Curve Sketching
N ( x)
Given f ( x) 
; ( N ( x) stands for the function in the numerator and D( x) the denominator)
D( x)
A) Domain of the function: Possible values of x.
B) Intercepts of the function: xint solve N(x) = 0; yint= f(0) .
C) Symmetry: If f ( x)  f ( x) the function is even (symmetric about the y-axis); if
f ( x)   f ( x) the function is odd (symmetric about the origin);
D) Asymptotes of the function:
a) Vertical (VA): solve D( x)  0 . If x=a is a vertical asymptote, evaluate
lim f ( x ) and lim f ( x) to find the direction of the curve at the asymptote.
xa
xa
b) Horizontal (HA): Evaluate lim f ( x) . This also tells you the behavior of the
x 
graph far to the right and left.
E) Interval of Increase/ Decrease Use the first derivative test to find the intervals
where f '(x) >0 (increasing) and where f '(x) <0 (decreasing). Test f '(x) in
regions given by zeros, CP and VA in the domain of the function.
F) Local Maximum and Minimum Values: Find the critical points (CP) by finding
points where f '(x) = 0 or f '(x) is undefined. Find the y coordinate by evaluating
f(x) at those points. Remember that not all critical points are relative extrema.
G) Concavity and Points of Inflection: Solve f "(x) = 0 for x, to find the possible
inflection points. Inflection points are points where concavity changes. Not all
points where f "(x) = 0 are inflection points.
Find where f (x) is concave up (CU) or concave down (CD) by testing f ''(x) in
regions given by zeros, CP and VA in the domain of the function. If you
evaluate f '' at the CP, you obtain f '' |xCP  0 (CU/relative min); f '' |xCP  0 (CD/
relative max).
H) Sketch the curve without the use of a graphics calculator. Check with a grapher.
Note: Start your preliminary sketch as you find the information about the graph in
parts A-G.
Examples:
1. Sketch the graph of f(x) = x3 -3x
A) Domain: all x.
B) xint  x(x2-3) = 0  x = 0, x=  3 ; yint = 0.
C) f ( x)   f ( x) the function is odd (symmetric about the origin)
D) No asymptotes. lim f ( x) =  lim f ( x) = 
x 
x 
E) Increasing when y' = 3x -3 > 0 on (,1)  (1, ) , and decreasing when
y' = 3x2 -3 < 0 on (-1,1).
F) Critical numbers at y' = 3x2 -3 = 0 (x = -1); (x = l,). f ( x) is changes from positive
to negative at x = -1 ( I(-1)=2 is a local maximum) and changes from negative
to positive at x = 1 ( f(1)=-2 is a local minimum)
2
G) y ''  6 x = 0 at (0,0). y ''  6 x > 0 for x>0 (CU); y ''  6 x < 0 for x<0 (CD). Since
concavity changes, (0,0) is an IP. y ''  6 x |1> 0 (CU/relative min.); y ''  6 x |-1< 0
(CD/relative max.)
H)
2. Sketch the graph of f ( x) 
x2
x2 1
A) Domain x  1
B) xint  x = 0; yint = 0.
C) f ( x)  f ( x) the function is even (symmetric about the y-axis)
D) VA, x  1 , lim f ( x) = lim f ( x) =  ; lim f ( x ) = lim f ( x) =  HA,
x 1
x 1
x 1
x 1
lim f ( x) =1 lim f ( x) =1
x 
x 
E) Increasing when y ' 
y'
F) y ' 
2 x
 x2  1
 x2  1
2
>0 on (, 0), x  1 , and decreasing when
2
<0 on (0, ), x  1 ,
2
= 0 (x = 0,y = 0); undefined x  1 (not in the domain of f(x)),
2 x
 x2  1
2 x
local maximum at f(0)=0, no local minimum.
3x 2  1
 0 ; no IP. y'' = > 0 for x>0 for x <-1 or x >1 (concave up)
G) y ''  2
3
2
x

1
 
y" = <0 for -1<x<1, (concave down).
H)
Exercises:
a)
b)
c)
d)
e)
f)
Find the critical points
Test for symmetry
Find the intervals on which f is increasing or decreasing
Find the intervals where f is concave up or concave down.
Find the inflection points if any.
Use the information in parts a-e to sketch the graph. Check with a grapher.
1) f ( x)  2 x3  3x 2  12 x
2) f ( x)  x 4  6 x 2
3) f ( x)  3x5  5 x3  3
4) f ( x)  x3  12 x  1
5) f ( x)  x 4  2 x 2  3
x
6) f ( x) 
x 1
1
7) f ( x )  2
x 9
1 x
8) f ( x) 
1 x
x
9) f ( x)  2
x 1
10) f ( x)  x 2e x
Answers:
1) Inc. (,1)  (2, ) ; Dec. (1, 2) ; local max (1, 7) ; local min (2, 20)
CU ( (1/ 2, ) ; CD (,1/ 2) ; IP (1/ 2, 13 / 2)
2) Inc. (  3, 0)  ( 3, ) ; Dec. ( ,  3)  (0, 3) ; local max (0, 0) ;
local min (  3, 9) ; CU (, 1)  (1, ) ; CD ( 1,1) ; IP (1, 5)
3) Inc. (,1)  (1, ) ; Dec. ( 1,1) ; local max (1,5) ;
local min (1,1) ; CU (1/ 2, 0)  (1/ 2, ) ; CD (, 1/ 2)  (0,1/ 2) ;
IP (1/ 2,3  7 2 / 8) , (1/ 2,3  7 2 / 8)
4) Inc. (, 2)  (2, ) ; Dec. ( 2, 2) ; local max (2,17) ; local min (2, 15)
CU ( (0, ) ; CD (, 0) ; IP (0,1)
5) Inc. (1, 0)  (1, ) ; Dec. (, 1)  (0,1) ; local max (0,3) ;
local min (1, 2) ; CU (,  3 / 3)  ( 3 / 3, ) ; CD ( 3, 3) ;
IP ( 3 / 3, 22 / 9)
6) y-int :0; x-int :0; VA: x=1; HA: y =1; Dec: (,1)  (1, ) ; CU (1,  ) ; CD
( ,1) ; IP: none
7) y-int :-1/9; VA: x  3 ; HA: y  0 ; Inc. (, 3)  (3, 0) ; Dec: (0,3)  (3, ) ;
local max (0, 1/ 9) ; CU (, 3)  (3, ) ; CD (3,3) ; IP: none
8) y-int :1; x-int :1; VA: x  1 ; HA: y  1 ; Dec: (, 1)  (1, ) ; CU (1, ) ;
CD (, 1) ; IP: none
9) y-int :0; x-int :0; VA: None; HA: None ; Inc: ( 1,1) ; Dec: (, 1)  (1, ) ; local
max (1,1/ 2) ; local min (1, 1/ 2) ;
CU ( 3, 0)  ( 3, ) ; CD
(,  3)  (0, 3) ; IP: (0, 0) ; ( 3, 3 / 4) ; ( 3,  3 / 4)
10) y-int :0; x-int :0; Inc: (, 2)  (0, ) ; Dec: ( 2, 0) ; local max (2, 4e2 ) ; local
min (0, 0) ;
CU (, 2  2)  (2  2, ) ; CD (2  2, 2  2) ; IP:

(0, 0) ; (2  2, 2  2


2
e
2
2 2
)  (.5858,.1910) ;
(2  2, 2  2 e2 2 )  (3.4142,.3826)