Polynomial and Rational Functions

Function ● (core concept) — A relation that assigns to each input value exactly one output value. The rule of a function, expressed graphically, numerically, analytically, or verbally, pairs each input value with its single corresponding output value (the image of the input); each output value has a set of input values that map to it (its preimage). Input values are controlled by the independent variable; the output values determined by the function are controlled by the dependent variable.

Domain ● (core concept) — The set of all input values that a function can accept.

Range ● (core concept) — The set of all output values that a function actually produces.

Equal functions ● (core concept) — Two functions f and g are equal when they share the same domain and give the same output for every input in that domain: f(a) = g(a) for every a in the domain.

Increasing function ● (core concept) — A function is increasing over an interval of its domain when larger inputs always give larger outputs: for all a and b in the interval, if a < b then f(a) < f(b). (Distinguishing whether the interval endpoints are included is outside the scope of the course.)

Decreasing function ● (core concept) — A function is decreasing over an interval of its domain when larger inputs always give smaller outputs: for all a and b in the interval, if a < b then f(a) > f(b). (Distinguishing whether the interval endpoints are included is outside the scope of the course.)

Concave up ● (core concept) — A graph is concave up on an interval when its rate of change is increasing over that interval; equivalently, its average rates of change over consecutive equal-length input intervals are increasing.

Concave down ● (core concept) — A graph is concave down on an interval when its rate of change is decreasing over that interval; equivalently, its average rates of change over consecutive equal-length input intervals are decreasing.

Zero of a function ● (core concept) — An input value a for which the output is zero, f(a) = 0. On a graph, zeros are the x-coordinates of the x-intercepts.

Average rate of change ● (core concept) — The constant rate of change that would produce the same change in output as the function did over an interval of its domain: the ratio of the change in output values to the change in input values, Δy/Δx = (f(b) − f(a))/(b − a). It is the slope of the secant line joining the points (a, f(a)) and (b, f(b)).

Rate of change at a point ● (core concept) — The rate at which the output values would change if the input values changed at that point; it can be approximated by average rates of change over small intervals containing the point, where such values exist. Comparing these values can indicate where a function is increasing, decreasing, or has local extrema.

Constant average rate of change (linear functions) ● (core concept) — A linear function's average rate of change is the same over any input interval of any length; its average rates of change change at a rate of zero.

Changing average rate of change (quadratic functions) ● (core concept) — A quadratic function's average rates of change over consecutive equal-length input intervals are given by a linear function, so they change at a constant rate. When these rates are increasing over small intervals the graph is concave up; when decreasing, it is concave down.

Polynomial function ● (core concept) — A nonconstant function whose analytical form is equivalent to p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ⋯ + a₁x + a₀, where n is a positive integer, each aᵢ is real, and aₙ ≠ 0. A polynomial's graph is continuous, with no sharp turns or abrupt changes.

Degree of a polynomial ● (core concept) — The highest power of x in the polynomial, n in the form above. The degree is also the least n for which the successive nth differences of output values over equal input intervals are constant and nonzero.

Leading term and leading coefficient ● (core concept) — The term aₙxⁿ with the highest power is the leading term; its coefficient aₙ (nonzero) is the leading coefficient. For inputs of large magnitude the leading term dominates all lower-degree terms, so the degree and the sign of the leading term determine the polynomial's end behavior.

Local maximum and minimum ● (core concept) — An output value where a polynomial switches from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum), or that occurs at an endpoint of a restricted domain. Between every two distinct real zeros of a nonconstant polynomial there is at least one local maximum or minimum.

Global maximum and minimum ● (core concept) — A local maximum greater than all other output values is the global (absolute) maximum; a local minimum less than all other output values is the global (absolute) minimum. A polynomial of even degree has either a global maximum or a global minimum; for a quadratic, that global extremum occurs at the vertex.

Point of inflection ● (core concept) — An input value where a polynomial's rate of change switches from increasing to decreasing or from decreasing to increasing — where the graph changes from concave up to concave down or the reverse.

Zero of a polynomial ● (core concept) — A complex number a with p(a) = 0 is a zero of p (a root of p(x) = 0). A real zero a gives an x-intercept at (a, 0); real zeros are endpoints of intervals satisfying polynomial inequalities p(x) ≥ 0 or p(x) ≤ 0.

Linear factor ● (core concept) — If a is a real zero of p, then (x − a) is a linear factor of p, and conversely: (x − a) is a linear factor of p if and only if a is a zero of p.

Multiplicity ● (core concept) — If the linear factor (x − a) is repeated n times in a factorization, the zero a has multiplicity n. If a real zero has even multiplicity, outputs have the same sign on both sides near a, so the graph is tangent to the x-axis there.

Fundamental theorem of algebra ● (core concept) — A polynomial function of degree n has exactly n complex zeros when multiplicities are counted.

Complex conjugate zeros ● (core concept) — Non-real zeros of a polynomial occur in conjugate pairs: if a + bi (with b ≠ 0) is a zero of p, then a − bi is also a zero.

Even function ● (core concept) — A function whose graph is symmetric over the line x = 0, satisfying f(−x) = f(x). Any monomial p(x) = aₙxⁿ with even n ≥ 1 and aₙ ≠ 0 is an even function.

Odd function ● (core concept) — A function whose graph is symmetric about the point (0, 0), satisfying f(−x) = −f(x). Any monomial p(x) = aₙxⁿ with odd n ≥ 1 and aₙ ≠ 0 is an odd function.

End behavior ● (core concept) — How a function's output values behave as inputs increase or decrease without bound. For a nonconstant polynomial, outputs go to ∞ or −∞ at each end — written lim(x→∞) p(x) = ±∞ and lim(x→−∞) p(x) = ±∞ — as determined by the degree and the sign of the leading term.

Rational function ● (core concept) — A function represented analytically as the quotient of two polynomial functions; for each input in its domain it measures the relative size of the numerator polynomial against the denominator polynomial.

Horizontal asymptote ● (core concept) — A horizontal line y = b that a rational function's outputs approach and stay near as inputs go to ±∞ (lim(x→±∞) r(x) = b). It occurs when neither the numerator nor the denominator polynomial dominates, so the quotient of the leading terms is a constant b. When the denominator polynomial dominates, the horizontal asymptote is y = 0.

Slant asymptote ● (core concept) — When the numerator polynomial dominates the denominator for inputs of large magnitude, the quotient of the leading terms is a nonconstant polynomial, and the rational function takes on that polynomial's end behavior. If that quotient is linear, the graph has a slant asymptote parallel to that line.

Zeros of a rational function ● (core concept) — The real zeros of a rational function are the real zeros of its numerator that are also in the function's domain. The real zeros of both the numerator and the denominator polynomials serve as endpoints or asymptotes for intervals satisfying rational inequalities r(x) ≥ 0 or r(x) ≤ 0.

Vertical asymptote ● (core concept) — A vertical line x = a that the graph approaches where the denominator's values get arbitrarily close to zero, so outputs increase or decrease without bound (lim(x→a⁺) r(x) = ±∞ and lim(x→a⁻) r(x) = ±∞). It occurs when a is a real zero of the denominator but not of the numerator — or more generally when a's multiplicity as a zero of the denominator is greater than its multiplicity as a zero of the numerator.

Hole ● (core concept) — A missing point in a rational function's graph. When the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, the graph has a hole at that input value rather than an asymptote. If outputs near x = c approach L, the hole is at (c, L), written lim(x→c) r(x) = L.

Analytic forms of polynomial and rational functions ● (core concept) — The factored form of a polynomial or rational function readily reveals its real zeros, and with them its x-intercepts, asymptotes, holes, domain, and range; the standard form reveals its end behavior.

Polynomial long division ● (core concept) — An algebraic process like numerical long division that divides a polynomial f by a polynomial g to get a quotient q and remainder r with f(x) = g(x)q(x) + r(x), where the degree of r is less than the degree of g. It is useful for finding equations of slant asymptotes.

Binomial theorem ● (core concept) — The expansion of (a + b)ⁿ using the entries of a single row of Pascal's Triangle, including polynomials of the form (x + c)ⁿ.

Vertical translation ● (core concept) — The additive transformation g(x) = f(x) + k shifts the graph of f vertically by k units.

Horizontal translation ● (core concept) — The additive transformation g(x) = f(x + h) shifts the graph of f horizontally by −h units; the graph moves left when h is positive.

Vertical dilation ● (core concept) — The multiplicative transformation g(x) = a·f(x), a ≠ 0, stretches or compresses the graph of f vertically by a factor of a; if a < 0 the graph is also reflected over the x-axis.

Horizontal dilation ● (core concept) — The multiplicative transformation g(x) = f(bx), b ≠ 0, stretches or compresses the graph of f horizontally by a factor of 1/|b|; if b < 0 the graph is also reflected over the y-axis.

Reflection ● (core concept) — A multiplicative transformation with a negative factor flips the graph: a < 0 in g(x) = a·f(x) reflects it over the x-axis; b < 0 in g(x) = f(bx) reflects it over the y-axis.

Preimage and image ● (core concept) — Under a transformation, the set of points before the transformation is the preimage; the set of points after is the image. Transformations can change a function's domain and range.

Piecewise-defined function ● (core concept) — A function made of a set of functions defined over nonoverlapping domain intervals, useful for modeling data or scenarios with different characteristics over different intervals.

Constant nth differences ● (core concept) — A polynomial function of degree n models data demonstrating roughly constant nonzero nth differences of output values over equal input intervals. As a guide: linear functions model roughly constant rates of change; quadratics model roughly linear rates of change, symmetric data with a unique maximum or minimum, or area/two-dimensional contexts; cubics model volume/three-dimensional contexts; and a polynomial of degree n or less fits n + 1 data points exactly.

Rational function model ● (core concept) — Data or contexts involving inversely proportional quantities can often be modeled by rational functions; for example, gravitational and electromagnetic forces are inversely proportional to the squared distance between the objects.