• Analyze and sketch the graph of a function.

Assignment

  • Vocabulary and teal boxes
  • p253 1, 12, 15, 22, 27, 31, 35, 36, 42, 43, 49, 53, 57, 64 90, 99–102

Additional Resources

Analyzing the Graph of a Function

Now that we’ve looked at the derivatives for increasing/decreasing and concavity, we wrap everything up into one package.

Guidelines for Analyzing the Graph of a Function

  1. Determine the domain and range of the function
  2. Determine the intercepts, asymptotes, and symmetry of the graph.
  3. Find critical points for $f’$ and $f^{\prime\prime}$ and determine extrema and points of inflection.

The book has a slew of examples which highlight some common things to look for, as well as organizational tips. When you read through them, here are some things to look for.

  • Example 1: The table they use here is helpful. I suggest adopting it or something similar.
  • Example 2: Slant asymptotes are mentioned. These occur in rational functions when the numerator is one degree higher than the denominator. This is not critical information for the AP exam.
  • Example 3: Two different horizontal asymptotes / end behaviors.
  • Example 4: Factoring with fractional exponents.
  • Example 5: A reminder of properties of polynomials, such as number of extrema and points of inflection.
  • Example 6: Mentions theorem 1.14 which is the definition of vertical asymptotes. Technically you need something not equal to zero in the numerator while the denominator is zero for it to be an asymptote. $0/0$ doesn’t fulfill this, so they rewrite to get another version to see what’s actually going on.
  • Example 7–10: Note that the tables are not used as justification. Instead, verbal explanations are given, with any tables there only to organize work done before the explanations. Examples 9 and 10 in particular showcase problems that will likely appear on the exam.