AP Precalculus - Core Concepts

Front

What is the definition of a function?

Back

A function is a mathematical relation that maps a set of input values (domain) to a set of output values (range) such that each input value is mapped to exactly one output value. Functions can be represented graphically, numerically, analytically, or verbally.

Front

How do you determine if a function is increasing over an interval?

Back

A function f is increasing over an interval if, as the input values increase, the output values always increase. Mathematically: for all a and b in the interval, if a < b, then f(a) < f(b). On a graph, the function rises as you move left to right.

Front

Define average rate of change of a function over an interval [a, b].

Back

The average rate of change of f over [a, b] is the constant rate that yields the same change in output values as the function: [f(b) - f(a)] / (b - a). Geometrically, this equals the slope of the secant line connecting points (a, f(a)) and (b, f(b)).

Front

What is unique about the average rate of change for a linear function?

Back

For a linear function, the average rate of change over ANY interval of the domain is constant. This constant rate equals the slope of the line. This property distinguishes linear functions from nonlinear functions, whose rates of change vary.

Front

Describe the pattern of average rates of change for consecutive equal-length intervals in a quadratic function.

Back

For a quadratic function, the average rates of change over consecutive equal-length input-value intervals form a linear pattern (arithmetic sequence). The first differences of output values change by a constant amount, revealing the quadratic nature.