1.3Rates of Change and Behavior of Graphs
Calculate average rate of change over an interval, identify increasing and decreasing intervals, and locate local maxima and minima from graphs and tables.
Average rate of change is the precalculus version of the derivative. Understanding how functions increase, decrease, and reach extrema is the conceptual foundation for all of differential calculus.
Essential Question
How can we use the average rate of change to describe how a function behaves over an interval, and what do increasing, decreasing, and local extrema tell us about a function's graph?
Lesson Overview
The average rate of change (AROC) of a function f over an interval [a, b] measures how much the output changes per unit of input change. It equals the slope of the secant line connecting the two endpoints on the graph. A function is increasing on an interval when its output values rise as x increases, and decreasing when they fall. A local maximum is a point where the function value is higher than all nearby points; a local minimum is lower than all nearby points. These extrema occur where a function switches from increasing to decreasing (or vice versa). Intervals of increase/decrease are always written using open interval notation with x-values only.
Average Rate of Change Formula
Average Rate of Change
AROC = f(b) − f(a)
b − a
over interval [a, b]
f(a) = function value at the left endpoint
f(b) = function value at the right endpoint
b − a = width of the interval (always positive)
This is the slope of the secant line connecting (a, f(a)) and (b, f(b)).
AROC = Slope of the Secant Line
AROC = rise / run = slope of the secant line connecting (a, f(a)) and (b, f(b))
Increasing / Decreasing Intervals & Local Extrema
Increasing
As x increases, f(x) increases. Intervals: (a, b) and (c, d)
Decreasing
As x increases, f(x) decreases. Intervals: (b, c) and (d, e)
Local Extrema
Local max at x = b, d. Local min at x = c. Always use open intervals.
Worked Examples
Find the average rate of change of f(x) = x² + 1 over the interval [1, 4].
Identify a = 1 and b = 4.
f(1) = (1)² + 1 = 2
f(4) = (4)² + 1 = 17
AROC = (f(4) − f(1)) / (4 − 1) = (17 − 2) / 3 = 15 / 3 = 5
Find the average rate of change of g(x) = −2x + 7 over [0, 3].
g(0) = −2(0) + 7 = 7
g(3) = −2(3) + 7 = 1
AROC = (1 − 7) / (3 − 0) = −6 / 3 = −2
A function has values: f(1) = 3, f(2) = 7, f(3) = 5, f(4) = 9, f(5) = 6. Identify all local maxima and minima.
Compare each value to its neighbors.
f(2) = 7 > f(1) = 3 and f(2) = 7 > f(3) = 5 → local maximum at x = 2.
f(3) = 5 < f(2) = 7 and f(3) = 5 < f(4) = 9 → local minimum at x = 3.
f(4) = 9 > f(3) = 5 and f(4) = 9 > f(5) = 6 → local maximum at x = 4.
From a graph, a function increases on (−∞, −2), decreases on (−2, 1), and increases on (1, ∞). Identify all local extrema.
Where the function switches from increasing to decreasing: x = −2 → local maximum.
Where the function switches from decreasing to increasing: x = 1 → local minimum.
Find the average rate of change of h(x) = √x over [4, 9].
h(4) = √4 = 2
h(9) = √9 = 3
AROC = (3 − 2) / (9 − 4) = 1 / 5 = 0.2
Guided Practice
External Supplemental Resource
Guided Practice Video: Rates of Change and Behavior of Graphs
Review how to identify where a function is increasing or decreasing and locate relative extrema from a graph.
Video by Mathispower4u on YouTube
Watch on YouTube ↗Find the average rate of change of f(x) = 3x² − 2 over [−1, 2].
Hint: Calculate f(−1) and f(2) first, then apply AROC = (f(b) − f(a)) / (b − a).
A function has values: f(0) = 4, f(1) = 6, f(2) = 3, f(3) = 7, f(4) = 5. Find the average rate of change over [0, 4] and identify any local extrema.
Hint: AROC uses only the endpoints. For extrema, compare each interior value to its neighbors.
A graph shows a function increasing on (−3, 0) and decreasing on (0, 5). What is the local behavior at x = 0?
Hint: When a function switches from increasing to decreasing, what type of extremum occurs?
Find the average rate of change of f(x) = x³ over [−2, 2]. What does the sign of the AROC tell you?
Hint: f(−2) = (−2)³ = −8. f(2) = 8. Positive AROC means the function generally increases over the interval.
A function decreases on (1, 4) and increases on (4, 7). State the local behavior at x = 4 and write the intervals using correct notation.
Hint: Decreasing then increasing → what type of extremum? Use open interval notation with x-values only.
Key Vocabulary
Average Rate of Change (AROC)
The ratio (f(b) − f(a)) / (b − a). Measures the average change in output per unit change in input over [a, b]. Equals the slope of the secant line.
Secant Line
A line connecting two points on a curve. Its slope equals the average rate of change between those two points.
Increasing on an Interval
A function is increasing on (a, b) if f(x₁) < f(x₂) whenever x₁ < x₂ in (a, b). The graph rises left to right.
Decreasing on an Interval
A function is decreasing on (a, b) if f(x₁) > f(x₂) whenever x₁ < x₂ in (a, b). The graph falls left to right.
Local Maximum
A point (c, f(c)) where f(c) ≥ f(x) for all x near c. The function switches from increasing to decreasing at x = c.
Local Minimum
A point (c, f(c)) where f(c) ≤ f(x) for all x near c. The function switches from decreasing to increasing at x = c.
Interactive Practice — 5 Questions
What is the average rate of change of f(x) = x² over [2, 5]?
A function switches from increasing to decreasing at x = 3. What occurs at x = 3?
Which notation correctly describes an interval of increase?
f(1) = 10, f(6) = −5. What is the AROC over [1, 6]?
The AROC of a linear function f(x) = mx + b over any interval equals:
Independent Practice
Independent Practice
Find the average rate of change of f(x) = 2x² − x over [0, 3].
A graph shows a function increasing on (−∞, −1), decreasing on (−1, 2), and increasing on (2, ∞). Identify all local extrema and their types.
Find the average rate of change of f(x) = 1/x over [1, 4]. Is the function increasing or decreasing on this interval? How does the AROC confirm this?
A table of values: x = 0, 1, 2, 3, 4 with f(x) = 5, 8, 6, 9, 7. Find the AROC over [0, 4] and identify all local maxima and minima.
Explain why the AROC of a function over [a, b] can be zero even if the function is not constant on [a, b]. Give an example.
Common Mistakes
Using closed brackets [ ] for intervals of increase/decrease — e.g., writing [2, 5].
Always use open brackets ( ) for intervals of increase and decrease: (2, 5). Endpoints are not included.
Confusing AROC with instantaneous rate of change — treating AROC as the slope at a single point.
AROC is the average over an interval [a, b]. It equals the slope of the secant line, not the tangent line.
Including y-values in interval notation — e.g., 'increasing from y = 2 to y = 8'.
Intervals of increase/decrease use x-values only: 'increasing on (2, 5)'.
Forgetting that a negative AROC means the function decreased on average — not that it decreased everywhere.
AROC = −3 means the net change was negative. The function could have gone up and down within the interval.
Math Tips
AROC = (f(b) − f(a)) / (b − a) = slope of the secant line. Always subtract in the same order: f(b) − f(a) over b − a.
For a linear function f(x) = mx + b, the AROC over any interval always equals m (the slope).
Local max: function switches from increasing → decreasing. Local min: decreasing → increasing. The switch is the key signal.
Always write increasing/decreasing intervals with open parentheses ( ) and x-values only — never y-values, never closed brackets at extrema.
AROC = 0 does not mean the function is constant — it means the starting and ending values are equal (like a round trip).