Sketching Graphs of Functions and Their Derivatives - AP Calculus BC Study Guide

The Core Idea: Sketching Graphs of Functions and Their Derivatives

This topic explores the fundamental relationship between the graph of a function and the graphs of its derivatives. The central idea is that the graphical features of one function provide explicit information about the analytical properties of another. For instance, by observing where the graph of a derivative, $f^{\prime }(x)$, is positive, negative, or zero, we can determine where the original function, $f(x)$, is increasing, decreasing, or has a local extremum. Similarly, the behavior of the second derivative's graph, $f^{\prime \prime }(x)$, reveals the concavity and points of inflection of $f(x)$.

The process also works in reverse and between derivatives. The increasing or decreasing behavior of the $f^{\prime }(x)$ graph tells us about the sign of its derivative, $f^{\prime \prime }(x)$. This network of connections allows us to analyze and sketch a function's graph with precision, even without knowing its explicit equation. The core skill is translating graphical information—such as slopes, intercepts, and extrema—into a complete analytical description of a related function.

Key Graphical Relationships

The connection between a function and its derivatives is defined by a set of core principles that link graphical features to function behavior. These are not formulas to be memorized but conceptual links to be understood.

• From the Graph of $f(x)$:

  • The sign of $f^{\prime }(x)$ is determined by the direction of the graph of $f(x)$.

    • If $f(x)$ is increasing, then $f^{\prime }(x)>0$.

    • If $f(x)$ is decreasing, then $f^{\prime }(x)<0$.

    • If $f(x)$ has a horizontal tangent (local extremum or stationary point), then $f^{\prime }(x)=0$.

  • The sign of $f^{\prime \prime }(x)$ is determined by the concavity of the graph of $f(x)$.

    • If $f(x)$ is concave up, then $f^{\prime \prime }(x)>0$.

    • If $f(x)$ is concave down, then $f^{\prime \prime }(x)<0$.

    • If $f(x)$ has a point of inflection, then $f^{\prime \prime }(x)$ changes sign.

• From the Graph of $f^{\prime }(x)$:

  • The behavior of $f(x)$ is determined by the sign (y-values) of the graph of $f^{\prime }(x)$.

    • If the graph of $f^{\prime }(x)$ is above the x-axis ($f^{\prime }(x)>0$), then $f(x)$ is increasing.

    • If the graph of $f^{\prime }(x)$ is below the x-axis ($f^{\prime }(x)<0$), then $f(x)$ is decreasing.

    • If the graph of $f^{\prime }(x)$ crosses the x-axis from negative to positive, then $f(x)$ has a local minimum.

    • If the graph of $f^{\prime }(x)$ crosses the x-axis from positive to negative, then $f(x)$ has a local maximum.

  • The sign of $f^{\prime \prime }(x)$ is determined by the direction (slope) of the graph of $f^{\prime }(x)$.

    • If the graph of $f^{\prime }(x)$ is increasing, then $f^{\prime \prime }(x)>0$, and $f(x)$ is concave up.

    • If the graph of $f^{\prime }(x)$ is decreasing, then $f^{\prime \prime }(x)<0$, and $f(x)$ is concave down.

    • If the graph of $f^{\prime }(x)$ has a local extremum (a "peak" or "valley"), then $f^{\prime \prime }(x)$ changes sign, and $f(x)$ has a point of inflection.

• From the Graph of $f^{\prime \prime }(x)$:

  • The concavity of $f(x)$ is determined by the sign (y-values) of the graph of $f^{\prime \prime }(x)$.

    • If the graph of $f^{\prime \prime }(x)$ is above the x-axis ($f^{\prime \prime }(x)>0$), then $f(x)$ is concave up.

    • If the graph of $f^{\prime \prime }(x)$ is below the x-axis ($f^{\prime \prime }(x)<0$), then $f(x)$ is concave down.

    • If the graph of $f^{\prime \prime }(x)$ crosses the x-axis, then $f(x)$ has a point of inflection.

Understanding the "Graph of..." Problem

The most critical nuance in this topic is maintaining a clear distinction between the graph you are given and the function you are analyzing. AP Exam questions are frequently phrased as: "The graph of $f^{\prime }$, the derivative of $f$, is shown above. On what intervals is $f$ concave down?" A student's eyes are on the graph of $f^{\prime }$, but the question is about $f$.

The key is to translate the question into the language of the given graph.

• A question about $f$'s increase/decrease is a question about the sign (y-values) of $f^{\prime }$.

• A question about $f$'s local extrema is a question about the x-intercepts where the sign changes on the graph of $f^{\prime }$.

• A question about $f$'s concavity is a question about the slope (increasing/decreasing behavior) of $f^{\prime }$.

• A question about $f$'s points of inflection is a question about the local extrema on the graph of $f^{\prime }$.

Never describe a feature of the given graph to answer a question about a different function. For example, do not state that $f$ is concave down because the graph of $f^{\prime }$ is concave down. The correct reasoning is that $f$ is concave down because the graph of $f^{\prime }$ is decreasing. Always be deliberate in your analysis: identify what graph is provided, what function is being asked about, and what the connection is between them.

Core Concepts & Rules

• The function $f$ is increasing on intervals where its derivative $f^{\prime }$ is positive (i.e., the graph of $f^{\prime }$ is above the x-axis).

• The function $f$ is decreasing on intervals where its derivative $f^{\prime }$ is negative (i.e., the graph of $f^{\prime }$ is below the x-axis).

• A local minimum of $f$ occurs at $x=c$ if $f^{\prime }(c)=0$ and $f^{\prime }$ changes from negative to positive at $c$.

• A local maximum of $f$ occurs at $x=c$ if $f^{\prime }(c)=0$ and $f^{\prime }$ changes from positive to negative at $c$.

• The function $f$ is concave up on intervals where its second derivative $f^{\prime \prime }$ is positive. This corresponds to intervals where $f^{\prime }$ is increasing.

• The function $f$ is concave down on intervals where its second derivative $f^{\prime \prime }$ is negative. This corresponds to intervals where $f^{\prime }$ is decreasing.

• A point of inflection on the graph of $f$ occurs at a point where the concavity of $f$ changes. This corresponds to a point where $f^{\prime \prime }$ changes sign, which occurs where $f^{\prime }$ changes from increasing to decreasing or vice versa (i.e., at a local extremum of $f^{\prime }$).

Step-by-Step Example 1: Analyzing $f$ from the Graph of $f^{\prime }$

Problem: The graph of $f^{\prime }(x)$, the derivative of a function $f(x)$, is shown below on the interval $(0,8)$. Use the graph to determine the properties of $f(x)$.