Functions and Rate of Change

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 1 output value.

Range

The set of output values of a function, represented by the dependent variable.

Domain

The set of input values of a function, represented by the independent variable.

Average Rate of Change

In this course, it is what we used to think of as the slope.

Decreasing Function

If a < b, then function f is decreasing as long as f(a) > f(b).

Increasing Function

If a < b, then function f is increasing as long as f(a) < f(b).

Relative Minimum

A function has a relative minimum if at some point, the graph of the function changes from decreasing to increasing.

Relative Maximum

A function has a relative maximum if at some point, the graph of the function changes from increasing to decreasing.

Absolute Maximum

If a polynomial p has an even degree leading term and a negative leading coefficient, there is an absolute maximum on function p.

Absolute Minimum

If a polynomial p has an even degree leading term and a positive leading coefficient, there is an absolute minimum on function p.

Relative Extrema

Between 2 real zeros of a polynomial, there must be at least one relative maximum or relative minimum.

Concave Down

If the average rate of change over equal length interval values is decreasing, the function is concave down over that interval.

Concave Up

If the average rate of change over equal length interval values is increasing, the function is concave up over that interval.

Point of Inflection

Occurs when a function changes from concave down to concave up or from concave up to concave down.

Linear Function

Has a constant rate of change.

Quadratic Function

The rate of change of its rate of change is constant.

Reference Angles

All are positive.