Chabot Mathematics
§G Translate
Rational Plots
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Review §
G
MTH 55
 Any QUESTIONS About
• §G → Graphing Rational Functions
 Any QUESTIONS About HomeWork
• §G → HW-22
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
GRAPH BY PLOTTING POINTS
 Step1. Make a representative
T-table of solutions of the equation.
 Step 2. Plot the solutions as
ordered pairs in the Cartesian
coordinate plane.
 Step 3. Connect the solutions
(dots) in Step 2 by a smooth curve
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Translation of Graphs
 Graph y = f(x) = x2.
• Make T-Table & Connect-Dots
 Select integers for x, starting with −2 and
ending with +2. The T-table:
x
y  x2
2
y   2  4
1
y   1  1
2
0
y  02  0
1
y  12  1
2
y  22  4
Chabot College Mathematics
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2
Ordered Pair  x, y 
 2, 4 
 1,1
0, 0 
1,1
2, 4 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Translation of Graphs
 Now Plot
the Five
Points
and
connect
them
with a
smooth
Curve
6
5
(−2,4)
5
(2,4)
4
3
2
(−1,1)
(1,1)
1
x
0
-4
-3
-2
-1
0
(0,0)1
2
3
-1
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
y
-2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
4
Axes Translation
7
 Now Move
UP the graph
of y = x2 by
two units as
shown
 What is the
Equation
for the new
Curve?
(−2,6)
6
(2,6)
6
5
(−2,4)
(2,4)
4
(−1,3) (1,3)
3
2
(0,2)
(−1,1)
(1,1)
1
x
0
-4
-3
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
y
-1
0
(0,0)1
2
-1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
3
4
Axes Translation
 Compare ordered pairs on the graph of
with the corresponding ordered pairs on
the new curve:
7
(−2,6)
y
(2,6)
6
5
(−2,4)
(2,4)
4
(−1,3) (1,3)
3
2
(0,2)
(−1,1)
(1,1)
1
x
0
-4
-3
-2
M55_§JBerland_Graphs_0806.xls
-1
0
2
3
4
-1
Chabot College Mathematics
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(0,0)1
y  x2
 2, 4
 1,1
0, 0
1,1
2, 4
New Curve
(2 units up)
 2, 6
 1, 3
0, 2
1, 3
2, 6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
y x
 2, 4
 1,1
0, 0
1,1
2, 4
2
Axes Translation
New Curve
(2 units up)
 2, 6
 1, 3
0, 2
1, 3
2, 6
 Notice that the x-coordinates
on the new curve are the
same, but the y-coordinates
are 2 units greater
 So every point on the new curve makes
the equation y = x2+2 true, and every
point off the new curve makes the
equation y = x2+2 false.
 An equation for the new curve is thus
y = x2+2
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Axes Translation
5
 Similarly, if
every point on
the graph of
y = x2 were is
moved 2 units
down, an
equation of the
new curve is
y = x2−2
yx
4
9
2
3
2
1
x
0
-4
-3
-2
-1
0
-1
1
2
3
y x 2
2
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
y
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
4
Axes Translation
 When every point on a graph is moved
up or down by a given number of units,
the new graph is called a vertical
translation of the original graph.
7
5
y
6
y
4
y  x 2
2
5
3
4
2
y  x2
1
3
x
0
2
-4
1
yx
0
-4
-3
-2
-1
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
10
0
-1
1
2
-3
-2
-1
0
1
2
3
4
-1
2
x
3
4
y  x2  2
-2
M55_§JBerland_Graphs_0806.xls
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Vertical Translation
 Given the Graph of y = f(x),
and c > 0
1. The graph of y = f(x) + c is a
vertical translation c-units UP
2. The graph of y = f(x) − c is a
vertical translation c-units DOWN
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Horizontal Translation
 What if every point on the graph of
y = x2 were moved 5 units to the right
as shown below.
y
10
9
8
7
6
5
yx
(−2,4)4
2
(2,4)
(3,4)
(7,4)
y?
3
2
1
x
0
-3
-2
-1
-1
0
1
2
3
4
5
6
7
8
9
 What is the eqn of the new curve?
-2
Xlate_ABS_Graphs_1010.xls
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Horizontal Translation
 Compare ordered pairs on the graph of
with the corresponding ordered pairs on
the new curve:
y  x2
 2, 4
 1,1
0, 0
1,1
2, 4
Chabot College Mathematics
13
New Curve
(5 units right)
3, 4
4,1
5, 0
6,1
7, 4 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
y x
 2, 4
 1,1
0, 0
1,1
2, 4
2
Horizontal Translation
 Notice that the y-coordinates on the
new curve are the same, but the
x-coordinates are 5 units greater.
 Does every point on the new curve make the
equation y = (x+5)2 true?
New Curve
(5 units right)
3, 4
4,1
5, 0
6,1
7, 4 
• No; for example if we input (5,0) we get 0 = (5+5)2,
which is false.
• But if we input (5,0) into the equation y = (x−5)2 , we get
0 = (5−5)2 , which is TRUE.
 In fact, every point on the new curve makes the
equation y = (x−5)2 true, and every point off the new
curve makes the equation y = (x−5)2 false. Thus an
equation for the new curve is y = (x−5)2
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Horizontal Translation
 Given the Graph of y = f(x),
and c > 0
1. The graph of y = f(x−c) is a
horizontal translation c-units to
the RIGHT
2. The graph of y = f(x+c) is a
horizontal translation c-units to
the LEFT
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Use Translation to graph f(x) = (x−3)2−2
 LET y = f(x) → y = (x−3)2−2
 Notice that the graph of y = (x−3)2−2
has the same shape as y = x2, but is
translated 3-unit RIGHT and 2-units
DOWN.
 In the y = (x−3)2−2, call −3 and −2
translators
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 The graphs of y=x2 and y=(x−3)2−2 are
different; although they have the Same
shape they have different locations
 Now remove the translators by a
substitution of x’ (“x-prime”) for x,
and y’ (“y-prime”) for y
 Remove translators for an (x’,y’) eqn
y  x  3  2 
2
Chabot College Mathematics
17
y    x 
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Since the graph of y=(x−3)2−2 has the
same shape as the graph of y’ =(x’)2 we
can use ordered pairs of y’ =(x’)2 to
determine the shape
 T-table
y
Ordered Pair  x, y 
x
for
 1,1
1
1
y’ =(x’)2
0, 0
0
0
1
Chabot College Mathematics
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1
1,1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Next use the translation rules to find the
origin of the x’y’-plane. Draw the x’-axis
and y’-axis through the
2
2






y

x

3

2

y

x
translated origin
• The origin of the x’y’-plane is 3 units right
and 2 units down from the origin of the
xy-plane.
• Through the translated origin, we use
dashed lines to draw a new horizontal axis
(the x’-axis) and a new vertical axis
(the y’-axis).
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Locate the Origin of the Translated
Axes Set using the translator values
Move: 3-Right, 2-Down
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Now Plot the ordered pairs of the x’y’
equation on the x’y’-plane, and use the
points to draw an appropriate graph.
• Remember that this graph is smooth
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Perform a partial-check to determine
correctness of the last graph. Pick any
point on the graph and find its (x,y)
CoOrds; e.g., (4, −1) is on the graph
 The Ordered
Pair (4, −1)
should
make the xy
Eqn True
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 Sub (4, −1) into y=(x−3)2−2
y  x  3  2
2
:
 1  4  3  2
 1  1
2
 Thus (4, −1) does make y = (x−3)2−2
true. In fact, all the points on the
translated graph make the original Eqn
true, and all the points off the
translated graph make the original
Eqn false
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Plot by Translation
 What are the Domain &
Range of y=(x−3)2−2?
 To find the domain &
range of the xy-eqn,
examine the xy-graph
(not the x’y’ graph).
 The xy graph showns
7
y
y  x  3  2
2
6
5
4
3
2
1
x
0
-1
0
1
2
3
4
5
6
-1
-2
-3
M55_§JBerland_Graphs_0806.xls
• Domain of f is {x|x is any real number}
• Range of f is {y|y ≥ −2}
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
7
8
9
Graphing Using Translation
1. Let y = f(x)
2. Remove the x-value & y-value
“translators” to form an x’y’ eqn.
3. Find ordered pair solutions of the x’y’ eqn
4. Use the translation rules to find the origin
of the x’y’-plane. Draw dashed x’ and y’
axes through the translated origin.
5. Plot the ordered pairs of the x’y’ equation
on the x’y’-plane, and use the points to
draw an appropriate graph.
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example ReCall Graph y = |x|
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
y = |x |
6
6
5
4
3
2
1
0
1
2
3
4
5
6
5
4
3
2
x
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
Chabot College Mathematics
26
y
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
file =XY_Plot_0211.xls
-6
4
5
6
Example Graph y = |x+2|+3
 Step-1 f x  x  2  3
 Step-2 y  x
Xlators :
y  x2 3
x   2; y  3
 Step-3 → T-table in x’y’
x
1
0
1
Chabot College Mathematics
27
y
1
0
1
Ordered Pair  x, y 
 1,1
0, 0
1,1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example Graph y = |x+2|+3
 Step-4: the x’y’-plane origin is 2 units
LEFT and 3 units UP from xy-plane
y
y
6
5
4
x
3
Left 2
2
Up 3
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
Bruce Mayer, PE
Chabot College Mathematics
28
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
-2
6
Example Graph y = |x+2|+3
 Step-5: Remember that the graph of
y = |x| is V-Shaped:
y
y
y
6
5
4
x
3
Left 2
2
Up 3
1
x
0
-6
-5
-4
Chabot College Mathematics
29
-3
-2
-1
0
1
2
3
4
5
6
Bruce Mayer, PE
-1 BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Rational Function Translation
 A rational function is
a function f that is a
quotient of two
polynomials, that is,
p( x)
f ( x) 
,
q( x)
 Where
• where p(x) and q(x) are polynomials and
where q(x) is not the zero polynomial.
• The domain of f consists of all inputs x
for which q(x) ≠ 0.
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example 
1
f x  
x
 Find the DOMAIN
and GRAPH for f(x)
 Construct
T-table
 SOLUTION
 Next Plot
When the denom
points &
x = 0, we have
connect
x = 0, so the only
Dots
input that results in a
denominator of 0 is
0. Thus the domain
{x|x  0} or
(–, 0)  (0, )
Chabot College Mathematics
31
x y = f(x)
-8
-1/8
-4
-1/4
-2
-1/2
-1
-1
-1/2
-2
-1/4
-4
-1/8
-8
1/8
8
1/4
4
1/2
2
1
1
2
1/2
4
1/4
8
1/8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Plot
1
f x  
x
10
 Note that the Plot
approaches, but
never touches,
y
8
• the y-axis (as x ≠ 0)
6
– In other words the
graph approaches the
LINE x = 0
4
2
x
0
-10
-8
-6
-4
-2
0
-2
-4
-6
-8
M55_§JBerland_Graphs_0806.xls
4
6
8
10
– In other words the
graph approaches the
LINE y = 0
 A line that is
approached by a graph
is called an
ASYMPTOTE
-10
Chabot College Mathematics
32
2
• the x-axis (as 1/  0)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
ReCall Asymptotic Behavior
 The graph of a rational function
never crosses a vertical
asymptote
 The graph of a rational function
might cross a horizontal
asymptote but does not necessarily
do so
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Recall Vertical Translation
 Given the graph of the equation
y = f(x), and c > 0,
 the graph of y = f(x) + c
is the graph of
y = f(x) shifted UP
(vertically) c units;
 the graph of y = f(x) – c
is the graph of
y = f(x) shifted DOWN
(vertically) c units
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
y = 3x2+2
y = 3x2
y = 3x2−3
Recall Horizontal Translation
 Given the graph of the equation
y = f(x), and c > 0,
 the graph of y = f(x– c)
is the graph of
y = f(x) shifted RIGHT
(Horizontally) c units;
y = 3(x+2)
 the graph of y = f(x + c)
is the graph of
y = f(x) shifted LEFT
(Horizontally) c units.
Chabot College Mathematics
35
y = 3(x-2)2
2
y = 3x2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
ReCall Graphing by Translation
1.
2.
3.
4.
Let y = f(x)
Remove the translators to form an x’y’ eqn
Find ordered pair solutions of the x’y’ eqn
Use the translation rules to find the origin
of the x’y’-plane. Draw the dashed x’ and
y’ axes through the translated origin.
5. Plot the ordered pairs of the x’y’ equation
on the x’y’-plane, and use the points to
draw an appropriate graph.
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
1
Example  Graph f x   x  2  1
1
1
1
y
1
 Step-1 f x  
x2
x2
1
1
1
 y  1 
y 
 Step-2 y  x  2  1
x  2 
x
 Step-3 → T-table in x’y’
x'
y'
x'
y'
-4
-2
-1
-1/2
-1/4
-1/4
-1/2
-1
-2
-4
1/4
1/2
1
2
4
4
2
1
1/2
1/4
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Graph
 Step-4: The
origin of the
x’y’ -plane is 2
units left and 1
unit up from
the origin of
the xy-plane:
1
f x  
1
x2
y
6
y
5
4
3
x
2
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
38
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
4
5
x
6
1
f x  
1
x2
Example  Graph
 Step-5: We know
that the basic
shape of this
graph is
Hyperbolic. Thus
we can sketch the
graph using Fewer
Points on the
translated axis
using the T-Table
Chabot College Mathematics
39
y
6
y
5
4
3
x
2
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
M55_§JBerland_Graphs_0806.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
4
5
x
6
1
f x  
1
x2
Example  Graph
y
 Examination of
the Graph
reveals
y
5
4
3
• Domain →
{x|x ≠ −2}
• Range →
{y|y ≠ 1}
6
x
2
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
Chabot College Mathematics
40
M55_§JBerland_Graphs_0806.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
4
5
x
6
2
Example  Graph g x    x  3  1
2
2
1
y
1
 Step-1 g x   
x 3
x3
2
2
2
 y  1  
y 
1
 Step-2 y  
x  3
x
x3
 Step-3 → T-table in (x’y’) by y’ = −2/x’
x'
y'
x'
y'
-4
-2
-1
-1/2
-1/4
1/2
1
2
4
8
1/4
1/2
1
2
4
-8
-4
-2
-1
-1/2
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Graph
 Step-4: The
origin of the
x’y’ -plane is
3 units RIGHT
and 1 unit
DOWN from
the origin of
the xy-plane
4
2
g x   
1
x  3
y
y
3
2
x
1
0
-3
-2
-1
0
1
2
3
4
5
6
-1
-3
-4
-5
-7
-8
M55_§JBerland_Graphs_0806.xls
42
8
9
x
-2
-6
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
2
g x   
1
x  3
Example  Graph
 Step-5: We know
that the basic
shape of this
graph is
Hyperbolic. Thus
we can sketch the
graph using Fewer
Points on the
translated axis
using the T-Table
4
y
y
3
2
x
1
0
-3
-2
-1
0
1
2
3
4
5
6
-1
-3
-4
-5
-6
-7
-8
43
8
9
x
-2
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
2
g x   
1
x  3
Example  Graph
4
 Notice for this
Graph that the
Hyperbola is
3
2
Chabot College Mathematics
44
x
1
• “mirrored”, or -3
rotated 90°,
by the leading
Negative sign
• “Spread out”,
or expanded,
by the 2 in the
numerator
y
y
0
-2
-1
0
1
2
3
4
5
6
-1
7
8
9
x
-2
-3
-4
-5
-6
-7
-8
M55_§JBerland_Graphs_0806.xls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
2
g x   
1
x  3
Example  Graph
4
 Examination
of the Graph
reveals
• Domain →
{x|x ≠ 3}
• Range →
{y|y ≠ −1}
y
y
3
2
x
1
0
-3
-2
-1
0
1
2
3
4
5
6
-1
-3
-4
-6
-7
-8
M55_§JBerland_Graphs_0806.xls
45
8
9
x
-2
-5
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example  Graph


1
Step-1 hx    2x  1  2
1
Step-2 y   2x  1  2
1
h x   
2
2x  1
1
y
2
2x  1
1
1
y 
 y  2  
2x  1
2 x
 Step-3 → T-table in x’y’ by y’ = −1/(2x’)
x'
y'
x'
y'
-4
-2
-1
-1/2
-1/4
1/8
1/4
1/2
1
2
1/4
1/2
1
2
4
-2
-1
-1/2
-1/4
-1/8
Chabot College Mathematics
46
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
1
h x   
2
2x  1
Example  Graph
 Step-4: The
origin of the
x’y’ -plane is 1
unit RIGHT
and 2 units
UP from the
origin of the
xy-plane
y y
8
7
6
5
4
3
2
x
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
47
-4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
5
6
x
7
Example  Graph
1
h x   
2
2x  1
y y
8
 Step-5: We know
that the basic
shape of this
graph is
Hyperbolic. Thus
we can sketch the
graph using Fewer
Points on the
translated axis
using the T-Table
7
6
5
4
3
2
0
-5
Chabot College Mathematics
48
x
1
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
M55_§JBerland_Graphs_0806.xls
-4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
5
6
x
7
Example  Graph
1
h x   
2
2x  1
 Notice for this
Graph that the
Hyperbola is
8
• “mirrored”, or
rotated 90°,
by the leading
Negative sign
4
7
6
5
3
2
0
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
M55_§JBerland_Graphs_0806.xls
49
x
1
• “Pulled in”, or -5
contracted, by
the 2 in the
Denominator
Chabot College Mathematics
y y
-4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
5
6
x
7
Example  Graph
1
h x   
2
2x  1
y y
8
 Examination
of the Graph
reveals
7
6
5
• Domain →
{x|x ≠ 1}
4
3
2
• Range →
{y|y ≠ 2}
x
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
50
-4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
5
6
x
7
WhiteBoard Work
 Problems From
§G1 Exercise Set
• G8, G10, G12

A Function with
TWO Vertical
Asymptotes
x3
f ( x)  2
.
2 x  5x  3
Chabot College Mathematics
51
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
All Done for Today
Another
Cool Design
by
Asymptote
Architecture
Chabot College Mathematics
52
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
53
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Translate Up or Down
 Make f (x)  x , g(x)  x  3 and h(x)  x  5
Graphs
h(x)  x  5
for
 
Notice:
g x   f x   3
h x   f  x   5
• Of the form of
VERTICAL
Translations
Chabot College Mathematics
54


f (x)  x
g(x)  x  3

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Translate Left or Right
 Make Graphs for
f (x)  x , g(x)  x  3 and h(x)  x  5
 Notice: g x   f x  3
hx  f x  5
• Of the form of HORIZONTAL Translations
h(x)  x  5
g(x)  x  3


Bruce Mayer, PE
Chabot College Mathematics
55
f (x)  x

BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example Graph y = |x+2|+3
 Step-4: The origin of the x’y’ -plane is 2
units LEFT and 3 units UP from the
origin of the xy-plane:
Chabot College Mathematics
56
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
Example Graph y = |x+2|+3
 Step-5: Remember that the graph of
y = |x| is V-Shaped:
Chabot College Mathematics
57
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
y
y
6
5
4
x
3
Over 22
Up 3
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
Chabot College Mathematics
58
-4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
6
y
y
y
6
5
4
x
3
Over 2
2
Up 3
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
Chabot College Mathematics
59
-3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
10
9
y
y
8
7
6
5
4
3
2
1
0
-3
-2
-1
-1
0
1
2
3
4
5
6
7
8
-2
Xlate_ABS_Graphs_1010.xls
Chabot College Mathematics
60
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
9
x
x
10
9
y
8
7
6
5
4
3
2
1
x
0
-3
-2
-1
-1
0
1
2
3
4
5
6
7
8
-2
Xlate_ABS_Graphs_1010.xls
Chabot College Mathematics
61
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-28_sec_Jb_Graph_Rational_Functions.ppt
9