Graphing without Calculators for AP Calculus

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:

1. Determine the domain and if possible the range of the function f(x).
2. 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)).
3. Find f '(x) and f ''(x).
4. 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).
5. Using the numbers in Step 4, determine the intervals on which to analyze f(x).
6. Set up a table using the intervals, to
   1. determine where f(x) is increasing or decreasing.
   2. find relative and absolute extrema.
   3. find points of inflection.
   4. determine the concavity of f(x) on each interval.
7. Find any horizontal, vertical, or slant asymptotes.
8. If necessary, find the x-intercepts, the y-intercepts, and a few selected points.
9. 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 –42x = 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