Praxis Math 5165 flashcards

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

Rational number

A number expressible as p/q where p and q are integers and q is not zero; its decimal form either terminates or eventually repeats.

Complex conjugate

For a complex number a + bi, the conjugate is a − bi. Multiplying a complex number by its conjugate yields the real number a² + b².

Absolute value equation

|x| is the distance from x to 0 on the number line; the equation |x| = k has the two solutions x = k and x = −k when k ≥ 0.

GCD and LCM relationship

For any two positive integers a and b, the greatest common divisor multiplied by the least common multiple equals the product a·b.

Scientific notation

A number written as a × 10ⁿ where 1 ≤ |a| < 10; it compactly expresses very large or very small quantities and simplifies multiplication and division.

Rationalizing a denominator

Rewriting a fraction so no radical remains in the denominator, done by multiplying numerator and denominator by the radical itself or by its conjugate.

Irrational number

A real number that cannot be written as a ratio of two integers; its decimal expansion is infinite and non-repeating, as with √2 and π.

Fractional exponent rule

For a positive base a, a^(m/n) equals the n-th root of a raised to the m power, which also equals the n-th root of aᵐ.

Quadratic formula

The solutions of ax² + bx + c = 0, with a not zero, are x = (−b ± √(b² − 4ac)) divided by 2a.

Discriminant

The quantity b² − 4ac from the quadratic formula; positive gives two real roots, zero gives one real root, and negative gives two complex roots.

Vertex of a parabola

For y = ax² + bx + c the vertex occurs at x = −b/(2a); the parabola opens upward when a > 0 and downward when a < 0.

Remainder theorem

Dividing a polynomial p(x) by x − c leaves remainder p(c); therefore c is a zero of the polynomial exactly when p(c) = 0.

Change of base formula

log_b x = (ln x)/(ln b), which equals (log x)/(log b); it lets logarithms in any base be evaluated with common or natural logs.

Inverse function property

If f and g are inverse functions, then f(g(x)) = x and g(f(x)) = x on suitable domains, and their graphs mirror across the line y = x.

Horizontal asymptote rule

For a rational function compare degrees: numerator lower gives y = 0, equal degrees give the ratio of leading coefficients, numerator higher gives none.

Exponential growth model

A(t) = A₀(1 + r)ᵗ or A(t) = A₀e^(kt) models growth when r > 0 or k > 0, and models decay when those values are negative.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is the square root of (x₂ − x₁)² plus (y₂ − y₁)², derived from the Pythagorean theorem.

Circle equation

A circle whose center is (h, k) and whose radius is r has the equation (x − h)² + (y − k)² = r².

30-60-90 triangle sides

In a 30°-60°-90° triangle the sides opposite those angles are in the ratio x : x√3 : 2x, so the hypotenuse is twice the shortest side.

Law of cosines

c² = a² + b² − 2ab·cos C relates the three sides of a triangle to the angle opposite side c; it reduces to the Pythagorean theorem at 90°.

Arc length and sector area

For a central angle θ measured in radians, arc length is s = rθ and the area of the corresponding sector is one-half r²θ.

Unit circle coordinates

A point on the unit circle at angle θ has coordinates (cos θ, sin θ), which produces the Pythagorean identity sin²θ + cos²θ = 1.

Similar figure ratios

If two figures are similar with scale factor k, then corresponding lengths scale by k, areas scale by k², and volumes scale by k³.

Polygon angle sums

A convex polygon with n sides has interior angles summing to (n − 2)·180°, while its exterior angles always sum to 360°.

Mean versus median

The mean is the sum of values divided by the count; the median is the middle ordered value, and skewed data pulls the mean toward the longer tail.

Empirical rule

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

z-score

The formula z = (x − μ)/σ states how many standard deviations a value x lies above or below the mean of its distribution.

Conditional probability

P(A|B) = P(A and B)/P(B) gives the probability of event A given that event B has occurred, defined whenever P(B) > 0.

Combinations versus permutations

Combinations count unordered selections using n!/(r!(n − r)!), while permutations count ordered arrangements using n!/(n − r)! for choosing r items from n.

Binomial probability

P(X = k) = C(n, k)·pᵏ·(1 − p)^(n − k) gives the probability of exactly k successes in n independent trials with success probability p.

Chain rule

The derivative of f(g(x)) is f′(g(x))·g′(x): differentiate the outer function, keep the inner function, then multiply by the inner derivative.

Fundamental theorem of calculus

If F is any antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a), the net signed area under the curve.