5.4: 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
25
.
x
a) Find the domain of f , the intervals on which f is continuous, VA, HA, any intercepts.
f (x) = x +
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+2
3