Unit 1: Polynomial and Rational Functions

Function

A relationship that maps each input value to exactly one output value.

Domain

The set of all allowable input values of a function (often the independent variable x).

Range

The set of all output values a function produces (often the dependent variable y).

Independent variable

The input variable of a function (commonly x), chosen freely from the domain.

Dependent variable

The output variable of a function (commonly y), determined by the function rule from the input.

Increasing (on an interval)

A function is increasing on an interval if whenever a<b in that interval, f(a)<f(b).

Decreasing (on an interval)

A function is decreasing on an interval if whenever af(b).

x-intercept

A point where a graph meets the x-axis; it occurs when the output is 0.

y-intercept

The point where a graph meets the y-axis; it occurs at x=0 if x=0 is in the domain.

Zero (of a function)

An input r such that f(r)=0; real zeros correspond to x-intercepts.

Concavity

A description of how the rate of change is changing (whether the rate is increasing or decreasing).

Concave up

Concavity where the rate of change is increasing.

Concave down

Concavity where the rate of change is decreasing.

Average Rate of Change (AROC)

The slope of the secant line over [a,b]: (f(b)-f(a))/(b-a).

Secant line

A line through two points (a,f(a)) and (b,f(b)) on a graph; its slope is the average rate of change.

AROC of a linear function

For a linear function, the average rate of change is constant (it does not change across intervals).

AROC pattern of a quadratic function

For equal-length x-intervals, a quadratic’s AROC changes at a constant rate; increasing AROC indicates concave up, decreasing AROC indicates concave down.

Polynomial function

A function of the form f(x)=an x^n + a{n-1}x^{n-1}+…+a1 x + a0 with n a nonnegative integer and a_n≠0.

Degree (of a polynomial)

The highest exponent n in a polynomial (with nonzero coefficient); it strongly influences end behavior and turning-point limits.

Leading coefficient

The coefficient an of the highest-degree term an x^n; it helps determine end behavior (up/down).

Leading term

The highest-degree term a_n x^n; it dominates the polynomial’s behavior for large |x|.

End behavior

How a function behaves as x→±∞; for polynomials it is determined by the leading term.

Even-degree polynomial end behavior

If degree n is even, both ends go the same way: an>0 means both up; an<0 means both down.

Odd-degree polynomial end behavior

If degree n is odd, ends go opposite ways: an>0 gives left down/right up; an<0 gives left up/right down.

Turning point

A point where a function changes from increasing to decreasing or decreasing to increasing; a degree-n polynomial has at most n−1 turning points.

Local (relative) maximum/minimum

A peak (maximum) or valley (minimum) relative to nearby points; can also occur at endpoints of a restricted domain.

Global (absolute) maximum/minimum

The greatest function value (absolute max) or least function value (absolute min) on the domain under consideration.

Point of inflection

A point where a graph changes concavity (rate of change switches from increasing to decreasing or vice versa).

Standard form (polynomial)

Expanded form an x^n + … + a0; useful for identifying degree, leading coefficient, and often the y-intercept.

Factored form (polynomial)

A product form such as a(x-r1)^{m1}(x-r2)^{m2}…; useful for zeros and multiplicities.

Linear factor (from a real zero)

If r is a real zero of f, then (x−r) is a factor of the polynomial.

Multiplicity

If f(x) contains (x−r)^k with g(r)≠0, then r is a zero with multiplicity k.

Odd multiplicity (zero behavior)

If a zero has odd multiplicity, the graph crosses the x-axis at that x-value (the function changes sign).

Even multiplicity (zero behavior)

If a zero has even multiplicity, the graph touches the x-axis and turns around (no sign change); larger multiplicity looks flatter.

Finite differences (degree from a table)

With evenly spaced x-values, the polynomial’s degree is the least n for which the nth successive differences of outputs are constant.

Binomial Theorem

Expansion rule: (a+b)^n = Σ_{k=0}^{n} (n choose k) a^{n-k} b^k.

Pascal’s Triangle

A triangular array that gives binomial coefficients (n choose k) used in expansions like (a+b)^n.

Polynomial long division

An algorithm to divide polynomials to rewrite P(x)/D(x) as Q(x) + R(x)/D(x); used for analyzing rational end behavior (e.g., slant asymptotes).

Quotient-Remainder form

When dividing P(x) by D(x) (D≠0): P(x)=D(x)Q(x)+R(x), with degree(R) < degree(D).

Remainder Theorem

When P(x) is divided by (x−c), the remainder equals P(c).

Factor Theorem

(x−c) is a factor of P(x) if and only if P(c)=0.

Synthetic division

A shortcut method for dividing a polynomial by a linear divisor (x−c), producing quotient coefficients and a remainder efficiently.

Complex conjugate pairs (real coefficients)

If a polynomial has real coefficients and a+bi (b≠0) is a zero, then a−bi is also a zero.

Real quadratic factor from conjugates

If zeros are a±bi, then the polynomial has factor (x-(a+bi))(x-(a-bi)) = (x-a)^2 + b^2.

Rational function

A function of the form f(x)=P(x)/Q(x), where P and Q are polynomials and Q(x)≠0.

Domain restrictions (rational)

Inputs that make the denominator zero must be excluded from the domain (solve Q(x)=0).

Hole (removable discontinuity)

A discontinuity created when a common factor cancels between numerator and denominator; the x-value is still excluded from the original function.

Vertical asymptote

A non-removable discontinuity where (after simplification) the denominator is zero and the function grows without bound near that x-value.

Horizontal asymptote (degree comparison)

For P/Q with degrees n and m: if nm then no horizontal asymptote.

Slant (oblique) asymptote

When degree(numerator) is exactly 1 more than degree(denominator), polynomial division gives f(x)=S(x)+R(x)/Q(x) with linear S(x); the slant asymptote is y=S(x).