Calculus- Early Transcendentals, 2021a

5.6. Curve Sketching 203
If there are any points where the derivative fails to exist (a cusp or corner), then we should take special
note of what the function does at such a point.
Finally, it is worthwhile to notice any symmetry. A function f (x) that has the same value for −x as
for x, i.e., f (−x) = f (x), is called an “even function.” Its graph is symmetric with respect to the y-axis.
Some examples of even functions are: x n when n is an even number, cosx, andsin 2 x. On the other hand,
a function that satisfies the property f (−x) =− f (x) is called an “odd function.” Its graph is symmetric
with respect to the origin. Some examples of odd functions are: x n when n is an odd number, sinx, and
tanx. Of course, most functions are neither even nor odd, and do not have any particular symmetry.
5.6.5 Summary of Curve Sketching
The following is a guideline for sketching a curve y = f (x) by hand. Each item may not be relevant to
the function in question, but utilizing this guideline will provide all information needed to make a detailed
sketch of the function.
GuidelineforCurveSketching
1. Domain of the function
2. x-andy-Intercepts
3. Symmetry
4. Vertical and Horizontal Asymptotes
5. Intervals of Increase/Decrease, and Local Extrema
6. Concavity and Points of Inflection
7. Sketch the Graph
Example 5.46: Graph Sketching
Sketch the graph of y = f (x) where f (x)=
2x2
x 2 − 1
Solution.
1. The domain is {x : x 2 − 1 ≠ 0} = {x : x ≠ ±1} =(−∞,−1) ∪ (−1,1) ∪ (1,∞)
2. There is an x-intercept at x = 0. The y intercept is y = 0.
3. f (−x)= f (x), so f is an even function (symmetric about y-axis)
2x 2
4. lim
x→±∞ x 2 − 1 = lim 2
= 2, so y = 2 is a horizontal asymptote.
x→±∞ 1 − 1/x2 Now the denominator is 0 at x = ±1, so we compute:
2x 2
2x 2
lim
x→1 + x 2 =+∞, lim
− 1 x→1 − x 2 − 1 = −∞,
lim 2x 2
x→−1 + x 2 − 1 = −∞,
lim
x→−1 − 2x 2
x 2 − 1 =+∞.