AP PreCalculus Flashcards: Polynomial Functions and Rates of Change
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 14 cards to help you master important concepts.
If a function's rate of change is increasing on (-∞, 1) and decreasing on (1, ∞), what is significant about the input value x=1?
The input value x=1 is a point of inflection for the function, as its rate of change switches from increasing to decreasing.
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If a function's rate of change is increasing on (-∞, 1) and decreasing on (1, ∞), what is significant about the input value x=1?
The input value x=1 is a point of inflection for the function, as its rate of change switches from increasing to decreasing.
A polynomial is defined only on the interval [-5, 5]. It is decreasing from x=-5 to x=0. What kind of extremum is at x=-5?
The point at x=-5 is a local maximum because it is an included endpoint of a polynomial with a restricted domain.
Where can local extrema of a polynomial function occur?
Local extrema occur where the function switches between increasing and decreasing, or at the included endpoints of a restricted domain.
A polynomial of degree 4 is known to have a global maximum. What must be true about its leading coefficient, a₄?
Since an even-degree polynomial has a global maximum, its graph must open downwards, meaning its leading coefficient (a₄) must be a non-zero negative number.
Does a nonconstant polynomial function always have at least one local maximum or minimum between any two points?
No, this is only guaranteed to be true between two distinct *real zeros* of the function.
A 6th-degree polynomial has a leading coefficient that is positive. What type of global extremum must it have?
Because it is an even-degree polynomial, it must have a global extremum; a positive leading coefficient means it will have a global minimum.
What is the relationship between the real zeros of a polynomial and its local extrema?
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.
Define a nonconstant polynomial function.
A function equivalent to the form p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}, where n is a positive integer and the leading coefficient a_{n} is nonzero.
What is a point of inflection of a polynomial function?
It is an input value where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing.
What are the two primary conditions for the terms in the standard analytical form of a polynomial p(x)?
The degree `n` must be a positive integer, and the leading coefficient `a_n` must be a nonzero real number.
What happens to the rate of change of a function at a point of inflection?
At a point of inflection, the rate of change of the function itself stops increasing and starts decreasing, or vice versa.
What is a key characteristic regarding the extreme values of an even-degree polynomial function?
Polynomial functions of an even degree will always have either a global maximum or a global minimum.
A polynomial function has distinct real zeros at x = 0 and x = 4. What must exist on the interval (0, 4)?
There must be at least one local maximum or local minimum on the interval between x = 0 and x = 4.
What is a local (or relative) maximum or minimum?
It is an output value where a polynomial function switches between increasing and decreasing, or at an included endpoint of a polynomial with a restricted domain.