Practice problems for these concepts can be found at: Graphs of Functions and Derivatives Practice Problems for AP Calculus

**General Procedure for Sketching the Graph of a Function**

Steps:

- Determine the domain and if possible the range of the function
*f*(*x*). - Determine if the function has any symmetry, i.e., if the function is even (
*f*(*x*) =*f*(–*x*)), odd (*f*(*x*) = –*f*(–*x*)), or periodic (*f*(*x*+*p*)=*f*(*x*)). - Find
*f*'(*x*) and*f*''(*x*). - Find all critical numbers (
*f*'(*x*) = 0 or*f*'(*x*) is undefined) and possible points of inflection (*f*''(*x*) = 0 or*f*''(*x*) is undefined). - Using the numbers in Step 4, determine the intervals on which to analyze
*f*(*x*). - Set up a table using the intervals, to
- determine where
*f*(*x*) is increasing or decreasing. - find relative and absolute extrema.
- find points of inflection.
- determine the concavity of
*f*(*x*) on each interval.

- determine where
- Find any horizontal, vertical, or slant asymptotes.
- If necessary, find the
*x*-intercepts, the*y*-intercepts, and a few selected points. - Sketch the graph.

### Example 1

Sketch the graph of *f*(*x*) = .

Step 1: Domain: all real numbers *x* ≠ ± 5.

Step 2: Symmetry: *f*(*x*) is an even function (*f*(*x*) = *f*(–*x*)); symmetrical with respect to the *y*-axis.

Step 3:

Step 4: Critical numbers:

*f*'(

*x*) = 0 –42

*x*= 0 or

*x*= 0

*f*'(

*x*) is undefined at

*x*= ± 5 which are not in the domain.

- Possible points of inflection:

*f*''(

*x*) ≠ 0 and

*f*''(

*x*) is undefined at

*x*= ± 5 which are not in the domain.

Step 5: Determine intervals:

- Intervals are (–∞, –5), (5, 0), (0, 5) & (5, ∞)

Step 6: Set up a table:

Step 7: Vertical asymptote: *x* = 5 and *x* = – 5

- Horizontal asymptote:

*y*=1

Step 8: *y*-intercept:

*x*-intercept: (–2, 0) and (2, 0)

- See Figure 7.3-1.

Practice problems for these concepts can be found at:

Graphs of Functions and Derivatives Practice Problems for AP Calculus

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