of zero.
Example: On a number line, -2 is positioned 2 units left of 0, and 3 is positioned 3 units right of 0.
One-Step Equations
Equations requiring exactly one inverse operation to isolate the variable and find the solution.
Example: Solving x + 7 = 15 requires one step: subtracting 7 from both sides to get x = 8.
Or Inequalities
Compound inequalities where at least one of the conditions must be satisfied, representing the union of solution sets.
Example: x < -3 OR x > 2 describes all values less than -3 or greater than 2.
Order of Operations
The standardized sequence for evaluating mathematical expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
Example: Evaluating 3 + 4 × 2² requires: 3 + 4 × 4 = 3 + 16 = 19, not 49.
Ordered Pair
A pair of numbers (x, y) that represents the coordinates of a point in a coordinate plane.
Example: The ordered pair (3, -2) locates a point 3 units right and 2 units down from the origin.
Origin
The point (0, 0) where the x-axis and y-axis intersect in a coordinate plane.
Example: The origin is the starting point for measuring distances in all directions.
Output
The dependent variable value produced by a function when given an input value.
Example: For f(x) = x² + 1, when the input is 3, the output is f(3) = 10.
Parabola
The U-shaped curve that forms the graph of a quadratic function.
Example: The graph of y = x² is a parabola opening upward with vertex at the origin.
Parent Function
The simplest form of a function family, serving as the basis for transformations.
Example: f(x) = x² is the parent function for all quadratic functions.
Perfect Square Trinomial
A trinomial that factors into a squared binomial, having the form a² ± 2ab + b² = (a ± b)².
Example: x² + 10x + 25 = (x + 5)².
Perimeter Problems
Word problems involving the distance around the boundary of geometric figures, often requiring equation setup and solving.
Example: If a rectangle has perimeter 30 and length 8, then 2(8) + 2w = 30, so width w = 7.
Piecewise Function
A function defined by different expressions for different intervals of the input domain.
Example: f(x) = {x² if x < 0; 2x if x ≥ 0} uses different rules based on x's value.
Plotting Points
The process of marking the location of ordered pairs on a coordinate plane.
Example: Plotting (2, 3) means starting at the origin, moving 2 units right and 3 units up.
Point-Slope Form
The linear equation form y - y₁ = m(x - x₁), where m is slope and (x₁, y₁) is a known point.
Example: A line with slope 3 passing through (2, 5) has equation y - 5 = 3(x - 2).
Polynomial
An algebraic expression consisting of one or more terms combined by addition or subtraction, with variables having whole number exponents.
Example: 4x³ - 2x² + 7x - 5 is a polynomial with four terms.
Positive Correlation
A relationship between variables where increases in one variable correspond to increases in the other.
Example: Study time and test scores typically show positive correlation.
Positive Slope
A slope value greater than zero, indicating a line that rises from left to right.
Example: The line y = 3x - 2 has positive slope 3.
Power
The result of raising a base to an exponent, or the entire exponential expression itself.
Example: 2³ = 8, where 8 is the power (result) or 2³ is called "2 to the third power."
Power Rule for Exponents
The rule stating that (aᵐ)ⁿ = aᵐⁿ when raising a power to another power.
Example: (x³)⁴ = x¹² and (2²)³ = 2⁶ = 64.
Prime Factorization
The expression of a number or polynomial as a product of prime numbers or irreducible factors.
Example: The prime factorization of 60 is 2² × 3 × 5.
Prime Polynomial
A polynomial that cannot be factored into polynomials of lower degree with integer coefficients.
Example: x² + x + 1 is prime over the integers.
Product Rule for Exponents
The rule stating that aᵐ × aⁿ = aᵐ⁺ⁿ when multiplying powers with the same base.
Example: x³ × x⁵ = x⁸ and 2² × 2³ = 2⁵ = 32.
Projectile Motion
The curved path of an object under gravity's influence, modeled by quadratic functions.
Example: A ball's height h(t) = -16t² + 32t + 6 shows projectile motion over time t.
Quadrants
The four regions of a coordinate plane created by the intersection of the x-axis and y-axis.
Example: Quadrant I contains points with positive x and y coordinates; Quadrant II has negative x and positive y.
Quadratic Equation
An equation that can be written in the form ax² + bx + c = 0, where a ≠ 0.
Example: x² - 5x + 6 = 0 is a quadratic equation with solutions x = 2 and x = 3.
Quadratic Formula
The formula x = (-b ± √(b² - 4ac))/(2a) used to solve any quadratic equation ax² + bx + c = 0.
Example: For x² - 3x - 4 = 0, using a = 1, b = -3, c = -4 yields solutions x = 4 and x = -1.
Quadratic Function
A function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0, producing a parabolic graph.
Example: f(x) = 2x² - 4x + 1 is a quadratic function.
Quotient Rule for Exponents
The rule stating that aᵐ/aⁿ = aᵐ⁻ⁿ when dividing powers with the same base (a ≠ 0).
Example: x⁷/x³ = x⁴ and 2⁵/2² = 2³ = 8.
Radical Expression
An expression containing a root symbol, such as square root, cube root, or higher roots.
Example: √(x² + 4), ∛8, and 2√5 are radical expressions.
Range
The complete set of possible output values that a function can produce.
Example: For f(x) = x², the range is y ≥ 0 because x² is never negative.
Rate of Change
The ratio describing how one quantity changes relative to another, represented by slope in linear relationships.
Example: A car traveling 60 miles in 2 hours has a rate of change of 30 miles per hour.
Rational Exponents
Exponents expressed as fractions, where the numerator indicates a power and the denominator indicates a root.
Example: x^(3/2) = (√x)³ = √(x³) and 8^(2/3) = (∛8)² = 2² = 4.
Rational Numbers
Numbers that can be expressed as the ratio of two integers, written as p/q where q ≠ 0.
Example: 3/4, -2, 0.5, and 7 (which equals 7/1) are all rational numbers.
Real Numbers
The complete set of numbers including all rational and irrational numbers, representing all points on the number line.
Example: -5, 3/4, √2, π, and 0 are all real numbers.
Reflection
A transformation that flips a function's graph across a line, such as the x-axis or y-axis.
Example: f(x) = -x² is a reflection of f(x) = x² across the x-axis.
Relation
Any set of ordered pairs or rule that pairs input values with output values.
Example: {(1, 2), (2, 4), (3, 6), (1, 8)} is a relation, though not a function.
Rise
The vertical change between two points on a line, used in calculating slope.
Example: From (1, 2) to (4, 8), the rise is 8 - 2 = 6.
Run
The horizontal change between two points on a line, used in calculating slope.
Example: From (1, 2) to (4, 8), the run is 4 - 1 = 3.
Scatterplot
A graph displaying individual data points as dots on a coordinate plane to reveal patterns or relationships.
Example: Plotting study hours versus test scores creates a scatterplot showing their relationship.
Scientific Notation
A compact way of writing very large or small numbers as a product of a number between 1 and 10 and a power of 10.
Example: 3,500,000 = 3.5 × 10⁶ and 0.00042 = 4.2 × 10⁻⁴.
Simplifying Expressions
The process of rewriting expressions in simpler form by combining like terms and applying algebraic properties.
Example: Simplifying 3x + 5x - 2 + 7 yields 8x + 5.
Simplifying Radicals
The process of rewriting radical expressions in simplest form by removing perfect square factors from under the radical.
Example: √50 = √(25 × 2) = 5√2.
Slope
A number representing the steepness and direction of a line, calculated as the ratio of vertical change to horizontal change.
Example: A line through (1, 2) and (3, 8) has slope m = (8 - 2)/(3 - 1) = 3.
Slope Formula
The formula m = (y₂ - y₁)/(x₂ - x₁) used to calculate the slope between two points (x₁, y₁) and (x₂, y₂).
Example: The slope between (2, 5) and (6, 13) is m = (13 - 5)/(6 - 2) = 2.
Slope-Intercept Form
The linear equation form y = mx + b, where m is the slope and b is the y-intercept.
Example: y = 3x - 2 has slope 3 and y-intercept -2.
Solution of a System
An ordered pair or set of values that satisfies all equations or inequalities in a system simultaneously.
Example: (2, 3) is the solution to the system y = x + 1 and y = 5 - x.
Solution Region
The area on a coordinate plane containing all points that satisfy a system of inequalities.
Example: The solution region for y > x and y < x + 4 is the area between these two boundary lines.
Solution Set
The collection of all values that satisfy an equation or inequality.
Example: The solution set for x² = 9 is {-3, 3}.
Solving by Factoring
The method of finding equation solutions by factoring one side to zero and applying the zero product property.
Example: Solving x² + 5x + 6 = 0 by factoring: (x + 2)(x + 3) = 0, so x = -2 or x = -3.
Solving by Square Roots
The method of solving quadratic equations by isolating the squared term and taking the square root of both sides.
Example: Solving x² = 25 yields x = ±5.
Solving Equations
The process of finding all values of a variable that make an equation true.
Example: Solving 2x + 7 = 15 involves subtracting 7 and dividing by 2 to get x = 4.
Solving for a Variable
The process of isolating a specific variable in an equation or formula containing multiple variables.
Example: Solving V = πr²h for h yields h = V/(πr²).
Solving Linear Inequalities
The process of finding all values that satisfy a linear inequality, similar to solving equations but preserving inequality direction.
Example: Solving 3x - 5 < 10 yields x < 5.
Solving Quadratics by Graphing
The method of finding quadratic equation solutions by identifying where the parabola intersects the x-axis.
Example: The solutions to x² - 4 = 0 are x = -2 and x = 2, where the graph crosses the x-axis.
Special Products
Recognizable polynomial multiplication patterns that produce predictable results, such as (a + b)² and (a + b)(a - b).
Example: (x + 5)² = x² + 10x + 25 follows the pattern (a + b)² = a² + 2ab + b².
Square Root
The value that, when multiplied by itself, produces the original number, denoted by the radical symbol √.
Example: √25 = 5 because 5 × 5 = 25.
Standard Form
The conventional way of writing expressions or equations with terms arranged in descending degree order.
Example: The standard form of a quadratic is ax² + bx + c; for a line, it's Ax + By = C.
Standard Form of Polynomial
A polynomial written with terms ordered from highest to lowest degree.
Example: 3x⁴ - 2x² + 5x - 7 is in standard form.
Standard Form of Quadratic
The form ax² + bx + c where a ≠ 0, arranging the quadratic expression in descending degree order.
Example: 2x² - 5x + 3 is a quadratic in standard form.
Step Function
A piecewise function consisting of horizontal line segments, creating a graph resembling steps or stairs.
Example: The greatest integer function f(x) = ⌊x⌋ is a step function.
Substitution
The process of replacing a variable with a specific value or expression.
Example: Substituting x = 3 into 2x + 5 gives 2(3) + 5 = 11.
Substitution Method
A technique for solving systems of equations by solving one equation for a variable and substituting into the other equation.
Example: From x + y = 5, get y = 5 - x, then substitute into 2x + y = 7: 2x + (5 - x) = 7.
Subtracting Polynomials
The process of finding the difference between polynomial expressions by subtracting coefficients of like terms.
Example: (4x² + 3x - 2) - (x² + 5x - 7) = 3x² - 2x + 5.
Sum of Cubes
A polynomial pattern of the form a³ + b³ that factors as (a + b)(a² - ab + b²).
Example: x³ + 27 = (x + 3)(x² - 3x + 9).
System of Equations
A set of two or more equations with the same variables that are solved simultaneously.
Example: The system {y = 2x + 1, y = -x + 4} has solution (1, 3).
System of Inequalities
A set of two or more inequalities with the same variables whose solution is the intersection of individual solution regions.
Example: The system {y > x, y < x + 4} describes a region between two boundary lines.
Term
A single number, variable, or product of numbers and variables in an algebraic expression, separated by addition or subtraction.
Example: In 3x² - 5x + 7, there are three terms: 3x², -5x, and 7.
Translation
A transformation that shifts a function's graph horizontally, vertically, or both without changing its shape.
Example: g(x) = (x - 3)² is f(x) = x² translated 3 units right.
Trinomial
A polynomial expression consisting of exactly three terms connected by addition or subtraction.
Example: x² + 5x - 6, 2a² - 3b + 4, and 3x² + 2xy + y² are all trinomials.
Two-Step Equations
Equations requiring exactly two inverse operations to isolate the variable and find the solution.
Example: Solving 3x + 5 = 17 requires subtracting 5 and then dividing by 3 to get x = 4.
Undefined Slope
The characteristic of vertical lines that have no defined slope because the run (horizontal change) is zero.
Example: The line x = 5 has undefined slope because all points have the same x-coordinate.
Variable
A symbol, typically a letter, used to represent an unknown or changing value in mathematical expressions and equations.
Example: In the equation 2x + 5 = 13, the letter x is a variable representing the unknown value 4.
Variables on Both Sides
Equations containing the variable on both sides of the equals sign, requiring strategic manipulation to collect terms.
Example: Solving 5x - 3 = 2x + 9 requires moving variables to one side: 3x = 12, so x = 4.
Vertex
The highest or lowest point on a parabola, representing the maximum or minimum value of a quadratic function.
Example: The parabola y = (x - 2)² + 3 has vertex at (2, 3).
Vertex Form
The quadratic equation form y = a(x - h)² + k, where (h, k) is the vertex.
Example: y = 2(x - 3)² + 1 has vertex (3, 1) and opens upward.
Vertical Line
A line with undefined slope that runs parallel to the y-axis, having constant x-coordinate for all points.
Example: The line x = -2 is vertical, passing through all points with x-coordinate -2.
Vertical Line Test
A method for determining if a graph represents a function by checking if any vertical line intersects the graph more than once.
Example: A circle fails the vertical line test because vertical lines through the center intersect it twice.
X-Coordinate
The first number in an ordered pair, indicating horizontal distance from the origin.
Example: In the point (5, -3), the x-coordinate is 5.
X-Intercept
The point where a graph crosses the x-axis, having y-coordinate equal to zero.
Example: The line y = 2x - 6 has x-intercept at (3, 0).
Y-Coordinate
The second number in an ordered pair, indicating vertical distance from the origin.
Example: In the point (5, -3), the y-coordinate is -3.
Y-Intercept
The point where a graph crosses the y-axis, having x-coordinate equal to zero.
Example: The line y = 3x + 4 has y-intercept at (0, 4).
Zero Exponent
An exponent of zero applied to any nonzero base, always resulting in a value of one.
Example: 5⁰ = 1, x⁰ = 1 (provided x ≠ 0), and (3x²y)⁰ = 1.
Zero Product Property
The principle stating that if the product of factors equals zero, then at least one factor must equal zero.
Example: If (x - 2)(x + 5) = 0, then x = 2 or x = -5.
Zero Slope
The characteristic of horizontal lines that have slope equal to zero because the rise (vertical change) is zero.
Example: The line y = 3 has zero slope because all points have the same y-coordinate.