NYS Math flashcards

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

Rational exponent rule

For a > 0, a^(m/n) equals the n-th root of a raised to the m power, or equivalently the n-th root of a^m. The denominator is the root and the numerator is the power.

Negative exponent

a^(−n) = 1/a^n for any nonzero base a. A negative exponent signals a reciprocal, not a negative value, so 2^(−3) equals 1/8.

Powers of i

i = √(−1) and i² = −1. The values i, −1, −i, 1 repeat with period 4, so reduce the exponent modulo 4 to evaluate any power of i.

Modulus of a complex number

For z = a + bi, the modulus |z| = √(a² + b²) is the distance from the origin in the complex plane. Multiplying z by its conjugate a − bi gives a² + b².

Scientific notation

A number written as a × 10^n with 1 ≤ |a| < 10. To multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents, then renormalize.

Percent error

Percent error = (|measured − actual| / |actual|) × 100%. It expresses the size of a measurement or estimate error relative to the true value.

Quadratic formula

x = (−b ± √(b² − 4ac))/(2a) gives the roots of ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac determines whether there are two, one, or no real roots.

Discriminant

For ax² + bx + c = 0, the discriminant is b² − 4ac. Positive gives two distinct real roots, zero gives one repeated real root, and negative gives no real roots.

Vertex of a parabola

The graph of y = ax² + bx + c has vertex at x = −b/(2a). Substitute that x to find the y-coordinate; the axis of symmetry is the vertical line through the vertex.

Difference of squares

a² − b² factors as (a − b)(a + b). Recognizing this pattern quickly factors expressions such as 9x² − 25 = (3x − 5)(3x + 5).

Exponential growth model

y = a·b^x, where a is the initial amount and b is the growth factor. Growth occurs when b > 1 and decay when 0 < b < 1; b = 1 + r for rate r.

Compound interest formula

A = P(1 + r/n)^(nt) gives the balance after t years when principal P earns annual rate r compounded n times per year. As n grows without bound the formula approaches A = Pe^(rt).

Logarithm product and quotient rules

log_b(xy) = log_b x + log_b y and log_b(x/y) = log_b x − log_b y. The power rule log_b(x^p) = p·log_b x lets you move exponents to the front.

Absolute value inequality

For k > 0, |x| < k means −k < x < k, while |x| > k means x < −k or x > k. The first gives a single interval; the second gives two rays.

Composition of functions

(f ∘ g)(x) = f(g(x)) means apply g first, then f. The domain of the composition excludes inputs where g(x) is undefined or lies outside the domain of f.

Inverse function

f⁻¹ reverses f, so f⁻¹(f(x)) = x. It exists only when f is one-to-one, and its graph is the reflection of f's graph across the line y = x.

Horizontal shift rule

Writing f(x + h) shifts the graph of f left by h units when h > 0, and right when h < 0. Changes inside the function affect the input and shift horizontally in the opposite direction.

Average rate of change

On the interval [a, b], the average rate of change of f is (f(b) − f(a))/(b − a). It equals the slope of the secant line joining the two points on the graph.

Even and odd functions

An even function satisfies f(−x) = f(x) and is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x) and is symmetric about the origin.

Arithmetic vs geometric sequences

An arithmetic sequence adds a constant difference: a_n = a₁ + (n−1)d. A geometric sequence multiplies by a constant ratio: a_n = a₁·r^(n−1).

Distance and midpoint formulas

The distance between two points is √((x₂−x₁)² + (y₂−y₁)²). Their midpoint is ((x₁+x₂)/2, (y₁+y₂)/2), the average of the coordinates.

Parallel and perpendicular slopes

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals, so their product is −1. A horizontal line is perpendicular to a vertical line.

Circle equation

(x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r. Complete the square on x and y terms to rewrite a general-form equation in this form.

Special right triangle ratios

A 45-45-90 triangle has sides in ratio 1 : 1 : √2. A 30-60-90 triangle has sides in ratio 1 : √3 : 2, with the hypotenuse twice the shorter leg.

Volume of a sphere

V = (4/3)πr³ gives the volume of a sphere of radius r. Surface area is 4πr²; both formulas use only the radius.

Similar figure scaling

When two figures are similar with scale factor k, corresponding lengths scale by k, perimeters scale by k, and areas scale by k². Volumes of similar solids scale by k³.

Reflection across y = x

Reflecting a point across the line y = x swaps its coordinates, mapping (x, y) to (y, x). This is the same relationship that links a function's graph to its inverse.

Effect of data transformations

Adding a constant c to every data value increases the mean and median by c but leaves the standard deviation unchanged. Multiplying every value by k multiplies the standard deviation by |k|.

Normal distribution rule

In a normal distribution about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. The z-score (x − μ)/σ measures distance in standard deviations.

Addition rule for probability

P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap so outcomes counted in both events are not double-counted; for mutually exclusive events the overlap is zero.

Conditional probability

P(A|B) = P(A and B)/P(B) gives the probability of A given that B occurred. Events A and B are independent exactly when P(A|B) equals P(A).

Permutations vs combinations

Permutations count ordered arrangements: nPr = n!/(n−r)!. Combinations count unordered selections: nCr = n!/(r!(n−r)!). Use combinations when the order of choices does not matter.