Curve Sketching
Putting together all the various tests and theorems from the past several lectures, we create the
following procedure to sketch the graph of f(x).
1. Find the domain of f(x)
2. Find x and g intercepts, if any
3. Find vertical and horizontal asymptotes or end behaviour (i.e. if f approaches ± infinity as x
approaches ± infinity.
4. Find f ’(x), critical points, and where f ‘(x) is discontinuous.
5. Find f ‘’(x) and potential inflection points.
6. Classify the points from steps 4 and 5 to identify local extrema and actual inflection points. With
these and any discontinuities, determine intervals of increase/decrease and concavity.
7. Sketch and label.
Ex. Sketch
Solution:
1. F is defined for all
2. When x = 0, y = 0, and vice versa.
3. Vertical asymptotes at since the numerator ≠ 0.
Horizontal asymptotes:
approaches infinity.
4.
, critical points at
Note: are not in the domain of f, so technically aren’t critical points.
5.
Possible inflection points at x=0 and once again, are not in the
domain but are nonetheless “point of interest”.
6. 3 1 0 1 3
f ‘’(x)
+

+

+

f ‘(x)
+



+
f(x)
Inc.
Up.
Inc.
Down.
Dec.
Down.
Dec.
Up.
Dec.
Down.
Dec.
Up.
Inc.
Up.
Inc.
Down.
Key
points
POI Local Max VA POI VA Local Min POI
7. Sketch on own, check on Desmos.
Exercise: Sketch