AP PreCalculus Flashcards: Rates of Change in Linear and Quadratic Functions
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 graph is concave up, what must be true about its average rate of change?
The average rate of change over equal-length input-value intervals must be increasing for all small-length intervals.
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If a function's graph is concave up, what must be true about its average rate of change?
The average rate of change over equal-length input-value intervals must be increasing for all small-length intervals.
Geometric Interpretation of Average Rate of Change
The average rate of change over the closed interval [a, b] is geometrically represented by the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
What is the average rate of change over a closed interval [a, b]?
It is the slope of the secant line from the point (a, f(a)) to (b, f(b)).
What is a key difference between the rates of change for linear and quadratic functions?
A linear function's average rate of change is constant, while a quadratic function's average rates of change are themselves changing at a constant rate.
What two key calculations can be performed regarding rates of change for functions?
One can determine the average rates of change for a function, and one can also determine the change in those average rates of change.
How does the average rate of change for a quadratic function change?
The average rates of change for a quadratic function are changing at a constant rate.
What is the relationship between a function's average rates of change and its concavity?
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.
If a function's average rates of change over consecutive equal-length intervals can be described by a linear function, what type of function is it?
The function is a quadratic function.
Change in the Average Rates of Change
This is a secondary calculation performed to determine how the average rates of change themselves are changing for functions like quadratics.
How can the average rates of change for a quadratic function be described?
For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.
If you calculate the average rate of change for a function over several different intervals and get the same value every time, what can you conclude?
You can conclude the function is linear, as its average rate of change over any length input-value interval is constant.
What does it mean for the graph of a function to be concave up?
The graph is concave up when the average rate of change over equal-length input-value intervals is increasing for all small-length intervals.
What is the key characteristic of the average rate of change for a linear function?
For a linear function, the average rate of change over any length input-value interval is constant.
Why is the change in average rates of change constant for a quadratic function?
It is constant because the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.