5.1 - 5.4: Curve Sketching 5.2: Additional Curve Sketching Steps for graphing a function: A) Use f (x) to find: 1) Domain. 2) All x-intercepts, y-intercepts. 3) Vertical Asymptotes (VA), Horizontal Asymtotes (HA). B) Use f 0 (x) to find: 1) Critical values c and f (c). 2) Intervals on which f (x) is increasing/decreasing. 3) Relative extrema. C) Use f 00 (x) to find: 1) Intervals on which f (x) is concave up/down. 2) Inflection points. D) Graph f (x) using information from A, B, C (starting from plot important points such as x-intercepts, relative extrema, inflection points, special points, and asymptotes, then sketch the graph). E) Check the graph by calculator. Example 1. Sketch the graph of a function with: • Domain: (−∞, −2) ∪ (−2, 0) ∪ (0, ∞) • Vertical Asymtotes: x = −2, x = 0 • Horizontal Asymtote: y = 0 • f (−3) = 1, f (−1) = 0, f (1) = 1 • f 0 (x) < 0 on (−2, −1) ∪ (0, ∞) • f 0 (x) > 0 on (−∞, −2) ∪ (−1, 0) • f 00 (x) > 0 on (−∞, −2) ∪ (−2, 0) ∪ (0, ∞) 1 Example 2. Given f (x) = x−1 . x+2 a) Find the domain of f , the intervals on which f is continuous, VA, HA, any intercepts. b) Find the critical points, intervals on which f is increasing/decreasing, local extrema. c) Find the intervals on which f is concave up/down, inflection points. d) Sketch the graph of f . 2 Example 3. Sketch the graph of function in example 4 of section 5.1. f (t) = (t − 1)2 √ . t 3