Chabot Mathematics
§3.3 Curve
Sketching
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Review §
3.2
 Any QUESTIONS About
• §3.2 → ConCavity & InflectionPoints
 Any QUESTIONS
About
HomeWork
• §3.2 →
HW-14
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
§3.3 Learning Goals
 Determine horizontal and vertical
asymptotes of a graph
 Use Algebra to find Axes InterCepts on
a Funciton Graph
 Use Derivatives to find
Significant Points on the graph
 Discuss and apply a
general procedure for
sketching graphs
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
T-Table Can Miss Features
 Consider the
10 x x  8


f
x

y

2
Function
x  10
 Make T-Table,
MTH15 • GraphSketching
Connect-Dots
3
4
2
y = f(x) = 10x(x+8)/(x+10)2
x
Y
-5
-6.00
-4
-4.44
-3
-3.06
-2
-1.88
-1
-0.86
0
0.00
1
0.74
2
1.39
3
1.95
4
2.45
5
2.89
Chabot College Mathematics
1
0
-1
-2
-3
-4
-5
-6
-5
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-4
-3
-2
-1
0
x
1
2
3
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
5
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 13Jul13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
% ref:
%
% The Limits
xmin = -35; xmax = 25;
ymin = -15; ymax = 40;
% The FUNCTION
x = linspace(xmin,xmax,500); y = 10*x.*(x+8)./(x+10).^2;
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) =
= 10x(x+8)/(x+10)^2'),...
title(['\fontsize{16}MTH15 • GraphSketching',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on',
'EdgeColor', 'none', 'String',
'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)
hold on
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
plot([-10 -10], [ymin, ymax], '-- m', [xmin xmax],[10 10], '-- m',
'LineWidth', 2)
set(gca,'XTick',[xmin:5:xmax]); set(gca,'YTick',[ymin:5:ymax])
T-Table Can Miss Features
 But Using Methods to be Discussed, Find
MTH15 • GraphSketching
40
y = f(x) = = 10x(x+8)/(x+10)2
35
30
25
20
15
10
5
0
-5
-10
-15
-35 -30 -25 -20 -15 -10
Chabot College Mathematics
6
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-5
x
0
5
10
15
20
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 23Jun13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
% ref:
%
% The Limits
xmin = -5; xmax = 5;
ymin = -6; ymax = 3;
% The FUNCTION
x = [-5 -4 -3 -2 -1 0
1
2
3
4
5];
y = [-6 -4.444444444
-3.06122449 -1.875 -0.864197531
0
0.743801653
1.388888889 1.952662722 2.448979592 2.888888889]
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) =
10x(x+8)/(x+10)^2'),...
title(['\fontsize{16}MTH15 • GraphSketching',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)
hold on
plot(x,y, 'x m', 'MarkerSize', 15, 'LineWidth', 3)
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax]
hold off
T-Table Can Miss Features
8
• x Span
• ∆x Increment Size
MTH15 • GraphSketching
MTH15 • GraphSketching
3
40
2
35
y = f(x) = = 10x(x+8)/(x+10)2
• Cover sufficiently
Wide values
• Have sufficiently
small increments
Chabot College Mathematics
 Unfortunately the
Grapher does NOT
know a-priori the
y = f(x) = 10x(x+8)/(x+10)2
 In Order for
T-Tables &
ConnectDots to
properly
Characterize the
Fcn Graph, the
Domain (x) Column
must
1
0
-1
-2
-3
-4
-5
-6
-5
30
25
20
15
10
5
0
-5
-10
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-4
-3
-2
-1
0
1
2
3
4
5
-15
-35 -30 -25 -20 -15 -10
x
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-5
x
x-Span
InSufficent
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
0
5
10
15
20
25
Better Graphing GamePlan
1. Find THE y-Intercept, if Any
a. Set x = 0, find y
b. Only TWO Functions do NOT have a
y-intercepts
–
–
Of the form 1/x
x = const; x ≠ 0
2. Find x-Intercept(s), if Any
a. Set y = 0, find x
b. Many functions do NOT have x-intercepts
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Better Graphing GamePlan
3. Find VERTICAL (↨) Asymptotes, If Any
a. Exist ONLY when fcn has a denom
b. Set Denom = 0, solve for x
–
These Values of x are the Vertical Asymptote
(VA) Locations
4. Find HORIZONTAL (↔) Asymptotes
(HA), If Any
a. HA’s Exist ONLY if the fcn has a finite
limit-value when x→+∞, or when x→−∞
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Better Graphing GamePlan
f x 
b. Find y-value for: yHA  xlim
 
–
These Values of y are the HA Locations
5. Find the Extrema (Max/Min) Locations
a. Set dy/dx = 0, solve for xE
b. Find the corresponding yE = f(xE)
c. Determine by 2nd Derivative, or
ConCavity, to test whether (xE, yE) is a
Max or a Min
–
See Table on Next Slide
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Better Graphing GamePlan
– Determine Max/Min By Concavity
𝒅𝟐 𝒚
Sign
Concavity
Max or Min
POSitive
NEGative
Neither (Zero)
Up ↑
Down ↓
No Information
Min
Max
Flat Spot
𝒅𝒙𝟐 𝒙𝑬
6. Find the Inflection Pt Locations
a. Set d2y/dx2 = 0, solve for xi
b. Find the corresponding yi = f(xi)
c. Determine by 3rd Derivative test The
Inflection-concavity form: ↑-↓ or ↓- ↑
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Better Graphing GamePlan
7. Find the Inflection Pt Locations
a. Set d2y/dx2 = 0, solve for xi
b. Find the corresponding yi = f(xi)
c. Determine by 3rd Derivative test The
Inflection form: ↑-↓ or ↓- ↑
–
𝒅𝟑 𝒚
𝒅𝒙𝟑 𝒙𝒊
Determine Inflection form by 3rd Derivative
Sign
POSitive
NEGative
Neither (Zero)
Chabot College Mathematics
13
ConCavity Change
Inflection Form
Down-to-Up
Up-to-Down ↓
No Information
↓-↑
↑-↓
↑-↑ OR ↓-↓
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Better Graphing GamePlan
8. Sign Charts for Max/Min and ↑-↓/↓-↑
a. To Find the “Flat Spot” behavior for dy/dx
= 0, when d2y/dx2 exists, but [d2y/dx2]xE =
0 use the Direction-Diagram
Slope
df/dx Sign
Critical (Break)
Points
Chabot College Mathematics
14
−−−−−−
++++++
a
Max
−−−−−−
b
NO
Max/Min
++++++
c
Min
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
x
Better Graphing GamePlan
9. Sign Charts for Max/Min and ↑-↓/↓-↑
a. To Find the ↑-↑ or ↓-↓ behavior for d2y/dx2
= 0, when d3y/dx3 exists, but [d3y/dx3]xi = 0
use the Dome-Diagram
ConCavity
Form
d2f/dx2 Sign
++++++
Critical (Break)
Points
Chabot College Mathematics
15
−−−−−−
a
Inflection
−−−−−−
b
NO
Inflection
++++++
c
Inflection
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
x
Example  Sketch Rational Fcn
2



2x 1 x  2
 Sketch y  f  x  
x  12 x  3
 Set x = 0 to Find y-intercept
2
2


2  0  10  2 
 12 
4 4
y 0  
 2


2
0  1 0  3 1  3  3 3
• Thus y-intercept → (0, 4/3)
 Set y = 0 to Find x-intercept(s), if any
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Example  Sketch Rational Fcn
2
2
2



2 x  1 x  2
2 x  1x  2  x  1 x  3
 y=0: 0  x  12 x  3  0  x  12 x  3 
1


0  2 x  1 x  2
2
 0  2 x  1 or 0  x  2
2
 Solving for x: x  1 2 or x  2
 Finding y(x):
2
2

0 5 2 
 1  21 2  11 2  2 
y  

0
2
2
1 2  1 1 2  3 3 2  5 2
2
y  2  
Chabot College Mathematics
17
2 2  1 2  2
 2  12  2  3
2
 50  0
 12  5
2

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Example  Sketch Rational Fcn
 The x-Intercepts
• (½,0); Multiplicity = 1 (Linear Form)
• (−2,0); Multiplicity = 2 (Parabolic Form)
 The Horizontal Intercept(s)
2



2 x  1x  2 
2 x  1 x  2  1 x 3 
lim y  lim
 lim 
 3
2
2
x  
x    x  1  x  3
x    x  1  x  3 1 x


2
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Example  Sketch Rational Fcn
 Continuing with the Limit
2

1  2  
 2 x  1 x  2 

2

1




 



2
x  x  

x
x

lim y  lim 
  lim
2
2
x  
x  
x



 x  1 x  3 
 1  3
 1    1   
 x 2

x 
 x  x
 2  01  02  2 1
lim y  lim 

2

2
x  
x   1  0  1  0 

 1 1
2
• Thus have a HORIZONTAL asymptote
at y = 2
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Example  Sketch Rational Fcn
 To Find VERTICAL asymptote(s) set
the DeNom/Divisor = 0
2

2 x  1 x  2 
2


x  3
yx  

0

x

1
2
x  1 x  3
• Using Zero
2
0  x  1 x  3  x  1 or x  3
Products
• Thus have VERTICAL Asymptotes at
– x = −1
–x = 3
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Example  Sketch Rational Fcn
 Use Computer
Algebra System,
MuPAD to find and
Solve Derivatives
 From the Derivatives Find
• Min at (−2,0) → ConCave UP
• Inflection Points
– ↓-to-↑ at (−2.63299, 0.16714)
– ↑-to-↓ at (0.63299, −0.29213)
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
MTH15 • GraphSketching
y = f(x) = = (2x+1)(x+2)2/(x+1)2(x-3)
20
16
12
The Graph
8
4
0
-4
-8
-12
-4
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
-3
-2
Chabot College Mathematics
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-1
0
1
x
2
3
4
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
6
WhiteBoard Work
 Problems From §3.3
• P46 → Inventory Cost
• P60 → Immunization
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
All Done for Today
A Graphic
Scaling
Machine
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
ConCavity Sign Chart
ConCavity
Form
d2f/dx2 Sign
++++++
Critical (Break)
Points
Chabot College Mathematics
26
−−−−−−
a
Inflection
−−−−−−
b
NO
Inflection
++++++
c
Inflection
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
x
ConCavity Sign Chart
ConCavity
Form
d2f/dx2 Sign
++++++
Critical (Break)
Points
Chabot College Mathematics
27
−−−−−−
a
Inflection
−−−−−−
b
NO
Inflection
++++++
c
Inflection
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
MTH15 • P3.3-46 • Bruce Mayer, PE
3
2
1.5
P33-46
y = f(x) = 2x + 80k/x
2.5
1
0.5
0
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
50
100
Chabot College Mathematics
38
150
200
250
x
300
350
400
450
500
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
P3.3-56
Chabot College Mathematics
43
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx
Chabot College Mathematics
44
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx