Georgia Unified Math flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rational exponent rule

a^(m/n) equals the n-th root of a raised to the m power. The denominator gives the root index and the numerator gives the exponent applied afterward.

Negative exponent

A base raised to a negative exponent equals the reciprocal of that base raised to the positive exponent, so a^(−n) = 1/a^n.

Imaginary unit i

The number i is defined so that i² = −1. Powers cycle through i, −1, −i, and 1, repeating every fourth power.

Rationalizing a denominator

Rewrite a fraction so no radical remains in the denominator. Multiply numerator and denominator by the radical or conjugate that clears it.

Scientific notation product

To multiply numbers in scientific notation, multiply the coefficients and add the exponents of ten, then adjust so the coefficient lies between 1 and 10.

Vector magnitude

For a vector with components x and y, its magnitude is the square root of x squared plus y squared, found by the distance formula.

Quadratic formula

For ax² + bx + c = 0 with a not zero, the solutions are x = (−b ± √(b² − 4ac)) / (2a).

Discriminant

The value b² − 4ac from a quadratic. Positive gives two real roots, zero gives one repeated root, negative gives two complex roots.

Vertex of a parabola

For y = ax² + bx + c, the vertex x-coordinate is −b/(2a). Substituting it back gives the maximum or minimum value.

Zero product property

If a product of factors equals zero, at least one factor must equal zero. This justifies solving factored equations by setting each factor to zero.

Logarithm product rule

The logarithm of a product equals the sum of the logarithms of the factors: log base b of xy equals log x plus log y.

Change of base formula

Any logarithm can be rewritten using common or natural logs: log base b of x equals ln x divided by ln b.

Exponential growth model

Continuous growth is modeled by A = Pe^(rt), where P is the initial amount, r the rate, and t the time. Negative r models decay.

Extraneous solution

A value produced by squaring both sides of an equation that does not satisfy the original equation. Always check answers in the original.

Vertical line test

A graph represents a function only if no vertical line crosses it more than once, meaning each input has exactly one output.

Composite function

The composition f(g(x)) applies g first, then f to that result. Composition order matters and is generally not commutative.

Inverse function

A function that reverses another, so f⁻¹(f(x)) = x. Its graph is the reflection of the original across the line y = x.

Even and odd functions

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis. Odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Average rate of change

Over an interval from a to b, it equals f(b) − f(a) divided by b − a. It is the slope of the secant line connecting the endpoints.

Arithmetic sequence formula

The n-th term of an arithmetic sequence is a₁ + (n − 1)d, where d is the constant difference between consecutive terms.

Geometric sequence formula

The n-th term of a geometric sequence is a₁ times r raised to the power n − 1, where r is the constant ratio.

Distance formula

The distance between two points equals the square root of the sum of the squared differences of their x-coordinates and y-coordinates.

Perpendicular slope relationship

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals negative one.

Similar figure scaling

Corresponding sides of similar figures are proportional. Areas scale by the square of the scale factor and volumes by its cube.

Sector area formula

The area of a circle sector equals the central angle divided by 360, multiplied by pi times the radius squared.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc. An angle inscribed in a semicircle is a right angle.

Sphere volume

The volume of a sphere is four-thirds times pi times the radius cubed. Its surface area is four times pi times the radius squared.

Mean versus median

The mean is the sum divided by the count and shifts toward outliers. The median is the middle ordered value and resists outliers.

Standard deviation

A measure of typical distance from the mean. Square deviations, average them, then take the square root of that average.

Empirical rule

In a normal distribution roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

Z-score

A z-score of (x − μ)/σ reports how many standard deviations a value lies above or below the mean of its distribution.

Conditional probability

The probability of A given B equals the probability of both events divided by the probability of B, provided B has nonzero probability.

Permutation versus combination

Permutations count ordered arrangements using n!/(n − r)!. Combinations count unordered selections using n!/(r!(n − r)!).