Understanding how the graph of a derivative reflects the behavior of the original function is essential for interpreting change, identifying trends, and analyzing intervals of increase or decrease.

Describing Behavior from a Graph of f′

When analyzing how a function behaves, interpreting the graph of its first derivative provides a powerful way to understand whether the function is rising, falling, or maintaining horizontal tangents. This subsubtopic focuses on translating the visual information from f′(x) into precise statements about the behavior of the original function f(x). Because the derivative represents the instantaneous rate of change, its graph directly reveals where the function increases, decreases, or has stationary points.

Understanding What a Graph of f′ Represents

The graph of f′(x) displays the numerical value of the derivative at each input x, which means it communicates slope information visually. When the graph lies above or below the x-axis, it reveals the sign of the derivative, which directly determines how the original function behaves.

A graph of the derivative therefore provides a continuous illustration of how steeply and in which direction the original function changes.

This figure shows four segments of a function with tangent lines labeled according to whether $f'(x)$ is positive or negative. It visually connects the sign of the derivative to increasing or decreasing behavior. The focus is exclusively on monotonicity without introducing techniques beyond this subsubtopic. Source.

Interpreting the Sign of f′ from Its Graph

The most essential technique for interpreting f′(x) is examining whether the graph is above the x-axis (positive) or below the x-axis (negative).

• When the graph of f′(x) lies above the x-axis, the derivative is positive, so f is increasing on that interval.

• When the graph of f′(x) lies below the x-axis, the derivative is negative, so f is decreasing on that interval.

• When the graph touches or crosses the x-axis at some value of x, we have f′(x) = 0, meaning the original function has a horizontal tangent at that point.

These observations allow us to describe behavior without manipulating algebraic expressions.

Identifying Horizontal Tangent Points

Points where the derivative crosses or touches the x-axis are extremely important because they indicate potential changes in the direction of the original function.

A horizontal tangent does not necessarily indicate a maximum or minimum; it simply marks a moment when the instantaneous rate of change pauses. Interpreting the sign of f′(x) on either side of such points helps determine how the function behaves near them.

Describing Increasing and Decreasing Behavior Visually

Using the graph of f′, we describe behavior of f by noting changes in the derivative’s sign and position relative to the x-axis.

This diagram compares $f(x)$ with its derivative $f'(x)$, showing increasing intervals where $f'(x)$ lies above the x-axis and decreasing intervals where it lies below. It demonstrates how the behavior of $f$ is read directly from the sign of $f'(x)$. Some additional features such as extrema and inflection behavior appear, though they are not required for this subsubtopic. Source.

Key steps for describing behavior:

• Identify intervals above the x-axis → f is increasing.

• Identify intervals below the x-axis → f is decreasing.

• Mark all x-intercepts of f′ → possible turning points of f.

• Connect sign intervals verbally to build a clear narrative about the function’s behavior.

These steps form the basis for writing mathematical justifications in AP Calculus.

Understanding Transitions in the Graph of f′

Transitions provide valuable information about change in behavior. A shift from positive to negative values of f′ means the original function transitions from increasing to decreasing, while a shift from negative to positive values reveals the opposite. However, identifying these transitions is only descriptive in this subsubtopic; classification of extrema belongs elsewhere in the syllabus.

Using f′ Graphs to Understand Trends in f

The graph of f′(x) can also help provide a deeper understanding of how the original function rises or falls.

Important insights include:

• Steeper positive values of f′ indicate faster increase in f.

• Steeper negative values of f′ indicate faster decrease in f.

• Portions of the derivative graph hovering close to the x-axis correspond to intervals where f changes very slowly.

These interpretations strengthen students’ ability to describe a function qualitatively.

Writing Clear Descriptions from f′ Graphs

When writing about behavior from a graph of f′, use precise statements grounded in the sign of the derivative. Strong AP-level descriptions reference where the derivative is positive or negative, not merely where the graph appears to rise or fall.

Helpful phrasing includes:

• “Because f′(x) > 0 on the interval…, the function f is increasing there.”

• “Since f′(x) < 0 on…, the function f decreases throughout that interval.”

• “At x = c, the derivative equals zero, so f has a horizontal tangent at that point.”

These descriptions maintain clarity, mathematical rigor, and alignment to the syllabus.