5.4 Graphing Functions

In this lesson, you will learn to sketch the graph of functions using information from the first and second derivatives and other properties (such as asymptotes or symmetry).

Lesson Objectives

Upon completion of the lesson 5.4, you will be able to:

• • Sketch a graph from properties given about the function's first and second derivative.

  • Utilize graphing guidelines to graph given functions. The guidelines include:

    1. Identify the domain.

    2. Determine symmetry about the y-axis or about the origin, if applicable.

    3. Find information about the graph provided by the first derivative.

    4. Find information about the graph provided by the second derivative.

    5. Find extreme values and inflection points.

    6. Find any asymptotes.

    7. Find intercepts.

  • Identify cusps, if they occur.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube video: Curve Sketching

• 1. YouTube videos: Curve Sketching Using Calculus (domain, x-y intercepts, symmetry of the function, intervals of increase/decrease, local maximums and minimums, concavity, inflection points, horizontal and vertical asymptotes)

  2. Part 1:

• 1. Part 2:

• 1. YouTube videos: Sketching the curve summary - Graphing Example 2

  2. Part 1:

• 1. Part 2:

• 1. Part 3:

• 2. Part 4:

• 1. Curve Sketching [University of Houston] (summary of graphical features, graphing y=f(x) using the first and second derivatives, infinite limits, and limits at ±∞)

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Curve Sketching Tutorial [Nipissing University]

  2. (general guidelines for all curves and examples are included. You will not be able to do the General Curve Sketching Test on iLn.)

  3. Overview of Curve Sketching [McGraw Hill]

  4. Note: Skip example 6.5.

  5. Chapter 4: 4. 3 - Using Derivatives for Curve Sketching

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Vertical and horizontal asymptotes [University of Houston] (problems with answers and solutions provided)

  2. Vertical tangents; vertical cusps [University of Houston]

  3. Some curve sketching [University of Houston] (problems with answers and solutions provided)

  4. Graphing of functions using first and second derivatives [UC Davis]

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. YouTube video: Sketching a Graph Given Conditions about Derivative Requirements

• 1. YouTube video: Graphing with Calculus

  2. Note: Only watch the first problem on the video (until 8:08). Skip the second problem.

• 1. The Graphical Relationship between First and Second Derivatives [Cengage]

  2. YouTube video: Calculus: Graphing Using Derivatives