# Sketch the Function (given its Derivative)

Objective: To provide instructors with interactive examples for the classroom or student assignments for sketching a function given a sketch of its derivative.

Level: First term calculus.

Prerequisites: The derivative as a function, the connection between the derivative and the tangent line to y = f(x), and graphing the function y = f(x) using properties of its derivative y' = f '(x).  In addition, the relationships between the sign of the derivative and its intercepts to the behavior of the  function y = f(x) are required. It is also recommended that a brief intuitive discussion of "sharp points" on the graph of y = f(x) and corresponding  "jumps" or "breaks" in the graph of the derivative be included.

Platform: Included are Excel worksheets that will work on both a PC and a MAC, and animations in both Flash and QuickTime.

Instructor's Notes: In many text books after the concept of a derivative has been discussed in terms of limits, tangent lines, and the rate of change there is a discussion involving the derivative function from a graphical point of view. A part of such a discussion involves the slope of the tangent line to the curve and the sign of the derivative. For example:

• If the slope of the tangent line is positive, then f ' is positive.
• If the slope of the tangent line is negative, then f ' is negative.
• If the tangent line is horizontal, then f ' is zero.

The discussions involve such behavior at points and over intervals. This leads to statements like the following:

If f ' > 0 on an interval, then we say f is increasing over that interval.

If f' < 0 on an interval, then we say f is decreasing over the interval.

In addition, it is noted that the magnitude of the derivative effects the rate at which the function increases or decreases over and interval. If f ' is large and positive, then f increases rapidly, while if f ' is a large negative value, then f decreases rapidly.

With such basic connections between the slope of the tangent line (alias the value of the derivative) and the geometric behavior of the graph of y' = f '(x) we can establish enough information to produce a reasonable sketch of the function y = f (x). We use the properties of the graph of y' = f '(x) to produce a sketch of y = f (x) that usually does not have a precise scale, but does reflect the qualitative aspect of the graph of the original function.

Here we do not assume that techniques for computing derivative formulas have been discussed. In fact, our examples provide only the graph of the function y' = f '(x).

Given the graph of a function y' = f '(x) a standard approach is to identity intervals over which its graph is positive, other intervals over which it is negative, and its intercepts. Next use this information to identify behavior of the graph of           y = f(x), and then provide a sketch based on the information recorded. We illustrate this approach in Example 1.

Example 1. Sketch a graph of the y = f(x) given the sketch of its derivative shown in Figure 1.

 Figure 1.

In Figure 2 we identity features of the graph of y' = f '(x) and connect them to properties of the graph of y = f(x).

 Figure 2.

In Figure 3 we use the information derived from Figure 2 about f(x) to sketch a graph that has the approximate shape of a graph of f(x), but not necessarily with the proper scale.

 Figure 3.

The turning points of the curve shown in Figure 3 may not be in the correct vertical positions, but such points must occur based on the properties of the derivative.

In the introductory animation at the top of this page and the Excel worksheets included in this demo the derivative curve is sketched and simultaneously three choices for an approximate graph of the function are sketched. The user is to select the correct choice for ( an approximate) curve y = f(x), but must perform their own analysis like that illustrated in Example 1. For instance, as Figure 1 would be sketched the three choices appearing below would be generated.

 A. B. C.

Example 2. In Animation #1 that follows we develop an easy method for determining the behavior of the derivative of function by generating a sketch of the derivative itself. As the derivative is sketched a point moves along the curve. By recording the sign of the y-coordinate as the points that are plotted a pattern of the behavior of the graph of the function y = f(x) is constructed. (The Flash files below are smaller and load faster.)

Start Animation #1 by clicking one of the following: Flash file   QuickTime file

After you view and understand the idea in Animation #1 use Animation #2. At the end Animation #2 decide which figure is a good approximation to the graph of y = f(x).

Start Animation #2 by clicking one of the following: Flash file    QuickTime file

We have developed a gallery of 10 Excel routines that sketch y' = f '(x) and simultaneously generate three choices for possible graphs of the function f(x). The user is asked to determine which of the three graphs is a good approximation to the graph of the function f(x). These demos can be used by instructors as part of classroom demonstrations or can be part of an assignment for students for sketching a function given the graph of its derivative.

Credits:  This demo, the Excel files, and the animations were developed by

Dr. David R. Hill
Department of Mathematics
Temple University

and is included in Demos with Positive Impact with his permission.

DRH 1/1/07   Last updated 1/12/07

Since 1/1/07