Basic Principles of Curve Sketching
Let y = f (x) be a function.
1. Find the Domain of the function.
2. Find f ′ (x) and f ′′ (x).
3. Solve f ′ (x) = 0 and also find the values of x where f ′ (x) is Undefined but
f (x) is Defined. These are Critical Values.
4. Use 1st Derivative test or 2nd Derivative test to Determine whether a
critical value is a Relative Maximum point or a Relative Minimum point
or a Horizontal Point of Inflection.
5. Solve f ′′ (x) = 0 and also find the values of x where f ′′ (x) is Undefined
but f (x) is Defined. These are potential Points of Inflection.
6. Let x = c be a point of these kind. If f ′′ (x) changes sign on two sides of
c, then x = c is a point of inflection.
7. Find the limit lim f (x) and
x→+∞
lim f (x):
x→−∞
g(x)
h(x) ,
where g(x) and h(x)
g(x)
are polynomials, then you already know how to find lim
and
x→+∞ h(x)
g(x)
lim
from Section 9.2.
x→−∞ h(x)
If f (x) is a Rational Function, i.e., f (x) =
g(x)
g(x)
= b or lim
= b, where b is a finite number,
x→−∞
h(x)
h(x)
then y = b is a horizontal asymptote to the curve y = f (x).
(a) If
lim
x→+∞
a
g(x)
g(x)
= , where a 6= 0, then we know that lim
does
x→+∞
h(x)
0
h(x)
not exist, but this much of information is not enough for us to sketch
g(x)
= +∞
the graph of y = f (x). We need to know whether lim
x→+∞ h(x)
or −∞. To find that, take the highest degree term of g(x) and the
highest degree term of h(x) and take their quotient, then put x = 1
g(x)
= +∞ and if negative
there, if this value is positive, then lim
x→+∞ h(x)
(b) If lim
x→+∞
1
then lim
x→+∞
g(x)
= −∞.
h(x)
a
g(x)
= , where a 6= 0, then do the similar calculation as
h(x)
0
above with x = −1 and draw the same conclusion.
If
lim
x→−∞
g(x)
8. Find the Vertical Asymptotes: If f (x) = h(x)
is a Rational Function,
then first solve for h(x) = 0. Let x = c be such a solution, i.e., h(c) = 0.
Now if g(c) 6= 0, then x = c is a Vertical Asymptote to the curve y = f (x).
g(x)
g(x)
g(x)
and limx→c− h(x)
. To find limx→c+ h(x)
,
We also need to find limx→c+ h(x)
choose a value d of x which is on the right of c, i.e., d > c and “very
g(d)
g(x)
g(d)
close to” c. If h(d)
> 0, then limx→c+ h(x)
= +∞ and if h(d)
< 0, then
limx→c+
g(x)
h(x)
= −∞.
g(x)
, choose a similar point b to the left of c, i.e., b < c
To find limx→c− h(x)
and do the similar calculation and draw the same conclusion.
9. If possible find the values of x where f (x) = 0 and also the point (0, f (0)).
You may need to plot some additional points to get a nice shape of the
graph.
• Informations obtained from Step 1 - 6 and Step 9 will be enough to sketch
the graph of a Polynomial Function.
• All the above informations will be needed to sketch the graph of a Rational
Function.
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