AP Precalculus 1.1-1.6

Increasing Interval

A function is increasing over its interval of its domain if, as the input values increase, the output values always increase. That is for all a and b in the interval, if a < b then f(a) < f(b)

Number of Complex zeros

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

Odd functions

An odd function is graphically symmetric about the point (0,0) and analytically has the property f(-x) = -f(x)

End behavior (cont.)

As input values of a nonconstant polynomial function decrease without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is lim p(x) = ∞ or lim p(x) = -∞

End behavior

As input values of a nonconstant polynomial function increase without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is lim p(x) = ∞ or lim p(x) = -∞

Function Definition

A function is 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.

Decreasing Interval

A function is decreasing over its interval of its domain if, as the input values increase, the output values always decrease. That is for all a and b in the interval, if a < b then f(a) > f(b)

Even functions

An even function is graphically symmetric over the line x = 0 and analytically has the property f(-x) = f(x)

Linear Functions

For a linear function, since the average rates of change over consecutive equal-length input value intervals can be given by a constant function, these average rates of change for a linear function are changing at a rate of zero.

Quadratic functions

For a quadratic function, since the average rates of change over consecutive equal-length input value intervals can be given by a linear function, these average rates of change for a quadratic function are changing at a constant rate.

Conjugate

If a + bi is a non-real zero of a polynomial function p, then its conjugate a - bi is also a zero of p.

Real Zero

If a is a real zero of a polynomial function p, then the graph of y=p(x) has an x-intercept at the point (a,0).

Multiplicity

If a linear factor (x-a) is repeated n time, the corresponding zero of the polynomial function has multiplicity n.

Leading Coefficient of a polynomial

In standard form, it is the coefficient of the term with the highest degree. In factored form, it is the product of all of the coefficients of each factor's highest degree term.

Degree of a Polynomial

In standard form, the degree of the polynomial is the degree of the highest degreed term. In factored form, the degree of the polynomial is the sum of the degrees of each factor.

Absolute extrema

Of all local maxima, the greatest is called the global, or absolute, maximum. Likewise, the least of all local minima is called the global, or absolute, minimum.

Point of Inflection

Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing. This occurs where the graph of a polynomial function changes from concave up to concave down or from concave down to concave up.

Using successive differences

The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial is equal to the least value of n for which the successive nth differences are constant.

Zeros

The graph intersects the x-axis when the output value is zero. The corresponding input values are said to be zeros of the function

Concave down

When the average rate of change over equal length input-value intervals is decreasing for all small-length intervals, the graph of the function is concave down

Concave up

When the average rate of change over equal length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up

Relative minimum

Where a polynomial function switches between decreasing and increasing or at the included endpoint of a polynomial with a restricted domain

Relative maximum

Where a polynomial function switches between increasing and decreasing or at the included endpoint of a polynomial with a restricted domain

Rate of Change

the ratio of the change in the output values of a function to the change in the input values over that interval

Domain

the set of input values of a function

Range

the set of output values of a function

Independent Variable

the variable representing the input values of a function

Dependent Variable

the variable representing the output values of a function