Unit 1a : Functions, Rate of Change, End Behavior, and Transformations

Vocabulary flashcards covering core concepts from Unit 1a and related sections (functions, rate of change, end behavior, asymptotes, and transformations).

Function

A mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.

Domain

Inputs (x)

Range

Outputs (y)

Function Positive

y-values above the x-axis.

Function Negative

y-values below the x-axis.

Function Increasing

As the input values increase, the output values increase; if a < b, then f(a) < f(b).

Function Decreasing

As the input values increase, the output values decrease; if a < b, then f(a) > f(b).

Concave Up

The rate of change is increasing.

Concave Down

The rate of change is decreasing.

Instantaneous Rate of Change (IROC)

The rate of change of the tangent line (at a point on a function.)

Average Rate of Change (AROC)

The rate of change of the secant line (between two points on the function.)

Positive AROC

As the input increases or decreases, the output changes in the same direction (slope is positive).

Negative AROC

As the input increases the output decreases, or vice versa (slope is negative).

Linear Function

The AROC over any length input intervals is constant.

Quadratic

The AROC over equal-length input intervals is linear (or the ROC of the AROC is constant).

AROC over [a,b]

AROC = [f(b) − f(a)] / (b − a).

Polynomial notation

If p(x) = an x^n + … + a1 x + a_0. Then A_n is the leading coefficient, n is the degree

Relative Max/Min

When a polynomial switches from increasing to decreasing or vice versa, or at the endpoint of a polynomial within a restricted domain.

Global Max/Min

The absolute/Global maximum is the greatest local maximum; the absolute minimum is the least local minimum

Point of Inflection (POI)

Occurs when the AROC changes from increasing to decreasing or vice versa ( where concavity changes).

Multiplicity

The number of times a linear factor (x − a) is repeated in the factored form of a polynomial.

Complex Conjugate

If a + bi is a non-real zero, then its conjugate a − bi is also a zero.

Fundamental Theorem of Algebra

A polynomial of degree n has exactly n complex zeros when counting multiplicity.

Degree of Polynomial

The number of successive differences needed for the difference to be constant = degree of the polynomial

Even Function

f(x) = f(−x).

Odd Function

f(−x) = −f(x).

Left End Behavior

Lim f(x)^n | x → −∞, As x-values decrease without bound the y values of f(x)….

Right End Behavior

Lim f(x)^n | x → ∞, As x-values increase without bound the y values of f(x)….

** local fact

Between 2 real zeros of a non-constant polynomial function there must be at least one local max or min

**Degree fact

Polynomials of Even Degree will have either a global min max