Topic 1.4 Polynomial Functions and Rates of Change: relate the degree of a polynomial to its number of local extrema and points of inflection, and analyze where its rate of change is positive, negative or zero.

What this topic is asking

The College Board (Topic 1.4) wants you to analyze how a polynomial changes by reading its local extrema (local maxima and minima) and its points of inflection. You should relate the degree of the polynomial to the maximum number of each, and describe the intervals where the rate of change is positive (increasing), negative (decreasing) or zero (at a turning point).

Local extrema

At every local extremum the rate of change is zero: the graph levels off momentarily before reversing direction. Between consecutive extrema the polynomial is strictly increasing or strictly decreasing.

Degree and the number of extrema

These bounds let you reject impossible graphs quickly: a degree-$3$ polynomial cannot have three turning points, and a parabola cannot have a point of inflection.

Points of inflection

A point of inflection is where the concavity changes, from concave up to concave down or vice versa. The rate of change need not be zero there; what changes is whether the rate of change is increasing or decreasing. Cubics always have exactly one point of inflection, which is also their center of symmetry.

Putting degree, extrema and inflection together

The power of this topic is reading a polynomial's behavior straight from its degree and a few features. If a graph shows three turning points, the degree is at least $4$, because $n - 1 \geq 3$ forces $n \geq 4$. If a table or context implies one change of concavity, the polynomial is at least degree $3$. Working backward from observed features to a minimum degree, and forward from a stated degree to the maximum possible features, is exactly the reasoning the exam expects.

A clarifying point is that the bounds are maxima, not exact counts. A degree-$5$ polynomial could have four turning points, or two, or none at all if the curve is monotonic, because turning points can coincide or vanish. What never happens is exceeding the bound. This asymmetry, an upper limit fixed by the degree but a lower count that depends on the specific coefficients, is the subtlety the exam tests when it asks for the maximum or minimum possible number of extrema for a given degree.

Try this

Q1. A polynomial has four local extrema. What is the smallest possible degree? [1 point]

• Cue. $n - 1 \geq 4$ gives $n \geq 5$, so the smallest degree is $5$.

Q2. At a local minimum, is the function changing from increasing to decreasing or decreasing to increasing? [1 point]

• Cue. Decreasing to increasing: the graph falls into the minimum and rises out of it.