NY Regents Algebra II key terms

The number i, defined so that i² = −1. It lets us write square roots of negative numbers and serves as the building block for complex numbers.

A number of the form a + bi, where a and b are real numbers and i² = −1. The real part is a and the imaginary part is b.

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

An exponent written as the fraction m/n. The base is raised to the mth power and the nth root is taken, linking powers and radicals.

Rewriting a fraction so that no radical or imaginary unit remains in the denominator, usually by multiplying numerator and denominator by a suitable radical or conjugate.

An expression such as a_n x^n + ... + a_1 x + a_0, where exponents are nonnegative integers and coefficients are real. Its degree is the greatest exponent with nonzero coefficient.

If a polynomial P(x) is divided by x − c, the remainder equals P(c). This provides a quick way to evaluate a polynomial or test a possible factor.

A polynomial P(x) has x − c as a factor exactly when P(c) = 0. It links zeros, factors, and x-intercepts of a graph.

For a polynomial with integer coefficients, every rational zero p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

A shortcut for dividing a polynomial by x − c using only its coefficients. It produces the quotient and remainder with less writing than long division.

A value obtained while solving an equation that does not satisfy the original equation. It often arises after squaring both sides or multiplying by an expression that may be zero.

A ratio of two polynomials. Its domain excludes any values that make the denominator zero, and its operations follow the same rules as numerical fractions.

A sequence in which consecutive terms differ by a constant amount d. The nth term is a_1 + (n − 1)d.

A sequence in which each term after the first is found by multiplying by a constant ratio r. The nth term is a_1 r^(n−1).

A compact notation for a sum, using Σ with an index, lower and upper limits, and a rule for each term, as in Σ from k=1 to n of a_k.

A rule that assigns exactly one output to each allowed input. The set of allowed inputs is the domain, and the graph of a function passes the vertical line test.

The domain of a function is the set of all allowed input values; the range is the set of all output values actually produced.

A function that reverses another function, so f⁻¹(f(x)) = x. It exists only when the original function is one-to-one, and its graph reflects across y = x.

The combination (f ∘ g)(x) = f(g(x)), where the output of g becomes the input of f. The domain requires g(x) to be in f's domain.

The change in a function's output divided by the change in its input over an interval: [f(b) − f(a)]/(b − a). It equals the slope of the secant line.

A line that a graph approaches arbitrarily closely as the input or output grows without bound. Exponential graphs have horizontal asymptotes; rational graphs may have vertical ones.

A function is even when f(−x) = f(x), giving symmetry about the y-axis. It is odd when f(−x) = −f(x), giving symmetry about the origin.

A change to a parent graph such as a translation, reflection, stretch, or compression, described by parameters in an equation like a·f(b(x − h)) + k.

A function of the form f(x) = a·b^x, where b > 0 and b ≠ 1. It models constant percent growth or decay.

The inverse of an exponential function, written f(x) = log_b(x). It is defined only for x > 0 and has a vertical asymptote at x = 0.

A function such as sine or cosine that relates an angle to a ratio of side lengths or to coordinates on the unit circle; its graph is periodic.

A unit of angle measure defined as arc length divided by radius. A full circle is 2π radians, so π radians equals 180 degrees.

The circle of radius 1 centered at the origin. The point at angle θ has coordinates (cos θ, sin θ), which defines trigonometric functions for all angles.

An equation involving trigonometric functions that is true for every value of the variable, such as sin²θ + cos²θ = 1. Identities are used to simplify expressions.

In any triangle, the ratio of a side length to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C.

A relationship generalizing the Pythagorean theorem: c² = a² + b² − 2ab cos C. It solves triangles when two sides and the included angle or all three sides are known.

A symmetric, bell-shaped probability distribution described by its mean and standard deviation. About 68% of values lie within one standard deviation of the mean.

A measure of how spread out data values are around their mean, equal to the square root of the variance. Larger values indicate greater variability.

The number of standard deviations a data value lies from the mean, calculated as z = (x − μ)/σ. It allows comparison of values from different normal distributions.

The probability that event A occurs given that event B has occurred, written P(A|B) = P(A and B)/P(B), where P(B) > 0.

Two events are independent when the occurrence of one does not change the probability of the other, so P(A and B) = P(A)·P(B).

An ordered arrangement of items selected from a group. The number of permutations of n items taken r at a time is n!/(n − r)!.

A selection of items in which order does not matter. The number of combinations of n items taken r at a time is n!/[r!(n − r)!].

A line used to model the trend in bivariate data, often found by least squares. It can make predictions, and the correlation coefficient measures how well it fits.

A value describing the uncertainty in a sample estimate; the estimate plus or minus the margin gives an interval likely to contain the true population value.