AP Syllabus focus:
‘Key features of the graphs of f, f′, and f″ are related, allowing us to match each derivative graph to its original function by comparing slopes and concavity.’
This section explains how the graphs of a function f, its first derivative f′, and its second derivative f″ relate visually, enabling students to match the correct graph to each expression through slope and concavity behavior.
Understanding the Purpose of Matching Graphs
Matching graphs of f, f′, and f″ requires recognizing how changes in a function appear in its derivatives. Because differentiation transforms geometric behavior into slope and concavity information, each graph encodes different but connected features. Identifying these connections helps students interpret complex graphs and confirm consistency across representations.
Key Relationships Between f, f′, and f″
The relationships among a function and its derivatives rest on how differentiation captures slope and curvature. When interpreting graphs, students should focus on how visual behaviors correspond to algebraic features.
Slope Behavior and f′
The first derivative records the instantaneous rate of change of f, which directly relates to the steepness of the graph of f.
Instantaneous Rate of Change: The slope of the tangent line to the graph of a function at a given point.
A function’s rise or fall is reflected in the sign and magnitude of its derivative. The transitions between positive and negative slopes reveal major structural features of the function.
Concavity Behavior and f″
The second derivative describes how the slope of f is changing and therefore indicates the concavity of the graph.
Concavity: The direction in which a graph bends, determined by whether its slope is increasing or decreasing.
Where f bends upward or downward determines whether f″ is positive or negative. This connection helps students verify matches across multiple graphs.
Concave-up graph of with tangent lines whose slopes rise from left to right. This illustrates that increasing tangent slopes imply is increasing and is positive. The figure does not graph or directly but visually supports the relationships used when matching graphs of , , and . Source.
Identifying Key Features on Each Graph
Matching graphs requires extracting essential details from each representation.
Features of the Graph of f
The graph of f exhibits visible behaviors that translate into derivative characteristics. Important features include:
Increasing and decreasing intervals, indicated by the direction the graph moves left to right
Horizontal tangent lines, where the slope appears flat
Sharp turns or cusps, indicating points where differentiability may fail
Changes in concavity, seen where the graph transitions between bending upward and downward
Each of these features corresponds to identifiable patterns in f′ and f″.
Features of the Graph of f′
Because f′ displays slope information, students should examine:
Where f′ crosses the x-axis, marking potential critical points of f
Regions where f′ is positive or negative, indicating increasing or decreasing behavior of f
Peaks or valleys on f′, which correspond to potential inflection points on f
The general shape of f′ tells a consistent story about the direction and steepness of f.
Features of the Graph of f″
The second derivative graph shows:
Where f″ is positive (indicating f is concave up)
Where f″ is negative (indicating f is concave down)
Where f″ crosses zero, a necessary condition for potential points of inflection
These features allow the student to confirm whether the concavity behavior is consistent with the proposed graphs of f and f′.
Using Patterns to Match Graphs
To match the graphs successfully, students should compare patterns across all three representations.
Step-by-Step Matching Approach
Identify increasing or decreasing behavior of f.
Match those intervals to positive or negative regions on f′.
Locate horizontal tangents on f and match these to zeros of f′.
Three related curves showing , , and on the same axes. By studying horizontal tangents, maxima and minima, and concavity changes, one can determine which curve corresponds to each derivative. The figure includes only the essential features needed to practice matching the graphs. Source.
Observe concavity changes on f and pair them with sign changes in f″.
Identify peaks and troughs on f′, which correspond to zeros of f″.
Verify that all three graphs are mutually consistent in behavior.
This layered identification ensures each function corresponds correctly to its derivative graphs.
Confirming Consistency Across All Representations
After forming a tentative match, students must ensure that every identified feature corresponds appropriately:
A local maximum on f must appear where f′ changes from positive to negative.
A concave-up region on f must align with a positive portion of f″.
A point where f′ has its own maximum should coincide with an inflection point of f.
Any nondifferentiable behavior on f must be reflected by undefined or discontinuous behavior in f′.
Consistency checking allows students to verify that no contradictory features appear across the three graphs.
How Graph Shape Translates Through Derivatives
Understanding how shape transforms through differentiation reinforces correct matching.
From f to f′
When f rises steeply, f′ has large positive values. When f flattens out, f′ approaches zero. Sudden changes in direction in f appear as discontinuities or sharp behaviors in f′.
From f′ to f″
Because f″ describes the slope of f′, its positive or negative values correspond to rising or falling segments of f′. Consequently, a peak on f′ aligns with f″ = 0, indicating a transition in concavity for f.
These relationships form the backbone of reliable graph matching, allowing students to interpret multigraph problems with clarity and confidence.
FAQ
Large or rapid changes in the steepness of f appear as prominent peaks or troughs in f′, while mild changes create gentle variations.
Sudden curvature changes on f become sharp sign switches or noticeable extrema in f″.
Overall, the more dramatic the visual behaviour on f, the more pronounced the corresponding feature on its derivatives.
Focus on the most distinctive features first:
• Where slopes are zero.
• Where concavity switches.
• Where behaviour is unusually steep or flat.
Once one graph is confidently identified as f′ or f″, the remaining matching becomes more straightforward.
Yes. Differentiation can amplify noise, oscillation, or subtle curvature.
A function with gentle waves might have a derivative with sharper peaks or multiple crossings, making f′ appear more complex.
Likewise, f″ may show rapid sign changes even if f appears smooth.
If f′ merely touches the axis, the slope of f becomes zero but the function does not change from increasing to decreasing.
If f′ crosses the axis, f changes monotonicity.
Checking the behaviour immediately to the left and right of the zero helps confirm which case applies.
Inflection points on f correspond to locations where f″ changes sign.
These appear as:
• Zeros of f″ (usually with a sign change), and
• Local extrema on f′.
Recognising these connections provides an additional anchor when graphs of derivatives appear visually similar.
Practice Questions
Question 1 (1–3 marks)
The graphs of three functions, labelled A, B, and C, represent a function f, its first derivative f′, and its second derivative f″, though not necessarily in that order.
Graph A is always above the x-axis and is increasing.
Graph B crosses the x-axis at x = 1 and x = 4.
Graph C is decreasing throughout the interval shown.
a) Identify which graph corresponds to f′.
b) Briefly justify your choice.
Question 1
a) 1 mark
Correct identification: Graph C is f′.
b) 1–2 marks
Up to 2 marks for:
• Stating that a decreasing graph indicates negative values of the second derivative of f (if interpreted as f″), or negative slope of f (if interpreted as f′).
• Explaining that because Graph A is increasing, its derivative must be positive, which is inconsistent with Graph C unless Graph C is f′ of some other graph.
• Any clear argument that Graph C best represents the rate of change of one of the other graphs, particularly noting that Graph A increasing matches a positive f′, so Graph C must belong to another curve.
Award full credit for logically consistent reasoning identifying C as f′.
Question 2 (4–6 marks)
A function f is differentiable, and the graphs of f′ and f″ are shown separately.
• The graph of f′ is above the x-axis on the interval 0 < x < 3, equals zero at x = 3, and is below the x-axis for x > 3.
• The graph of f″ is positive for 0 < x < 2, negative for 2 < x < 5, and positive again for x > 5.
a) State the interval(s) on which f is increasing and explain why.
b) Determine whether f has a local maximum or minimum at x = 3. Give a reason.
c) Identify all intervals where the graph of f is concave up.
d) Determine the x-value(s) at which f has a point of inflection and justify your answer.
Question 2
a) 1–2 marks
• 1 mark for identifying the correct interval of increase: f is increasing on 0 < x < 3.
• 1 mark for stating that this is because f′ is positive on this interval.
b) 1–2 marks
• 1 mark for identifying the correct extremum: f has a local maximum at x = 3.
• 1 mark for justification: f′ changes from positive to negative at x = 3.
c) 1 mark
• Correct concave-up intervals: 0 < x < 2 and x > 5, based on f″ positive.
d) 1–2 marks
• 1 mark for identifying x = 2 and x = 5 as points where concavity changes.
• 1 mark for justification that f″ changes sign at both x = 2 and x = 5, so f has points of inflection there.
