Sec 4.3
Curve Sketching
1
Curve Sketching Problems
Given: A function y = f(x).
Objective: To sketch its graph.
2
Steps
(1) Find a “Frame” for the graph
 Domain
 Asymptotes – Horizontal, Vertical, Slant
(2) Find out how the graph “wiggles”
 Derivative – intervals of increase/decrease;
max/min
 Second derivative – intervals for concave
up/down; point(s) of inflection
(3) Sketch
3
Example (1)
Sketch
x
f x   2
x 1
Frame:
Next Question:
How does the
graph wiggle
between the two
ends ?
Domain:
Asymptotes:
Starts here
Ends here
4
Wiggle:
Derivative:
f ' x  
2nd
1 x2
x
2

1
2
derivative:
f ' ' x  


1  x 1  x 

x

2x x  3 x 
f x 
 3
x
1
2
 1
2

1
f x 
3

f x 
1
1
–
+
 3
1
–
+
–
3
0
–
2
0
+
3
3
Final Step: Put the wiggly graph onto the Frame.
5
f x 
 3
1
0
1
3
Decreasing;
Concave down
Decreasing;
Increasing;
Concave up
Concave down
Decreasing;
Decreasing;
Concave up
Concave down
Increasing;
Concave up
Starts here
Local max
 3
A “twist” :
Concavity
changes –
a point of
inflection
1
Graph rebounds
after a dip – a
local min
0
1
A “twist” :
Concavity
changes –
a point of
inflection
3
Ends here
A “twist” :
Concavity
changes –
a point of
inflection
6
Example (2)
Sketch
Next Question:
How does the
graph wiggle
within each of the
three sections ?
x
f x   2
x 1
Frame:
?
Domain:
Asymptotes:
?
?
?
?
?
?
?
Starts here
?
?
Ends here
?
7
Wiggle:
Derivative:
f ' x  

x
2nd derivative:
f ' ' x  
f x 
1

 x2 1

1
2

2

2x x2  3
x
0
2

1
3
1
8
Example (3)
Sketch
Next Question:
How does the
graph wiggle
within each of the
three sections ?
x2  9
f x   2
x 4
Frame:
?
?
?
?
?
Domain:
?
?
Asymptotes:
?
Starts here
?
?
?
Ends here
9
Wiggle:
Derivative:
f ' x  
x
10 x
2
2nd derivative:
f ' ' x  
f x 
2
4

2

 10 3x 2  4
x
0
2
4


3
2
10
Example (4)
Sketch
f x   x
Frame:
Domain:
Asymptotes:
2/3
2 x  5
Next Question:
How does the
graph wiggle
between the two
ends ?
?
?
Ends here
?
Starts here
11
Wiggle:
Derivative:
10 1/ 3
x  1
f ' x  
x
3
2nd derivative:
10  4 / 3
2 x  1
f ' ' x  
x
9
f x 
1
0
1
2
12
Example (5)
Sketch
x2  x  2
f x  
x 3
Frame:
Domain:
Asymptotes:
Next Question:
How does the
graph wiggle
within the two
regions ?
?
?
?
Ends here
?
?
Starts here
?
13
Wiggle:
Derivative:

x  1 x  5
f ' x  
x  32
2nd derivative:
8
f ' ' x  
 x  33
f x 
1
3
5
14
Next Question:
How does the
graph wiggle in
one of the
regions ?
Repeat here
Example (6)
Sketch
cos x
f x  
1  sin x
Frame:
?
Domain:
?
Asymptotes:
Periodicity:
?
Repeat here
?
15
Wiggle:
Derivative:
1
f ' x  
1  sin x
2nd derivative:
cos x
f ' ' x  
1  sin x 2
f x 
 2

2
3
2
16