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AP Precalculus 1.4 Polynomial Functions and Rates of Change Overview:

Polynomial Functions

A polynomial function is made up of terms with whole-number exponents, such as quadratic, cubic, or quartic functions.

The degree of the polynomial is the largest exponent, which tells us the general shape and end behavior of the graph.

The leading coefficient (the number in front of the highest power) determines whether the graph opens upward or downward for even degrees, or rises to the right versus left for odd degrees.

The zeros of a polynomial are the values of x that make the function equal to zero. These appear as the x-intercepts on the graph.

Average Rate of Change

The average rate of change describes how much the function’s output changes compared to the change in input over an interval.

It is the slope of the line connecting two points on the graph.

For example, if the function output increases quickly between two points, the average rate of change is large; if it decreases, the rate of change is negative.

Instantaneous Rate of Change (a preview to Calculus)

At a single point, the rate of change is harder to measure.

It can be estimated by looking at slopes of secant lines over smaller and smaller intervals.

This idea leads to the concept of a tangent line and is the foundation for derivatives in calculus.

Connections Between Polynomial Graphs and Rates of Change

When the rate of change is positive, the graph is increasing.

When the rate of change is negative, the graph is decreasing.

Turning points occur where the function switches from increasing to decreasing, or decreasing to increasing.

A polynomial of degree n can have at most n minus one turning points.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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