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