Chabot Mathematics
§J Graph
Rational Fcns
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Review §
5.7
MTH 55
 Any QUESTIONS About
• §5.7 → PolyNomical Eqn Applications
 Any QUESTIONS About HomeWork
• §5.7 → HW-21
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.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 in
Step 2 by a smooth curve
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Making Complete Plots


1.

2.

3.




 5.


Chabot College Mathematics
4
4.
Arrows in
POSITIVE
Direction Only
Label x & y axes
on POSITIVE ends
Mark and label at
least one unit on
each axis
Use a ruler for
Axes &
Straight-Lines
Label significant
points or quantities
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Rational Function
 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
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Visualizing Domain and Range
 Domain = the set of a function’s
Inputs, as found on the horizontal
axis (the x-Axis)
 Range = the set of a function’s
OUTputs , found on the vertical
axis (the y-Axis).
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Find Rational Function Domain
1. Write an equation that sets the
DENOMINATOR of the rational
function equal to 0.
2. Solve the equation.
3. Exclude the value(s) found in
step 2 from the function’s
domain.
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Domain & Range
 Graph y = f(x) = x2. Then State the
Domain & Range of the function
 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-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Domain & Range
 Now Plot
the Five
Points
and
connect
them
with a
smooth
Curve
6
5
(−2,4)
9
(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-06_sec_1-3_Graph_Functions.ppt.ppt
4
Example  Domain & Range
 The DOMAIN of a function is the set of
all first (or “x”) components of the
Ordered Pairs.
 Projecting on the X-axis the
x-components of ALL POSSIBLE
ordered pairs displays the DOMAIN of
the function just plotted
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Domain & Range
6
 Domain of
y = f(x) = x2
Graphically
 This
Projection
Pattern
Reveals a
Domain of
x
x is a real number
5
4
3
2
1
x
0
-4
-3
-2

11
-1
0
1
2
-1
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
y
-2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
3
4
Example  Domain & Range
6
 The RANGE of a
function is the set
of all second (or
“y”) components
of the ordered
pairs. The
projection of the
graph onto the
y-axis shows the
range
5
4
3
2
1
x
0
-4
Chabot College Mathematics
12
y
-3
-2
-1
0
1
2
RANGE   y y  0 
-1
M55_§JBerland_Graphs_0806.xls
-2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
3
4
Domain Restrictions
 EVERY element, x, in a functional Domain
MUST produce a VALID Range output, y
 ReCall the Real-Number Operations that
Produce INvalid Results
• Division by Zero
• Square-Root of a Negative Number
 x-values that Produce EITHER of
the above can NOT be in the
Function Domain
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  find
6
Domain: f ( y)  y3  5 y 2  4 y .
 SOLUTION  Avoid Division by Zero
y3  5 y 2  4 y  0
Set the DENOMINATOR equal to 0.
y  y2  5 y  4  0
Factor out the monomial GCF, y.
FOIL Factor by Guessing
y  y  4 y  1  0
y  0 or y  4  0 or y  1  0 Use the zero-products theorem.
y  4
y  1
Solve the MiniEquations for y
 The function is UNdefined if y is
replaced by 0, −4, or −1,
so the domain is {y|y ≠ −4, −1, 0}
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.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 a
connect
Div-by-Zero, so the
Dots
only input that results
in a denominator of 0
is 0. Thus the domain
{x|x  0} or
(–, 0) U (0, )
Chabot College Mathematics
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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-06_sec_1-3_Graph_Functions.ppt.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
16
2
• the x-axis (as 1/  0)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Vertical Asymptotes
 The VERTICAL asymptotes of a
rational function f(x) = p(x)/q(x) are found
by determining the ZEROS of q(x) that
are NOT also ZEROS of p(x).
• If p(x) and q(x) are polynomials with no
common factors other than constants, we
need to determine only the zeros of the
denominator q(x).
 If a is a zero of the denominator, then
the Line x = a is a vertical asymptote for
the graph of the function.
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Vertical Asymptote
 Determine the
vertical asymptotes
of the function
2x  3
f ( x)  2
x 4
 Thus the vertical
asymptotes are the
lines x = −2 & x = 2
 Factor to find the
zeros of the
denominator:
x2 − 4 = 0
= (x + 2)(x − 2)
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Horizontal Asymptotes
 When the numerator and the
denominator of a rational function have
the same degree, the line y = a/b is the
horizontal asymptote,
• where a and b are the leading
coefficients of the numerator and the
denominator, respectively.
 In This case The line y = c = a/b is a
horizontal asymptote.
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Horiz. Asymptote
 Find the horizontal asymptote for
6 x 4  3x 2  1
f ( x)  4
9 x  3x  2
 The numerator and denominator have
the same degree. The ratio of the
leading coefficients is 6/9, so the line
y = 2/3 is the horizontal asymptote
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Finding a Horizontal Asymptote
 When the numerator and the denominator of a
rational function have the same degree, the line
y = a/b is the horizontal asymptote, where a and
b are the leading coefficients of the numerator
and the denominator, respectively.
 When the degree of the numerator of a rational
function is less than the degree of the
denominator, the x-axis, or y = 0, is the
horizontal asymptote.
 When the degree of the numerator of a rational
function is greater than the degree of the
denominator, there is no horizontal asymptote.
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
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
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Graph
2x
h( x ) 
x3
 SOLUTION
 Vertical asymptotes: x + 3 = 0, so x = −3
 The degree of the numerator and
denominator is the same. Thus y = 2 is
the horizontal asymptote
 Graph Plan
• Draw the asymptotes with dashed lines.
• Compute and plot some ordered pairs and
connect the dots to draw the curve.
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
Example  Graph
 Construct T-Table
x
7
5
4
2
0
2
Chabot College Mathematics
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2x
h( x ) 
x3
 Plot Points, “Dash
In” Asymptotes
h(x)
3.5
5
8
4
0
4/5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
WhiteBoard Work
 Problems From
§J1 Exercise Set
• J2, J4, J6

Watch the
DENOMINATOR
PolyNomial; it can
Produce
Div-by-Zero
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt
All Done for Today
Asymptote
Architecture
wins competition
for WBCB Tower,
to be tallest
building in Asia
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.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
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt