Chapter 5
Graphing and
Optimization
Section 4
Curve Sketching
Techniques
Objectives for Section 5.4
Curve Sketching Techniques
■ The student will add
information about holes and
asymptotes to the curve
sketching strategy.
Barnett/Ziegler/Byleen Business Calculus 12e
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Modifying the Graphing Strategy
Up to this point, we haven’t considered the graphs of rational
functions.
We will recall the method for finding holes and asymptotes and
add that to our “toolbox” for sketching graphs of functions.
Barnett/Ziegler/Byleen Business Calculus 12e
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Graphing Strategy
 Step 1. Analyze f (x)
• Determine the domain of f.
• Determine the intercepts.
• Determine the holes & asymptotes
 Step 2. Analyze f (x)
• Find the partition points of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where f is increasing and
decreasing
• Find local maxima and minima (at critical points)
Barnett/Ziegler/Byleen Business Calculus 12e
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Graphing Strategy
 Step 3. Analyze f (x).
• Find the partition points of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where the graph of f is
concave upward and concave downward.
• Find inflection points.
 Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts, local max
and min, and inflection points.
• Plot additional points as needed and complete the
sketch
Barnett/Ziegler/Byleen Business Calculus 12e
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Review
 Find the holes, vertical and horizontal asymptotes of f(x).
 𝑓 𝑥 =
 𝑓 𝑥 =
 𝑓 𝑥 =
 𝑓 𝑥
2𝑥−4
𝑥+2
1
𝑥−5
=
2(𝑥−2)
𝑥+2
6𝑥 2
1+2𝑥 2
No holes
VA: x = -2 HA: y = 2
No holes
VA: x = 5
No holes
VA: none
HA: y = 0
HA: y = 3
𝑥 2 +𝑥−6
= 2
𝑥 −𝑥−12
(𝑥+3)(𝑥−2)
=
(𝑥−4)(𝑥+3)
=
𝑥−2
𝑥−4
Hole: −3, 57 VA: x = 4
Barnett/Ziegler/Byleen Business Calculus 12e
HA: y = 1
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Asymptotes and Limits
 Horizontal asymptotes determine the end behavior of a
function. Recall that end behavior can be described using
limits.
 Vertical asymptotes can also be described using limits.
Horizontal asymptote:
lim 𝑓 𝑥 = 1
lim 𝑓 𝑥 = 1
𝑥→∞
𝑥→−∞
Vertical asymptote:
lim− 𝑓 𝑥 = −∞
𝑥→2
Barnett/Ziegler/Byleen Business Calculus 12e
lim+ 𝑓 𝑥 = ∞
𝑥→2
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Example 1
 Use the information to sketch the graph of f(x).
HA: y = 3
Incr: (-, -1), (1, )
Decr: (-1, 1)
Vertical tangent at x=0
Max when x=-1
Min when x=1
CC up: (-, -2), (0, 2)
CC down: (-2, 0), (2, )
Inflect. Pt. when x= -2, 0, 2
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Example 1 (cont.)
y
HA: y = 3
Incr: (-, -1), (1, )
x
Decr: (-1, 1)
Max when x=-1
Min when x=1
CC up: (-, -2), (0, 2)
CC down: (-2, 0), (2, )
Inflect. Pt. when x = -2, 0, 2
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 2
 Use the information to sketch a graph of f(x).
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 2 (cont.)
y
(D) (-2,1) (2,1)
Barnett/Ziegler/Byleen Business Calculus 12e
x
11
Example 2 (cont.)
y
x
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 3
 Use the graphing strategy to sketch the graph of
𝑥 2 − 3𝑥
𝑓 𝑥 = 2
𝑥 − 5𝑥 + 6
 Analyze 𝑓 𝑥 :
𝑥(𝑥 − 3)
𝑥
𝑓 𝑥 =
=
𝑥 − 2 (𝑥 − 3) 𝑥 − 2
Domain: −∞, 2 ∪ 2,3 ∪ 3, ∞
Y-intercept: y = 0
X-intercept: 𝑥 = 0
Hole: (3, 3)
Vertical asymptote: x=2 Horizontal asymptote: y=1
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 3 (cont.)
 Analyze 𝑓′ 𝑥 :
𝑓 𝑥 =
𝑥(𝑥−3)
𝑥−2 (𝑥−3)
=
𝑥
𝑥−2
𝑓′
𝑥 =
𝑓′ 𝑥 =
𝑥−2 1 −𝑥(1)
𝑥−2 2
−2
𝑥−2 2
Partition points: x=2
𝑓′(𝑥) − − − 𝑁𝐷 − − −
2
Decreasing: −∞, 2 ∪ 2, ∞
No local extrema
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 3 (cont.)
 Analyze 𝑓′′ 𝑥 :
𝑓′
𝑓 ′′
−2
𝑥 =
𝑥−2
𝑥−2
2
2
0 − (−2)(2 𝑥 − 2 )
𝑥 =
𝑥−2 4
4
4(𝑥 − 2)
=
=
4
𝑥−2 3
𝑥−2
Partition points: x=2
− − − 𝑁𝐷 + + +
2
Barnett/Ziegler/Byleen Business Calculus 12e
CC down: −∞, 2
CC up: 2, ∞
No inflection points
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Example 3 (cont.)
Y-intercept: y = 0
X-intercept: 𝑥 = 0
Hole: (3, 3)
Vertical asymptote: x=2
Horizontal asymptote: y=1
y
x
Decreasing: −∞, 2 ∪ 2, ∞
No local extrema
CC down: −∞, 2
CC up: 2, ∞
No inflection points
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Homework
#5-4: Pg 319
1, 5, 7,
17, 31, 65
Barnett/Ziegler/Byleen Business Calculus 12e
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