AP Precalculus

~ 1.3-1.4: Polynomials & Rates of Change ~

1.3 Course Objectives

Determine the average rates of change for sequences and functions, including linear, quadratic, and other function types.
Determine the change in the average rates of change for linear, quadratic, and other function types.

For a linear function, the average rate of change over any length input-value interval is constant.
For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.
The average rate of change over the closed interval [a, b] is the slope of the secant line from the point (a, f(a)) to (b, f(b)).
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.
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.
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.
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.

Identify key characteristics of polynomial functions related to rates of change.

Know the analytical representation of a generic polynomial.
A constant is also a polynomial function of degree zero.
Where a polynomial function switches between increasing and decreasing, or at the included endpoint of a polynomial with a restricted domain, the polynomial function will have a local (or relative) maximum or minimum output value.
The greatest of all local maxima is called the global maximum, or absolute maximum.
The least of all local minima is called the global minimum, or absolute minimum.
Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum.
Polynomial functions of an even degree will have either a global maximum or a global minimum.

The video for AP Precalculus, 1.3-1.4: Polynomials and Rates of Change will come out on Thursday, September 7, 2023.