1.4 Polynomial Functions and Rates of Change

Average rate of change and concavity

For a polynomial function, the average rate of change over an interval is the slope of the secant line connecting its endpoints. When average rates of change over consecutive equal-width intervals are increasing, the graph is concave up; when decreasing, it's concave down. This connects local behavior to the shape of the curve.

Rate of change and end behavior

As inputs get large in magnitude, a polynomial's average rate of change over consecutive intervals of equal width itself changes in a predictable way tied to the degree of the polynomial. Higher-degree terms dominate far from the origin, so end behavior matches the leading term: odd-degree functions rise/fall in opposite directions, even-degree functions match directions on both ends.

Local extrema and inflection points

A polynomial's graph changes from increasing to decreasing (or vice versa) at local extrema, where the average rate of change switches sign. Points where concavity changes are called inflection points. Between consecutive zeros or turning points, the function maintains consistent increasing/decreasing behavior, useful for sketching accurate graphs.

Key terms

Average rate of change (polynomial)

Slope of the secant line between two points on the graph; equals change in output divided by change in input.

Concave up

Graph shape where average rates of change over consecutive equal intervals are increasing; curve bends upward.

Concave down

Graph shape where average rates of change over consecutive equal intervals are decreasing; curve bends downward.

Inflection point

A point on the graph where concavity changes from up to down or down to up.

End behavior of a polynomial

Determined by the degree and sign of the leading coefficient; describes how f(x) behaves as x approaches ±infinity.

Try an AP-style question

What is the greatest number of turning points a degree-4 polynomial can have?

1. 5
2. 3
3. 2
4. 4