3.6 A Summary of Curve
Sketching
So far, you have studied several concepts that
are useful in analyzing the graph of a function.
• x-intercepts and y-intercepts
(P.1)
• Symmetry
(P.1)
• Domain and Range
(P.3)
• Continuity
(1.4)
• Vertical Asymptotes
(1.5)
• Differentiability
(2.1)
• Relative Extrema
(3.1)
• Concavity
(3.4)
• Points of Inflection
(3.4)
• Horizontal Asymptotes
(3.5)
3.6 A Summary of Curve
Sketching
When you are sketching the graph of a function,
either by hand or with a calculator, remember that
normally you cannot show the entire graph.
Your window is critical!!!!
Different viewing rectangles
of the same graph
3.6 A Summary of Curve
Sketching
Guidelines for Analyzing the Graph of a Function
1. Determine the domain of the function.
2. Determine the intercepts and asymptotes of the graph.
3. Find the critical numbers and inflections points.
Use these results to determine relative extrema,
whether the function is increasing or decreasing,
and concavity.
4. Check the function for symmetry.
Domain
Intercepts
Asymptotes
Derivatives
Symmetry
Print for Students
Intervals
f
(
x
)
& Inputs
f '( x)
f ''( x)
Graph
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
Domain
x in , x  2
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
Intercepts ?
x-intercepts:
2( x 2  9)
0 2
x 4
 x  3
 (3,0),(3,0)
y-intercept:
2(02  9) 18 9
 9
   0, 

y 2
2
4
0 4
 2
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
Vertical Asymptotes ?
x  4  0  x  2
2
Horizontal Asymptote  s  ?
2
2x
lim f ( x)  2  2  y  2
x
x  
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
f ' and f ''?
2[(2 x)( x  4)  ( x  9)(2 x)]
f '( x) 
( x 2  4)2
2
2
4 x( x 2  4  x 2  9)

( x 2  4)2
4 x(5)
 2
( x  4)2
Do on
Whiteboard
20 x
f '( x)  2
2
( x  4)
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
20 x
f '( x)  2
( x  4) 2
20( x 2  4)2  20 x(2)( x 2  4)(2 x)
f ''( x) 
( x 2  4)4
20( x 2  4)2  80 x 2 ( x 2  4)

( x 2  4)4
20( x 2  4)  80 x 2

( x 2  4)3
80  60 x 2

( x 2  4)3
20(4  3x 2 )
f ''( x) 
( x 2  4)3
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
Critical Numbers
20 x
f '( x)  2
0 x0
2
( x  4)
( x  2 is not in the domain of f )
Inflection Points
20(4  3x 2 )
2

4

3
x
0
f ''( x) 

0
2
3
( x  4)
4
 None
x 
3
2
Guidelines
3.6 A Summary of Curve
Sketching
2( x 2  9)
Analyze the graph of f ( x) 
x2  4
Symmetry Tests for Symmetry
1. y - axis  f (- x)  f ( x)
2. x - axis  - f ( x)  f ( x)
3. (0,0)  - f (- x)  f ( x)
f ( x)  f ( x)
 f has symmetry with respect to the y - axis.
Guidelines
Summary :
Domain  x
in , x  2
 9
Intercepts  (3, 0), (3, 0),  0, 
 2
Asymptotes  x  2, y  2
Critical Numbers  Extrema ?  @ x  0
Inflection Points  None
Symmetry
 f has symmetry with respect to the y - axis.
Graph
2( x 2  9)
f ( x)  2
x 4
f ( x)
1
x
1
The concavity may only change between vertical asymptotes!!
Our final graph better have symmetry in the y-axis.
Summary
Intercepts: 3,0 ,  0,9 / 2  ;Asymptotes:x  2, y  2;Critical #'s:x  0;
Determined by CN’s, IP’s, &
Vertical Asymptotes.
Inflection Pts:None;Symmetry:y-axis
Intervals
& Inputs
f ( x)
(3, 2)
x
(2,
5 0)
9/2 20
x
2
Und.
15

(2,9)
Use the table to finish the graph!
Graph
11
Decr.
12
Und.
16



Conc. 
Vert
13.
Asymp.
Decr.
19
17
18
Conc. 
0 21
22
Rel.23
Min.
25
Incr
26 .
24
Und.
28
f ''( x)


(0,72)
8

10
14
4 2 Und.
x 6 0
f '( x)
Und.
29

Und.
30
31
2( x 2  9)
f ( x)  2
x 4

32
20 x
f '( x)  2
( x  4) 2
Conc. 
Vert
27 .
Asymp.
Incr
33 .
Conc. 
20(4  3x2 )
f ''( x) 
( x2  4)3
2( x 2  9)
f ( x)  2
x 4
f ( x)
1
x
1
The concavity may only change between vertical asymptotes!!
Our final graph better have symmetry in the y-axis.
Table
Using Calculus, you
can be certain that
you have determined
all characteristics of
the graph of f .
3.6 A Summary of Curve
Sketching
Slant Asymptote :
The graph of a rational function (having no common factors)
has a slant asymptote if the degree of the numerator exceeds
the degree of the denominator by 1.
x  2x  4
f ( x) 
x2
4
f ( x)  x 
x2
2
4
lim
0
x  x  2
x
x  2 x  2x  4
x 2 2x
2
0
4
 y  x is a slant asymptote
Dividend  Quotient  Remainder/Divisor
3.6 A Summary of Curve
Sketching
Slant Asymptote :
The graph of a rational function (having no common factors)
has a slant asymptote if the degree of the numerator exceeds
the degree of the denominator by 1.
2 1 -2 4
x  2x  4
f ( x) 
2 0
x2
4
1 0 4
f ( x)  x 
x2
lim f ( x)  x  y  x is still a slant asymptote
2
x  
3.6 A Summary of Curve
Sketching
An nth degree polynomial can have at most:
n  1 relative extrema
n  2 points of inflection
3.6 A Summary of Curve
Sketching
We need 5 groups.
Group
1
2
3
p.202-3 7,27,29 9,25,31 11,28,
33
4
5
13,26,
35
21,39
37
Use the procedure demonstrated today to sketch a graph of each
function without the use of a calculator. You may use a calculator to
check your results.
Choose one of your problems and create a poster to present.
(25 points)
Do your 3 problems tonight; create posters tomorrow; presentations
begin next week.