- Integer
- A whole number or its opposite, including zero: …, −2, −1, 0, 1, 2, …. No fractional part.
- Rational number
- Any number writable as a fraction a/b of two integers (b ≠ 0): integers, terminating decimals, and repeating decimals.
- Irrational number
- A real number that cannot be written as a fraction of integers; its decimal never terminates or repeats. Examples: π and √2.
- Order of operations (PEMDAS)
- Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
- Greatest common factor (GCF)
- The largest whole number that divides two or more numbers evenly. GCF of 8 and 12 is 4.
- Least common multiple (LCM)
- The smallest number that two or more numbers all divide into. LCM of 8 and 12 is 24.
- Prime number
- A whole number greater than 1 whose only factors are 1 and itself: 2, 3, 5, 7, 11, ….
- Composite number
- A whole number greater than 1 with more than two factors (it is not prime), such as 4, 6, 9.
- Prime factorization
- Writing a number as a product of primes, e.g. 24 = 2³ × 3. Used to find GCF and LCM.
- Absolute value
- The distance of a number from zero on the number line; always non-negative. |−8| = 8.
- Adding integers (same sign)
- Add the absolute values and keep the common sign: −5 + (−3) = −8.
- Adding integers (different signs)
- Subtract the smaller absolute value from the larger and keep the sign of the larger: −7 + 4 = −3.
- Subtracting integers
- Add the opposite: a − b = a + (−b). So 5 − (−3) = 5 + 3 = 8.
- Multiplying/dividing signs
- Same signs give a positive result; different signs give a negative result.
- Proportion
- An equation stating two ratios are equal, a/b = c/d. Solve by cross-multiplying: a·d = b·c.
- Ratio
- A comparison of two quantities by division, written a:b or a/b.
- Unit rate
- A rate with a denominator of 1, like miles per hour or cost per item.
- Percent
- A ratio out of 100. 35% means 35 per 100, or 0.35.
- Percent to decimal
- Divide by 100 (move the decimal two places left): 7.5% = 0.075.
- Decimal to percent
- Multiply by 100 (move the decimal two places right): 0.625 = 62.5%.
- Fraction to decimal
- Divide the numerator by the denominator: 5/8 = 0.625.
- Percent change
- Change ÷ original value × 100. Always divide by the starting amount.
- Percent increase by 25%
- Multiply the original by 1.25.
- Percent decrease by 25%
- Multiply the original by 0.75.
- Exponent rule: product of powers
- xᵐ · xⁿ = xᵐ⁺ⁿ — add exponents when multiplying like bases.
- Exponent rule: quotient of powers
- xᵐ ÷ xⁿ = xᵐ⁻ⁿ — subtract exponents when dividing like bases.
- Exponent rule: power of a power
- (xᵐ)ⁿ = xᵐⁿ — multiply the exponents.
- Zero exponent
- x⁰ = 1 for any x ≠ 0.
- Negative exponent
- x⁻ⁿ = 1/xⁿ — take the reciprocal.
- Fractional exponent
- A fractional exponent means a root: a number raised to the one-nth power is its nth root. Example: 8 to the two-thirds power = (³√8)² = 2² = 4.
- Square root
- √a is the non-negative value whose square is a: √49 = 7.
- Cube root
- ³√a is the value whose cube is a: ³√(−125) = −5.
- Perfect square
- A number that is the square of an integer: 1, 4, 9, 16, 25, 36, ….
- Scientific notation
- Writing a number as a × 10ⁿ with 1 ≤ a < 10, e.g. 45,000 = 4.5 × 10⁴.
- Distributive over fractions
- To compare fractions, find a common denominator or convert to decimals.
- Mixed number to improper fraction
- Multiply whole × denominator, add the numerator, keep the denominator: 2⅓ = 7/3.
- Reciprocal
- The multiplicative inverse: the reciprocal of a/b is b/a; a number times its reciprocal is 1.
- Dividing fractions
- Multiply by the reciprocal of the divisor: (a/b) ÷ (c/d) = (a/b) × (d/c).
- Factorial
- n! is the product of all positive integers up to n: 4! = 4·3·2·1 = 24.
- Rounding
- Look at the digit to the right of the target place: 5 or more rounds up, less rounds down.
- Distributive property of multiplication
- Multiplication distributes over addition: a(b + c) = ab + ac.
- Estimating with rounding
- Round numbers to convenient values to estimate a sum, difference, or product quickly.
- Variable
- A symbol (usually a letter) that represents an unknown or changing quantity.
- Coefficient
- The number multiplied by a variable. In 7x, the coefficient is 7.
- Like terms
- Terms with the same variable raised to the same power; only like terms can be combined.
- Distributive property
- a(b + c) = a·b + a·c — multiply the outside factor by each term inside.
- Combining like terms
- Add or subtract the coefficients of like terms: 7a − 2a = 5a.
- Linear equation
- An equation whose graph is a straight line; the variable appears only to the first power.
- Solving one-step equations
- Undo the single operation with its inverse on both sides.
- Solving two-step equations
- Undo addition/subtraction first, then multiplication/division: 3x − 7 = 11 → x = 6.
- Balancing equations
- Whatever operation you do to one side, do to the other to keep equality.
- Checking a solution
- Substitute it back into the original equation; both sides should be equal.
- Slope-intercept form
- y = mx + b, where m is the slope and b is the y-intercept.
- Slope formula
- m = (y₂ − y₁)/(x₂ − x₁) — the change in y over the change in x.
- Standard form (linear)
- Ax + By = C, where A, B, and C are constants.
- Point-slope form
- y − y₁ = m(x − x₁), using a known point and the slope.
- y-intercept
- The point where a graph crosses the y-axis (x = 0); the value of b in y = mx + b.
- x-intercept
- The point where a graph crosses the x-axis (y = 0).
- Inequality
- A comparison with <, >, ≤, or ≥. Solve like an equation.
- Inequality sign-flip rule
- Flip the inequality sign when multiplying or dividing both sides by a negative.
- Graphing inequalities
- Open circle for < or >, closed circle for ≤ or ≥; shade the solution direction.
- System of equations
- Two or more equations solved together; the solution satisfies all of them.
- Solving systems by substitution
- Solve one equation for a variable and substitute into the other.
- Solving systems by elimination
- Add or subtract equations to cancel a variable, then solve.
- System with no solution
- Lines are parallel — same slope, different intercept.
- System with infinite solutions
- The equations are multiples of each other (same line).
- Factoring x² + bx + c
- Find two numbers that multiply to c and add to b. x² − 5x − 6 = (x − 6)(x + 1).
- Difference of squares
- a² − b² = (a + b)(a − b).
- Perfect-square trinomial
- (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².
- Quadratic equation
- An equation of the form ax² + bx + c = 0.
- Quadratic formula
- x = (−b ± √(b² − 4ac)) ÷ (2a).
- Discriminant
- b² − 4ac; positive → two real roots, zero → one, negative → none.
- Translating words to algebra
- 'A number decreased by 4' is n − 4; 'twice a number' is 2n.
- Evaluating an expression
- Substitute the given values for the variables, then simplify.
- Inverse operations
- Operations that undo each other: + and −, × and ÷, power and root.
- Direct variation
- y = kx; y changes in constant proportion to x, with constant k.
- Inverse variation
- y = k/x; as x increases, y decreases proportionally.
- Constant of proportionality
- The fixed ratio k in y = kx (or y/x = k).
- Two-variable expression
- An expression with two variables, like 3x + 2y, evaluated by substituting both values.
- Solving for a variable
- Isolate the target variable using inverse operations, treating others as constants.
- Literal equation
- An equation with several variables (a formula) solved for one variable in terms of the others.
- Consecutive integers
- Integers in a row: n, n + 1, n + 2, …; used to model word problems.
- Function
- A rule assigning exactly one output to each input. Written f(x).
- Function notation
- f(x) names the output for input x; f(3) means evaluate at x = 3.
- Domain
- The set of all valid inputs (x-values) of a function.
- Range
- The set of all outputs (y-values) a function produces.
- Vertical line test
- A graph is a function if no vertical line crosses it more than once.
- Domain restriction (denominator)
- Exclude any input that makes a denominator zero.
- Domain restriction (radical)
- A square root's radicand must be ≥ 0, so x must satisfy that condition.
- Linear function
- A function whose graph is a straight line: f(x) = mx + b.
- Slope of a function
- The constant rate of change m in f(x) = mx + b.
- Constant function
- A horizontal line, f(x) = c; the output is the same for every input.
- Quadratic function
- f(x) = ax² + bx + c; its graph is a parabola.
- Parabola vertex
- The maximum or minimum point of a parabola; the x-value is −b/(2a).
- Zeros of a function
- Inputs where f(x) = 0 — the x-intercepts of the graph.
- Exponential function
- y = a·bˣ; multiplies by a constant factor each step.
- Exponential growth vs decay
- Growth when b > 1; decay when 0 < b < 1.
- Linear vs exponential
- Linear adds a constant amount each step; exponential multiplies by a constant factor.
- Inverse function
- Swaps inputs and outputs; written f⁻¹(x). For f(x) = 3x + 2, f⁻¹(x) = (x − 2)/3.
- Composition of functions
- (f ∘ g)(x) = f(g(x)) — apply g first, then f.
- Average rate of change
- (f(b) − f(a))/(b − a) over an interval [a, b] — the slope of the secant line.
- Increasing function
- Outputs rise as inputs increase (graph goes up left to right).
- Decreasing function
- Outputs fall as inputs increase (graph goes down left to right).
- Maximum value
- The largest output a function reaches; for a downward parabola, the vertex's y-value.
- Minimum value
- The smallest output a function reaches; for an upward parabola, the vertex's y-value.
- Absolute value function
- f(x) = |x − a| graphs as a V with vertex at (a, 0).
- Evaluating f(2a)
- Substitute 2a for x and simplify: if f(x) = x² − 4x + 3, f(2a) = 4a² − 8a + 3.
- Rate of growth (y = 2ˣ)
- The base 2 doubles the output for each unit increase in x — a 100% growth rate.
- Mapping diagram
- Shows each input arrow pointing to its single output; a function has one arrow per input.
- Table of values
- A function can be represented by a table pairing inputs with their outputs.
- Linear function from a table
- Constant first differences in y (for equal x-steps) indicate a linear function.
- Exponential from a table
- A constant ratio between successive y-values indicates an exponential function.
- Output of a function
- The value f(x) produces for a given input x; the dependent variable.
- Input of a function
- The value x fed into a function; the independent variable.
- Recursive pattern
- A rule that defines each term using the previous term.
- Function family
- A group of functions sharing a form, such as linear, quadratic, or exponential.
- Reading a graph for f(a)
- Find x = a on the horizontal axis, go up to the curve, and read the y-value.
- Acute angle
- An angle measuring less than 90°.
- Right angle
- An angle measuring exactly 90°.
- Obtuse angle
- An angle measuring between 90° and 180°.
- Straight angle
- An angle measuring exactly 180° (a straight line).
- Complementary angles
- Two angles whose measures sum to 90°.
- Supplementary angles
- Two angles whose measures sum to 180°.
- Vertical angles
- Opposite angles formed by two intersecting lines; they are equal.
- Angles in a triangle
- The interior angles of any triangle sum to 180°.
- Angles around a point
- Angles meeting at a point sum to 360°.
- Perimeter
- The distance around a two-dimensional figure (sum of the side lengths).
- Rectangle area
- A = l × w (length times width).
- Triangle area
- A = ½ × base × height.
- Trapezoid area
- A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the parallel sides.
- Parallelogram area
- A = base × height.
- Circle area
- A = πr², where r is the radius.
- Circle circumference
- C = 2πr = πd, where d is the diameter.
- Cylinder volume
- V = πr²h (base area times height).
- Rectangular prism volume
- V = l × w × h.
- Cube surface area
- SA = 6s², where s is the side length.
- Pythagorean theorem
- a² + b² = c² for the legs a, b and hypotenuse c of a right triangle.
- Pythagorean triple
- Whole numbers satisfying a² + b² = c², such as 3-4-5 and 5-12-13.
- Hypotenuse
- The side opposite the right angle in a right triangle; the longest side.
- Congruent figures
- Same shape and same size — all corresponding sides and angles equal.
- Similar figures
- Same shape, proportional sides; corresponding angles equal, sides scaled by a factor.
- Scale factor
- The ratio of corresponding side lengths between similar figures.
- Area scaling
- Under a scale factor k, area scales by k².
- Volume scaling
- Under a scale factor k, volume scales by k³.
- Translation
- A transformation that slides a figure without rotating or resizing it.
- Reflection
- A transformation that flips a figure over a line of symmetry.
- Rotation
- A transformation that turns a figure about a fixed point by a given angle.
- Dilation
- A transformation that resizes a figure by a scale factor from a center point.
- Coordinate plane
- A grid formed by the x-axis and y-axis, divided into four quadrants.
- Quadrant locations
- I (+,+), II (−,+), III (−,−), IV (+,−), counterclockwise from upper right.
- Distance formula
- d = √((x₂ − x₁)² + (y₂ − y₁)²), from the Pythagorean theorem.
- Midpoint formula
- M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- SOH-CAH-TOA
- sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
- Sum of polygon angles
- (n − 2) × 180° for a polygon with n sides.
- Surface area vs volume
- Surface area is in square units (covering); volume is in cubic units (filling).
- Radius vs diameter
- The diameter is twice the radius: d = 2r.
- Unit conversion
- Multiply by a conversion factor written as a ratio equal to 1 (e.g. 12 in / 1 ft).
- Line of symmetry
- A line that divides a figure into two mirror-image halves.
- Equilateral triangle
- A triangle with all three sides equal and all angles 60°.
- Isosceles triangle
- A triangle with two equal sides and two equal base angles.
- Mean
- The average: the sum of values divided by how many there are.
- Median
- The middle value of an ordered data set; resists outliers.
- Mode
- The value that appears most often in a data set.
- Range (statistics)
- The difference between the largest and smallest values.
- Outlier
- A value far from the rest of the data; it pulls the mean but not the median.
- Interquartile range (IQR)
- Q3 − Q1, the spread of the middle 50% of the data.
- First quartile (Q1)
- The median of the lower half of the data; the 25th percentile.
- Third quartile (Q3)
- The median of the upper half of the data; the 75th percentile.
- Standard deviation
- A measure of spread around the mean; larger means more spread.
- Bar graph
- A display that compares categories with bars.
- Histogram
- A display of a numeric variable's distribution using bins.
- Box plot
- A display showing the minimum, Q1, median, Q3, and maximum (five-number summary).
- Scatter plot
- A display of the relationship between two numeric variables.
- Two-way table
- A table showing frequencies across two categorical variables.
- Positive correlation
- As one variable increases, the other tends to increase.
- Negative correlation
- As one variable increases, the other tends to decrease.
- Correlation vs causation
- A relationship between variables does not prove one causes the other.
- Probability
- Favorable outcomes ÷ total equally likely outcomes; a value from 0 to 1.
- Probability of 0
- An impossible event.
- Probability of 1
- A certain event.
- Independent events
- One outcome does not affect the other; multiply their probabilities for 'and.'
- Mutually exclusive events
- Events that cannot both happen; add their probabilities for 'or.'
- Complement of an event
- P(not A) = 1 − P(A).
- Theoretical probability
- Based on equally likely outcomes (what should happen).
- Experimental probability
- Based on observed trial results (what did happen).
- Law of large numbers
- As trials increase, experimental probability tends toward theoretical probability.
- Counting principle
- If one choice has m options and another n, together there are m × n outcomes.
- Permutation
- An arrangement where order matters; choosing 3 roles from 8 = 8·7·6 = 336.
- Combination
- A selection where order does not matter.
- Sample space
- The set of all possible outcomes of an experiment.
- Empirical rule
- For a normal distribution, ~68%, 95%, 99.7% of data lie within 1, 2, 3 standard deviations.
- Normal distribution
- A symmetric, bell-shaped distribution centered on the mean.
- Mean of a frequency table
- Multiply each value by its frequency, sum, and divide by the total frequency.
- Weighted average
- An average where some values count more, each multiplied by its weight before summing.
- Skewed distribution
- Right-skewed: mean > median; left-skewed: mean < median.
- Five-number summary
- Minimum, Q1, median, Q3, maximum — the basis of a box plot.
- Frequency
- The number of times a value or category occurs in a data set.
- Relative frequency
- A category's frequency divided by the total — its proportion of the data.
- Random sample
- A sample chosen so every member of the population has an equal chance of selection.
- Biased sample
- A sample that does not fairly represent the population, skewing conclusions.
